Abstract group 230400.ca: $S_5^2.\SD_{16}$

• Label: 230400.ca
• Order: \( 2^{10} \cdot 3^{2} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $18$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$S_5^2.\SD_{16}$
Order:	\(230400\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2\times D_4.A_5^2.D_4$, of order \(921600\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \)
Composition factors:	$C_2$ x 6, $A_5$ x 2
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	40	60
Elements	1	4055	440	33512	624	21640	41280	10320	45760	960	26688	19200	10560	11520	3840	230400
Conjugacy classes	1	20	2	28	2	32	9	13	21	1	10	4	7	2	2	154
Divisions	1	20	2	28	2	32	5	13	21	1	10	2	7	1	2	147
Autjugacy classes	1	15	2	23	2	21	5	9	18	1	8	2	4	1	2	114

Dimension	1	2	4	8	10	12	16	20	24	25	32	36	40	48	50	60	64	72	80	96	100	120	144
Irr. complex chars.	8	10	1	16	16	8	12	4	2	8	10	4	16	8	10	8	1	3	4	2	1	2	0	154
Irr. rational chars.	8	6	3	16	16	8	12	4	2	8	6	4	16	8	6	8	3	1	4	2	3	2	1	147

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$40$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $18$ $\langle(1,3)(2,6,4,7,8,10)(5,9)(11,13,15,18)(12,16,14,17), (1,4,8,5)(3,6,7,9)(11,14)(12,15)(16,17), (1,2,5,4)(6,9)(7,10)(11,12,15,14)(13,17,18,16)\rangle$
Transitive group:	40T106324	40T106325			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$S_5^2$ . $\SD_{16}$	$(S_5^2\times D_4)$ . $C_2$	$(C_2\times S_5^2)$ . $D_4$	$(C_2.S_5^2)$ . $D_4$	all 33

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 13818115 subgroups in 27349 conjugacy classes, 35 normal (31 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_5^2:D_4$
Commutator:	$G' \simeq$ $C_4\times \PSOPlus(4,5)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $S_5^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $D_4$	$G/\operatorname{Fit} \simeq$ $A_5^2.D_4$
Radical:	$R \simeq$ $D_4$	$G/R \simeq$ $A_5^2.D_4$
Socle:	$\operatorname{soc} \simeq$ $C_2\times A_5^2$	$G/\operatorname{soc} \simeq$ $C_2^2\wr C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.C_2^5.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 225 (order at least 1024) are shown, as well as all normal subgroups of any index.

Order 230400: $S_5^2.\SD_{16}$

Order 115200: $S_5^2\times D_4$, $D_4.A_5^2.C_4$, $D_4.A_5^2.C_2^2$, $C_4.A_5^2.D_4$, $C_4.A_5^2.D_4$, $C_4.A_5^2.D_4$, $C_4.A_5^2.D_4$

Order 57600: $A_5^2:\SD_{16}$ x 2, $(C_2^2\times A_5).S_5.C_2$ x 2, $S_5^2:C_4$, $C_4\times S_5^2$, $A_5:(D_4\times S_5)$, $C_4:S_5\times S_5$, $D_4\times A_5\times S_5$, $(C_2^2\times A_5).S_5.C_2$, $D_4\times \PSOPlus(4,5)$, $C_2^2\times S_5^2$, $A_5^2:(C_2\times Q_8)$, $C_4.A_5^2.C_2^2$, $A_5^2:\SD_{16}$, $(C_4\times A_5^2):C_4$, $(C_2\times A_5^2).D_4$, $A_5^2:(C_2\times C_8)$, $A_5^2:(C_2\times C_8)$

Order 28800: $C_2\times S_5^2$ x 8, $A_5^2:D_4$ x 2, $A_5^2:D_4$ x 2, $A_5^2:C_2^3$ x 2, $A_5^2:D_4$ x 2, $A_5^2:Q_8$, $A_5^2:C_8$, $A_5^2:(C_2\times C_4)$, $C_2\times A_5^2:C_4$, $C_2.S_5^2$, $C_4\times A_5\times S_5$, $A_5^2:D_4$, $C_2.S_5^2$, $D_4\times A_5^2$, $C_4\times \PSOPlus(4,5)$, $A_5^2:C_2^3$, $A_5^2:D_4$, $A_5^2:D_4$, $A_5^2:C_8$, $A_5^2:Q_8$

Order 23040: $D_4\times S_4\times S_5$

Order 19200: $D_4\times F_5\times S_5$

Order 14400: $S_5^2$ x 9, $A_5^2:C_2^2$ x 7, $A_5^2:C_2^2$ x 4, $A_5^2:C_4$, $C_4\times A_5^2$, $A_5^2:C_4$, $C_2^2\times A_5^2$, $A_5^2:C_4$, $A_5^2:C_4$

Order 11520: $S_5\times \GL(2,\mathbb{Z}/4)$ x 2, $S_4\times C_2^2:S_5$ x 2, $\GL(2,\mathbb{Z}/4):S_5$ x 2, $C_2^2\times S_4\times S_5$ x 2, $D_4\times D_6\times S_5$, $C_4:(S_4\times S_5)$, $S_4\times C_4:S_5$, $C_4:S_4\times S_5$, $C_4\times S_4\times S_5$, $D_4\times S_4\times A_5$, $D_4\times A_4\times S_5$, $D_4\times A_4:S_5$

Order 9600: $C_2^2\times F_5\times S_5$ x 2, $D_{10}:(C_4\times S_5)$ x 2, $F_5\times C_2^2:S_5$ x 2, $D_{10}:C_4\times S_5$ x 2, $C_{20}:(C_4\times S_5)$, $F_5\times C_4:S_5$, $C_{20}:C_4\times S_5$, $C_4\times F_5\times S_5$, $D_4\times D_5\times S_5$, $D_4\times F_5\times A_5$, $D_4\times A_5:F_5$

Order 9216: $S_4^2:\SD_{16}$

Order 7680: $D_4^2\times S_5$

Order 7200: $A_5\times S_5$ x 4, $\PSOPlus(4,5)$ x 3, $C_2\times A_5^2$ x 2

Order 6400: $F_5^2:\SD_{16}$

Order 5760: $S_3\times D_4\times S_5$ x 16, $C_2\times S_4\times S_5$ x 15, $S_3\times C_2^3:S_5$ x 2, $C_6:(D_4\times S_5)$ x 2, $C_6:D_4\times S_5$ x 2, $\GL(2,4):C_2^5$ x 2, $A_5:\GL(2,\mathbb{Z}/4)$ x 2, $A_4\times C_2^2:S_5$ x 2, $(C_2\times S_4):S_5$ x 2, $A_5:\GL(2,\mathbb{Z}/4)$ x 2, $A_5:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2\times A_4:S_5$ x 2, $C_2^2\times A_4\times S_5$ x 2, $C_2^2\times S_4\times A_5$ x 2, $A_5\times \GL(2,\mathbb{Z}/4)$ x 2, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $D_4\times A_4\times A_5$, $A_5\times C_4:S_4$, $C_4\times S_4\times A_5$, $C_6:S_5\times D_4$, $(C_4\times A_4):S_5$, $A_4\times C_4:S_5$, $C_4\times A_4\times S_5$, $C_4\times A_4:S_5$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $A_4:(C_4\times S_5)$, $A_4:C_4\times S_5$, $D_4\times D_6\times A_5$, $A_5:C_4\times S_4$

Order 4800: $C_2\times F_5\times S_5$ x 16, $(C_{10}\times S_5):C_4$ x 4, $C_5:D_4\times S_5$ x 2, $C_5:(D_4\times S_5)$ x 2, $A_5\times D_{10}:C_4$ x 2, $C_2\times D_{10}\times S_5$ x 2, $D_5\times C_2^2:S_5$ x 2, $C_{10}.(C_4\times S_5)$ x 2, $A_5:C_4\times F_5$ x 2, $C_2^2\times F_5\times A_5$ x 2, $C_2\times D_{10}.S_5$ x 2, $(D_{10}\times A_5):C_4$ x 2, $D_{20}:S_5$, $C_4\times D_5\times S_5$, $D_{20}\times S_5$, $D_5\times C_4:S_5$, $D_4\times C_5:S_5$, $C_5\times D_4\times S_5$, $A_5\times C_{20}:C_4$, $D_4\times D_5\times A_5$, $C_4\times F_5\times A_5$, $C_4\times A_5:F_5$, $(C_{20}\times A_5):C_4$

Order 4608: $D_4\times S_4^2$, $A_4^2.D_4:C_4$, $A_4^2:(C_2\times \SD_{16})$, $(C_4\times A_4^2).D_4$, $S_4^2:C_8$, $A_4^2:(C_2\times \SD_{16})$, $S_4^2:Q_8$

Order 3840: $C_4:D_4\times S_5$ x 4, $A_5.D_4^2$ x 4, $A_5:D_4^2$ x 4, $A_5:D_4^2$ x 4, $C_2^2\times D_4\times S_5$ x 4, $S_5\times C_2^2\wr C_2$ x 4, $C_4\times D_4\times S_5$ x 2, $A_5:D_4^2$ x 2, $C_4:D_4\times S_5$, $A_5:D_4^2$, $D_4\times F_5\times S_4$, $A_5\times D_4^2$

Order 3600: $A_5^2$

Order 3200: $D_4\times F_5^2$, $(C_4\times D_5^2).C_2^3$, $(D_4\times C_5^2):\OD_{16}$, $(C_4\times D_5^2).D_4$, $F_5^2:Q_8$, $F_5^2:C_8$, $(C_4\times D_5^2).D_4$

Order 2880: $\GL(2,4):C_2^4$ x 39, $C_3:(D_4\times S_5)$ x 16, $S_3\times C_2^2:S_5$ x 16, $C_3:D_4\times S_5$ x 16, $S_4\times S_5$ x 16, $D_{12}:S_5$ x 8, $S_3\times C_4:S_5$ x 8, $D_{12}\times S_5$ x 8, $S_3\times D_4\times A_5$ x 8, $C_4\times S_3\times S_5$ x 8, $D_4\times C_3:S_5$ x 8, $C_3\times D_4\times S_5$ x 8, $C_2\times S_4\times A_5$ x 7, $C_2\times A_4\times S_5$ x 7, $C_2\times A_4:S_5$ x 7, $\GL(2,4):C_2^4$ x 2, $C_6:D_4\times A_5$ x 2, $C_2^4:\GL(2,4)$ x 2, $\GL(2,4):C_2^4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2^2:S_5\times C_6$ x 2, $\GL(2,4):C_2^4$ x 2, $C_2\times D_{12}\times A_5$, $C_4\times D_6\times A_5$, $C_4\times A_4\times A_5$, $A_5:C_4\times D_6$, $A_4:C_4\times A_5$, $C_6\times D_4\times A_5$, $C_2\times C_{12}:S_5$, $C_6:(C_4\times S_5)$, $C_6:C_4\times S_5$, $C_2\times C_{12}:S_5$, $C_2\times C_{12}\times S_5$, $C_2\times C_{12}:S_5$, $(A_4\times A_5):C_4$, $A_4\times A_5:C_4$

Order 2400: $F_5\times S_5$ x 18, $D_{10}\times S_5$ x 15, $C_2\times F_5\times A_5$ x 8, $D_{10}.S_5$ x 8, $(C_{10}\times S_5):C_2$ x 2, $D_{10}:S_5$ x 2, $D_{10}:S_5$ x 2, $C_5\times C_2^2:S_5$ x 2, $C_2\times C_{10}:S_5$ x 2, $(C_2\times C_{10}):S_5$ x 2, $C_2\times C_{10}\times S_5$ x 2, $C_2\times D_{10}\times A_5$ x 2, $C_5:D_4\times A_5$ x 2, $C_5:(C_4\times S_5)$, $C_5:C_4\times S_5$, $C_{20}:S_5$, $C_{20}:S_5$, $C_{20}:S_5$, $C_{20}\times S_5$, $C_4\times D_5\times A_5$, $D_{20}\times A_5$, $D_{10}.S_5$, $C_5\times D_4\times A_5$

Order 2304: $S_4\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4^2:\SD_{16}$ x 2, $(C_3\times C_6).C_2^6.C_2$, $C_2^4:D_6^2$, $C_4:S_4^2$, $C_4:S_4^2$, $C_4\times S_4^2$, $D_4\times \PSOPlus(4,3)$, $\GL(2,\mathbb{Z}/4):S_4$, $D_4\times A_4\times S_4$, $A_4^2:\SD_{16}$, $A_4^2:(C_2\times C_8)$, $(C_4\times A_4^2):C_4$, $A_4^2:(C_2\times C_8)$, $A_4^2:\SD_{16}$, $A_4^2:(C_2\times Q_8)$, $C_2^5.\SOPlus(4,2)$, $C_2^2\times S_4^2$, $S_4^2:C_4$

Order 1920: $C_2\times D_4\times S_5$ x 56, $C_2.(D_4\times S_5)$ x 12, $(C_2\times D_4).S_5$ x 12, $C_2^4:S_5$ x 8, $(C_2\times S_5):D_4$ x 8, $(C_2\times D_4):S_5$ x 8, $C_2^2:C_4\times S_5$ x 8, $(C_2\times D_4):S_5$ x 6, $C_2^4\times S_5$ x 4, $C_2^2\times D_4\times A_5$ x 4, $C_2^2\times C_4:S_5$ x 4, $C_4\times C_2^2\times S_5$ x 4, $C_4:D_4\times A_5$ x 4, $(C_2^2\times C_4):S_5$ x 4, $(C_2\times S_5):D_4$ x 4, $(C_2\times D_4):S_5$ x 4, $A_5\times C_2^2\wr C_2$ x 4, $C_2^4.S_5$ x 4, $C_2^2:(C_4\times S_5)$ x 4, $C_4\times C_2^2:S_5$ x 4, $C_4\times D_4\times A_5$ x 2, $C_4:(C_4\times S_5)$ x 2, $A_5:C_4\times D_4$ x 2, $C_4^2:S_5$ x 2, $C_4:C_4\times S_5$ x 2, $C_2^2\times F_5\times S_4$ x 2, $\GL(2,\mathbb{Z}/4):F_5$ x 2, $D_{10}:C_4\times S_4$ x 2, $F_5\times \GL(2,\mathbb{Z}/4)$ x 2, $\SD_{16}\times S_5$, $C_4:D_4\times A_5$, $C_4^2:S_5$, $C_4^2\times S_5$, $\GL(2,\mathbb{Z}/4):D_{10}$, $D_4\times A_4\times F_5$, $D_4\times A_4:F_5$, $C_{20}:(C_4\times S_4)$, $C_{20}:C_4\times S_4$, $F_5\times C_4:S_4$, $C_4\times F_5\times S_4$, $D_4\times D_6\times F_5$

Order 1600: $C_{10}^2:C_4^2$ x 2, $F_5^2:C_4$ x 2, $D_5^2.Q_{16}$ x 2, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_5^2.(C_2\times Q_8)$, $(C_5\times C_{20}):\OD_{16}$, $(C_4\times D_5^2).C_4$, $C_{10}^2:C_4^2$, $C_4:F_5^2$, $C_4:F_5^2$, $C_4\times F_5^2$, $(D_4\times C_5^2):Q_8$, $D_5^2:\SD_{16}$, $(D_4\times C_5^2).D_4$, $C_5:(\SD_{16}\times F_5)$, $(D_4\times C_5^2):C_8$, $D_5^2.\SD_{16}$, $D_{10}:D_5:Q_8$, $C_5:F_5:\SD_{16}$, $C_5^2:(C_4\times \SD_{16})$, $D_5^2.(C_2\times C_8)$, $(C_5\times C_{20}).\OD_{16}$, $C_2^2\times F_5^2$

Order 1536: $S_4\times D_4^2$, $(C_2^2\times \SD_{16}):S_4$

Order 1440: $D_6\times S_5$ x 124, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):D_4$ x 8, $\GL(2,4):D_4$ x 8, $\GL(2,4):D_4$ x 8, $D_6:S_5$ x 8, $\GL(2,4):D_4$ x 8, $D_6:S_5$ x 8, $S_4\times A_5$ x 4, $A_4:S_5$ x 4, $A_4\times S_5$ x 4, $D_4\times \GL(2,4)$ x 4, $D_{12}\times A_5$ x 4, $C_4\times S_3\times A_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}\times S_5$ x 4, $C_3:C_4\times S_5$ x 4, $D_6.S_5$ x 4, $C_3:(C_4\times S_5)$ x 4, $C_2^3:\GL(2,4)$ x 3, $C_2^3\times \GL(2,4)$ x 2, $C_2\times C_{12}\times A_5$, $C_6:C_4\times A_5$, $C_2\times \GL(2,4):C_4$, $A_5:C_4\times C_6$

Order 1280: $F_5\times D_4^2$

Order 1200: $D_5\times S_5$ x 16, $D_{10}\times A_5$ x 7, $C_{10}\times S_5$ x 7, $C_{10}:S_5$ x 7, $F_5\times A_5$ x 5, $A_5:F_5$ x 5, $C_2\times C_{10}\times A_5$ x 2, $C_{20}\times A_5$, $C_5:C_4\times A_5$, $C_{10}.S_5$, $A_5:C_{20}$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 16, $C_2\times S_4^2$ x 8, $C_2\times C_3:S_3:\SD_{16}.C_2$ x 4, $C_4:(C_2.S_3^2).C_2^2$ x 4, $C_2^3:D_6^2$ x 2, $A_4^2:C_2^3$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $C_2^5:S_3^2$ x 2, $D_6\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4:\GL(2,\mathbb{Z}/4)$ x 2, $A_4:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times C_3^2:Q_8).C_2^3$, $(C_6\times C_{12}).C_2^4$, $C_6^2.C_2^4.C_2$, $(C_3\times C_6).(C_2^5.C_2)$, $(C_3\times C_6).(C_2^5.C_2)$, $C_6^2.(C_2^2\times C_4).C_2$, $A_4^2:C_2^3$, $C_2\times A_4^2:C_4$, $D_4\times A_4\times D_6$, $C_6:S_4\times D_4$, $C_6\times D_4\times S_4$, $D_4\times A_4^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $A_4:\GL(2,\mathbb{Z}/4)$, $A_4^2:D_4$, $C_4\times \PSOPlus(4,3)$, $A_4^2:Q_8$, $A_4^2:D_4$, $C_4\times A_4\times S_4$, $A_4^2:Q_8$, $C_2^5.S_3^2$, $C_2.S_4^2$, $C_2.S_4^2$, $A_4^2:C_8$, $A_4^2:C_8$, $D_4\times D_6^2$

Order 1024: $C_2^2.C_2^5.C_2^3$

Order 25: $C_5^2$

Order 9: $C_3^2$

Order 8: $D_4$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 225 (order at least 1024) are shown, as well as all normal subgroups of any index.

Order 230400: $S_5^2.\SD_{16}$

Order 115200: $S_5^2\times D_4$, $D_4.A_5^2.C_4$, $D_4.A_5^2.C_2^2$, $C_4.A_5^2.D_4$, $C_4.A_5^2.D_4$, $C_4.A_5^2.D_4$, $C_4.A_5^2.D_4$

Order 57600: $A_5^2:\SD_{16}$, $S_5^2:C_4$, $C_4\times S_5^2$, $A_5:(D_4\times S_5)$, $C_4:S_5\times S_5$, $D_4\times A_5\times S_5$, $(C_2^2\times A_5).S_5.C_2$, $(C_2^2\times A_5).S_5.C_2$, $D_4\times \PSOPlus(4,5)$, $C_2^2\times S_5^2$, $A_5^2:(C_2\times Q_8)$, $C_4.A_5^2.C_2^2$, $A_5^2:\SD_{16}$, $(C_4\times A_5^2):C_4$, $(C_2\times A_5^2).D_4$, $A_5^2:(C_2\times C_8)$, $A_5^2:(C_2\times C_8)$

Order 28800: $C_2\times S_5^2$ x 5, $A_5^2:Q_8$, $A_5^2:C_8$, $A_5^2:(C_2\times C_4)$, $C_2\times A_5^2:C_4$, $C_2.S_5^2$, $C_4\times A_5\times S_5$, $A_5^2:D_4$, $C_2.S_5^2$, $A_5^2:D_4$, $A_5^2:D_4$, $D_4\times A_5^2$, $A_5^2:C_2^3$, $C_4\times \PSOPlus(4,5)$, $A_5^2:C_2^3$, $A_5^2:D_4$, $A_5^2:D_4$, $A_5^2:D_4$, $A_5^2:C_8$, $A_5^2:Q_8$

Order 23040: $D_4\times S_4\times S_5$

Order 19200: $D_4\times F_5\times S_5$

Order 14400: $S_5^2$ x 5, $A_5^2:C_2^2$ x 4, $A_5^2:C_2^2$ x 3, $A_5^2:C_4$, $C_4\times A_5^2$, $A_5^2:C_4$, $C_2^2\times A_5^2$, $A_5^2:C_4$, $A_5^2:C_4$

Order 11520: $D_4\times D_6\times S_5$, $C_4:(S_4\times S_5)$, $S_4\times C_4:S_5$, $C_4:S_4\times S_5$, $C_4\times S_4\times S_5$, $S_5\times \GL(2,\mathbb{Z}/4)$, $S_4\times C_2^2:S_5$, $D_4\times S_4\times A_5$, $D_4\times A_4\times S_5$, $D_4\times A_4:S_5$, $\GL(2,\mathbb{Z}/4):S_5$, $C_2^2\times S_4\times S_5$

Order 9600: $C_{20}:(C_4\times S_5)$, $F_5\times C_4:S_5$, $C_{20}:C_4\times S_5$, $C_4\times F_5\times S_5$, $D_4\times D_5\times S_5$, $D_4\times F_5\times A_5$, $C_2^2\times F_5\times S_5$, $D_{10}:(C_4\times S_5)$, $F_5\times C_2^2:S_5$, $D_{10}:C_4\times S_5$, $D_4\times A_5:F_5$

Order 9216: $S_4^2:\SD_{16}$

Order 7680: $D_4^2\times S_5$

Order 7200: $\PSOPlus(4,5)$ x 3, $C_2\times A_5^2$ x 2, $A_5\times S_5$ x 2

Order 6400: $F_5^2:\SD_{16}$

Order 5760: $S_3\times D_4\times S_5$ x 12, $C_2\times S_4\times S_5$ x 6, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $S_3\times C_2^3:S_5$, $D_4\times A_4\times A_5$, $A_5\times C_4:S_4$, $C_4\times S_4\times A_5$, $C_6:(D_4\times S_5)$, $C_6:S_5\times D_4$, $(C_4\times A_4):S_5$, $A_4\times C_4:S_5$, $C_4\times A_4\times S_5$, $C_4\times A_4:S_5$, $C_6:D_4\times S_5$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $\GL(2,4):C_2^5$, $A_5:\GL(2,\mathbb{Z}/4)$, $A_4\times C_2^2:S_5$, $A_4:(C_4\times S_5)$, $(C_2\times S_4):S_5$, $A_4:C_4\times S_5$, $A_5:\GL(2,\mathbb{Z}/4)$, $A_5:\GL(2,\mathbb{Z}/4)$, $C_2^2\times A_4:S_5$, $C_2^2\times A_4\times S_5$, $C_2^2\times S_4\times A_5$, $D_4\times D_6\times A_5$, $A_5\times \GL(2,\mathbb{Z}/4)$, $A_5:C_4\times S_4$

Order 4800: $C_2\times F_5\times S_5$ x 7, $(C_{10}\times S_5):C_4$ x 2, $A_5:C_4\times F_5$ x 2, $D_{20}:S_5$, $C_5:D_4\times S_5$, $C_4\times D_5\times S_5$, $D_{20}\times S_5$, $C_5:(D_4\times S_5)$, $D_5\times C_4:S_5$, $D_4\times C_5:S_5$, $A_5\times D_{10}:C_4$, $C_5\times D_4\times S_5$, $A_5\times C_{20}:C_4$, $C_2\times D_{10}\times S_5$, $D_5\times C_2^2:S_5$, $D_4\times D_5\times A_5$, $C_4\times F_5\times A_5$, $C_{10}.(C_4\times S_5)$, $C_2^2\times F_5\times A_5$, $C_4\times A_5:F_5$, $(C_{20}\times A_5):C_4$, $C_2\times D_{10}.S_5$, $(D_{10}\times A_5):C_4$

Order 4608: $D_4\times S_4^2$, $A_4^2.D_4:C_4$, $A_4^2:(C_2\times \SD_{16})$, $(C_4\times A_4^2).D_4$, $S_4^2:C_8$, $A_4^2:(C_2\times \SD_{16})$, $S_4^2:Q_8$

Order 3840: $C_4:D_4\times S_5$ x 3, $A_5:D_4^2$ x 3, $A_5:D_4^2$ x 3, $C_2^2\times D_4\times S_5$ x 3, $C_4\times D_4\times S_5$ x 2, $A_5.D_4^2$ x 2, $A_5:D_4^2$ x 2, $S_5\times C_2^2\wr C_2$ x 2, $C_4:D_4\times S_5$, $A_5:D_4^2$, $D_4\times F_5\times S_4$, $A_5\times D_4^2$

Order 3600: $A_5^2$

Order 3200: $D_4\times F_5^2$, $(C_4\times D_5^2).C_2^3$, $(D_4\times C_5^2):\OD_{16}$, $(C_4\times D_5^2).D_4$, $F_5^2:Q_8$, $F_5^2:C_8$, $(C_4\times D_5^2).D_4$

Order 2880: $\GL(2,4):C_2^4$ x 16, $C_3:(D_4\times S_5)$ x 8, $S_3\times C_2^2:S_5$ x 8, $C_3:D_4\times S_5$ x 8, $C_4\times S_3\times S_5$ x 8, $D_{12}:S_5$ x 6, $S_3\times C_4:S_5$ x 6, $D_{12}\times S_5$ x 6, $S_3\times D_4\times A_5$ x 6, $D_4\times C_3:S_5$ x 6, $C_3\times D_4\times S_5$ x 6, $S_4\times S_5$ x 6, $C_2\times A_4:S_5$ x 4, $C_2\times S_4\times A_5$ x 3, $C_2\times A_4\times S_5$ x 3, $C_2\times D_{12}\times A_5$, $C_4\times D_6\times A_5$, $\GL(2,4):C_2^4$, $C_4\times A_4\times A_5$, $A_5:C_4\times D_6$, $C_6:D_4\times A_5$, $A_4:C_4\times A_5$, $C_2^4:\GL(2,4)$, $C_6\times D_4\times A_5$, $C_2\times C_{12}:S_5$, $\GL(2,4):C_2^4$, $C_2\times D_6:S_5$, $C_6:(C_4\times S_5)$, $C_2\times \GL(2,4):D_4$, $C_2\times D_6:S_5$, $C_6:C_4\times S_5$, $C_2\times \GL(2,4):D_4$, $C_2\times C_{12}:S_5$, $C_2^2:S_5\times C_6$, $\GL(2,4):C_2^4$, $C_2\times C_{12}\times S_5$, $C_2\times C_{12}:S_5$, $(A_4\times A_5):C_4$, $A_4\times A_5:C_4$

Order 2400: $F_5\times S_5$ x 7, $D_{10}\times S_5$ x 6, $D_{10}.S_5$ x 5, $C_2\times F_5\times A_5$ x 4, $C_5:(C_4\times S_5)$, $(C_{10}\times S_5):C_2$, $C_5:C_4\times S_5$, $C_{20}:S_5$, $C_{20}:S_5$, $C_{20}:S_5$, $C_{20}\times S_5$, $C_4\times D_5\times A_5$, $D_{20}\times A_5$, $D_{10}.S_5$, $D_{10}:S_5$, $D_{10}:S_5$, $C_5\times C_2^2:S_5$, $C_2\times C_{10}:S_5$, $(C_2\times C_{10}):S_5$, $C_5\times D_4\times A_5$, $C_2\times C_{10}\times S_5$, $C_2\times D_{10}\times A_5$, $C_5:D_4\times A_5$

Order 2304: $(C_3\times C_6).C_2^6.C_2$, $C_2^4:D_6^2$, $C_4:S_4^2$, $C_4:S_4^2$, $C_4\times S_4^2$, $D_4\times \PSOPlus(4,3)$, $S_4\times \GL(2,\mathbb{Z}/4)$, $\GL(2,\mathbb{Z}/4):S_4$, $D_4\times A_4\times S_4$, $A_4^2:\SD_{16}$, $A_4^2:(C_2\times C_8)$, $(C_4\times A_4^2):C_4$, $A_4^2:(C_2\times C_8)$, $A_4^2:\SD_{16}$, $A_4^2:(C_2\times Q_8)$, $A_4^2:\SD_{16}$, $C_2^5.\SOPlus(4,2)$, $C_2^2\times S_4^2$, $S_4^2:C_4$

Order 1920: $C_2\times D_4\times S_5$ x 30, $C_2.(D_4\times S_5)$ x 6, $(C_2\times D_4).S_5$ x 6, $C_2^2:C_4\times S_5$ x 5, $C_2^4:S_5$ x 4, $(C_2\times D_4):S_5$ x 4, $(C_2\times S_5):D_4$ x 4, $(C_2\times D_4):S_5$ x 4, $C_2^2\times D_4\times A_5$ x 3, $C_2^2\times C_4:S_5$ x 3, $C_4\times C_2^2\times S_5$ x 3, $C_4:D_4\times A_5$ x 3, $(C_2^2\times C_4):S_5$ x 3, $(C_2\times S_5):D_4$ x 3, $(C_2\times D_4):S_5$ x 3, $C_2^2:(C_4\times S_5)$ x 3, $C_4\times C_2^2:S_5$ x 3, $C_2^4\times S_5$ x 2, $A_5\times C_2^2\wr C_2$ x 2, $C_2^4.S_5$ x 2, $C_4\times D_4\times A_5$ x 2, $C_4:(C_4\times S_5)$ x 2, $A_5:C_4\times D_4$ x 2, $C_4^2:S_5$ x 2, $C_4:C_4\times S_5$ x 2, $\SD_{16}\times S_5$, $C_4:D_4\times A_5$, $C_4^2:S_5$, $C_4^2\times S_5$, $C_2^2\times F_5\times S_4$, $\GL(2,\mathbb{Z}/4):D_{10}$, $D_4\times A_4\times F_5$, $D_4\times A_4:F_5$, $\GL(2,\mathbb{Z}/4):F_5$, $D_{10}:C_4\times S_4$, $F_5\times \GL(2,\mathbb{Z}/4)$, $C_{20}:(C_4\times S_4)$, $C_{20}:C_4\times S_4$, $F_5\times C_4:S_4$, $C_4\times F_5\times S_4$, $D_4\times D_6\times F_5$

Order 1600: $F_5^2:C_4$ x 2, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_5^2.(C_2\times Q_8)$, $(C_5\times C_{20}):\OD_{16}$, $(C_4\times D_5^2).C_4$, $C_{10}^2:C_4^2$, $C_{10}^2:C_4^2$, $C_4:F_5^2$, $C_4:F_5^2$, $C_4\times F_5^2$, $D_5^2.Q_{16}$, $(D_4\times C_5^2):Q_8$, $D_5^2:\SD_{16}$, $(D_4\times C_5^2).D_4$, $C_5:(\SD_{16}\times F_5)$, $(D_4\times C_5^2):C_8$, $D_5^2.\SD_{16}$, $D_{10}:D_5:Q_8$, $C_5:F_5:\SD_{16}$, $C_5^2:(C_4\times \SD_{16})$, $D_5^2.(C_2\times C_8)$, $(C_5\times C_{20}).\OD_{16}$, $C_2^2\times F_5^2$

Order 1536: $S_4\times D_4^2$, $(C_2^2\times \SD_{16}):S_4$

Order 1440: $D_6\times S_5$ x 46, $\GL(2,4):C_2^3$ x 8, $\GL(2,4):C_2^3$ x 8, $\GL(2,4):C_2^3$ x 8, $\GL(2,4):D_4$ x 4, $C_4\times S_3\times A_5$ x 4, $\GL(2,4):D_4$ x 4, $C_{12}:S_5$ x 4, $\GL(2,4):D_4$ x 4, $C_{12}\times S_5$ x 4, $D_6:S_5$ x 4, $\GL(2,4):D_4$ x 4, $C_3:C_4\times S_5$ x 4, $D_6:S_5$ x 4, $D_6.S_5$ x 4, $C_3:(C_4\times S_5)$ x 4, $A_4:S_5$ x 3, $D_4\times \GL(2,4)$ x 3, $D_{12}\times A_5$ x 3, $C_{12}:S_5$ x 3, $C_{12}:S_5$ x 3, $C_2^3:\GL(2,4)$ x 2, $S_4\times A_5$ x 2, $A_4\times S_5$ x 2, $C_2^3\times \GL(2,4)$, $C_2\times C_{12}\times A_5$, $C_6:C_4\times A_5$, $C_2\times \GL(2,4):C_4$, $A_5:C_4\times C_6$

Order 1280: $F_5\times D_4^2$

Order 1200: $D_5\times S_5$ x 6, $D_{10}\times A_5$ x 4, $A_5:F_5$ x 4, $C_{10}\times S_5$ x 3, $C_{10}:S_5$ x 3, $F_5\times A_5$ x 3, $C_2\times C_{10}\times A_5$, $C_{20}\times A_5$, $C_5:C_4\times A_5$, $C_{10}.S_5$, $A_5:C_{20}$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 12, $C_2\times S_4^2$ x 5, $C_2\times C_3:S_3:\SD_{16}.C_2$ x 4, $C_4:(C_2.S_3^2).C_2^2$ x 4, $(C_2\times C_3^2:Q_8).C_2^3$, $(C_6\times C_{12}).C_2^4$, $C_6^2.C_2^4.C_2$, $(C_3\times C_6).(C_2^5.C_2)$, $(C_3\times C_6).(C_2^5.C_2)$, $C_6^2.(C_2^2\times C_4).C_2$, $C_2^3:D_6^2$, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $C_2\times A_4^2:C_4$, $D_4\times A_4\times D_6$, $C_6:S_4\times D_4$, $C_6\times D_4\times S_4$, $D_4\times A_4^2$, $\GL(2,\mathbb{Z}/4):D_6$, $C_2^5:S_3^2$, $D_6\times \GL(2,\mathbb{Z}/4)$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $A_4:\GL(2,\mathbb{Z}/4)$, $A_4^2:D_4$, $C_4\times \PSOPlus(4,3)$, $A_4^2:Q_8$, $A_4\times \GL(2,\mathbb{Z}/4)$, $A_4^2:D_4$, $C_4\times A_4\times S_4$, $A_4^2:Q_8$, $C_2^5.S_3^2$, $A_4:\GL(2,\mathbb{Z}/4)$, $A_4:\GL(2,\mathbb{Z}/4)$, $C_2.S_4^2$, $C_2.S_4^2$, $A_4^2:C_8$, $A_4^2:C_8$, $D_4\times D_6^2$

Order 1024: $C_2^2.C_2^5.C_2^3$

Order 25: $C_5^2$

Order 9: $C_3^2$

Order 8: $D_4$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 230400: $S_5^2.\SD_{16}$ ($C_1$)

Order 115200: $S_5^2\times D_4$ ($C_2$), $D_4.A_5^2.C_4$ ($C_2$), $D_4.A_5^2.C_2^2$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$)

Order 57600: $C_4\times S_5^2$ ($C_2^2$), $A_5:(D_4\times S_5)$ ($C_2^2$), $D_4\times \PSOPlus(4,5)$ ($C_2^2$), $A_5^2:(C_2\times Q_8)$ ($C_2^2$), $(C_4\times A_5^2):C_4$ ($C_2^2$), $A_5^2:(C_2\times C_8)$ ($C_2^2$), $A_5^2:(C_2\times C_8)$ ($C_2^2$)

Order 28800: $A_5^2:D_4$ ($D_4$) x 2, $C_2.S_5^2$ ($D_4$), $D_4\times A_5^2$ ($D_4$), $C_4\times \PSOPlus(4,5)$ ($C_2^3$), $A_5^2:D_4$ ($D_4$), $C_2\times S_5^2$ ($D_4$)

Order 14400: $S_5^2$ ($\SD_{16}$) x 2, $C_4\times A_5^2$ ($C_2\times D_4$), $A_5^2:C_4$ ($C_2\times D_4$), $A_5^2:C_2^2$ ($C_2\times D_4$)

Order 7200: $\PSOPlus(4,5)$ ($D_8:C_2$), $\PSOPlus(4,5)$ ($C_2\times \SD_{16}$), $C_2\times A_5^2$ ($C_2^2\wr C_2$)

Order 3600: $A_5^2$ ($C_2^2:\SD_{16}$)

Order 8: $D_4$ ($A_5^2.D_4$)

Order 4: $C_4$ ($S_5^2:C_2^2$)

Order 2: $C_2$ ($S_5^2:D_4$)

Order 1: $C_1$ ($S_5^2.\SD_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 230400: $S_5^2.\SD_{16}$ ($C_1$)

Order 115200: $S_5^2\times D_4$ ($C_2$), $D_4.A_5^2.C_4$ ($C_2$), $D_4.A_5^2.C_2^2$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$), $C_4.A_5^2.D_4$ ($C_2$)

Order 57600: $C_4\times S_5^2$ ($C_2^2$), $A_5:(D_4\times S_5)$ ($C_2^2$), $D_4\times \PSOPlus(4,5)$ ($C_2^2$), $A_5^2:(C_2\times Q_8)$ ($C_2^2$), $(C_4\times A_5^2):C_4$ ($C_2^2$), $A_5^2:(C_2\times C_8)$ ($C_2^2$), $A_5^2:(C_2\times C_8)$ ($C_2^2$)

Order 28800: $C_2.S_5^2$ ($D_4$), $A_5^2:D_4$ ($D_4$), $D_4\times A_5^2$ ($D_4$), $C_4\times \PSOPlus(4,5)$ ($C_2^3$), $A_5^2:D_4$ ($D_4$), $C_2\times S_5^2$ ($D_4$)

Order 14400: $C_4\times A_5^2$ ($C_2\times D_4$), $A_5^2:C_4$ ($C_2\times D_4$), $A_5^2:C_2^2$ ($C_2\times D_4$), $S_5^2$ ($\SD_{16}$)

Order 7200: $\PSOPlus(4,5)$ ($D_8:C_2$), $\PSOPlus(4,5)$ ($C_2\times \SD_{16}$), $C_2\times A_5^2$ ($C_2^2\wr C_2$)

Order 3600: $A_5^2$ ($C_2^2:\SD_{16}$)

Order 8: $D_4$ ($A_5^2.D_4$)

Order 4: $C_4$ ($S_5^2:C_2^2$)

Order 2: $C_2$ ($S_5^2:D_4$)

Order 1: $C_1$ ($S_5^2.\SD_{16}$)

Series

Derived series

$S_5^2.\SD_{16}$

$\rhd$

$C_4\times \PSOPlus(4,5)$

$\rhd$

$A_5^2$

Chief series

$S_5^2.\SD_{16}$

$\rhd$

$S_5^2\times D_4$

$\rhd$

$D_4\times \PSOPlus(4,5)$

$\rhd$

$D_4\times A_5^2$

$\rhd$

$D_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$S_5^2.\SD_{16}$

$\rhd$

$C_4\times \PSOPlus(4,5)$

$\rhd$

$C_2\times A_5^2$

$\rhd$

$A_5^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $154 \times 154$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $147 \times 147$ rational character table (warning: may be slow to load).