Abstract group 230400.bf: $(C_2\times S_5^2).D_4$

• Label: 230400.bf
• Order: \( 2^{10} \cdot 3^{2} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{2} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $18$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$(C_2\times S_5^2).D_4$
Order:	\(230400\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^6.C_2^2.A_5^2.D_4$, of order \(7372800\)\(\medspace = 2^{15} \cdot 3^{2} \cdot 5^{2} \)
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	30
Elements	1	5407	440	44640	624	29000	28800	13968	57600	960	34560	14400	230400
Conjugacy classes	1	41	2	42	2	66	4	28	24	1	12	15	238
Divisions	1	41	2	36	2	66	2	28	20	1	10	15	224
Autjugacy classes	1	16	2	11	2	20	1	11	6	1	2	4	77

Dimension	1	2	8	10	12	16	25	32	36	40	48	50	60	72
Irr. complex chars.	16	12	32	32	16	16	16	12	8	32	16	12	16	2	238
Irr. rational chars.	8	16	32	32	16	8	8	16	4	32	16	16	16	4	224

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	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $18$ $\langle(1,3)(2,4,5,9)(6,10), (1,2,3,5,6,8)(4,7)(9,10)(11,12,13,15)(14,16,17,18), (1,3,6)(2,4,8,9,5)(11,13)(12,14)(15,17)(16,18)\rangle$
Transitive group:	40T106279	40T106307			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:	$(C_2\times S_5^2)$ . $D_4$	$C_2$ . $(S_5^2:D_4)$	$(C_2^2\times S_5^2)$ . $C_4$	$S_5^2$ . $(C_2^2:C_4)$	all 31

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 73 normal subgroups (17 characteristic).

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

Special subgroups

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

Order 115200: $C_2\times S_5^2:C_4$ x 4, $C_2^3\times A_5.S_5.C_2$, $C_2^3.\POPlus(4,5)$, $C_2^3.A_5^2.C_4$

Order 57600: $C_2^2\times S_5^2$ x 17, $S_5^2:C_4$ x 8, $(C_2^2\times A_5^2):C_4$ x 2, $A_5^2:(C_2^2\times C_4)$ x 2, $A_5^2:C_4\times C_2^2$ x 2, $(C_2^3\times A_5).A_5.C_2$, $(C_2^3\times A_5).S_5$, $(C_2^2\times A_5^2):C_4$, $(C_2\times A_5^2).D_4$

Order 28800: $C_2\times S_5^2$ x 62, $A_5^2:C_2^3$ x 14, $A_5^2:C_2^3$ x 10, $C_2\times A_5^2:C_4$ x 6, $A_5^2:(C_2\times C_4)$ x 4, $C_2\times A_5^2:C_4$ x 2, $(C_2^3\times A_5).A_5$

Order 23040: $C_2^3\times S_5\times S_4$

Order 19200: $(C_2^3\times F_5).S_5$

Order 14400: $S_5^2$ x 36, $A_5^2:C_2^2$ x 28, $A_5^2:C_2^2$ x 18, $C_2^2\times A_5^2$ x 5, $A_5^2:C_4$ x 2, $A_5^2:C_4$ x 2

Order 11520: $C_2^2\times S_4\times S_5$ x 28, $(C_2^4\times S_3).S_5$, $(C_2^3\times A_4).S_5$, $A_5.(C_2^3\times S_4)$, $C_2^3\times A_4:S_5$

Order 9600: $C_2^2\times F_5\times S_5$ x 28, $(C_2^3\times F_5).A_5$, $(C_2^3\times D_5).S_5$, $(C_2^3\times D_5).S_5$

Order 9216: $(C_2\times S_4^2).D_4$

Order 7680: $C_2^3\times S_5\times D_4$

Order 7200: $A_5\times S_5$ x 8, $\PSOPlus(4,5)$ x 6, $C_2\times A_5^2$ x 5

Order 6400: $(C_2^2\times F_5^2):C_4$

Order 5760: $C_2\times S_4\times S_5$ x 112, $\GL(2,4):C_2^5$ x 60, $C_2^2\times A_4:S_5$ x 14, $C_2^2\times A_4\times S_5$ x 14, $C_2^2\times S_4\times A_5$ x 14, $(C_2^3\times C_6).S_5$, $(C_2^3\times C_6).S_5$, $(C_2^4\times S_3).A_5$, $C_2^5:\GL(2,4)$

Order 4800: $C_2\times F_5\times S_5$ x 112, $C_2\times D_{10}\times S_5$ x 28, $C_2^2\times F_5\times A_5$ x 14, $C_2\times D_{10}.S_5$ x 14, $(C_2^2\times C_{10}).S_5$, $(C_2^2\times C_{10}).S_5$, $(C_2^3\times D_5).A_5$

Order 4608: $C_2\times S_4^2:C_4$ x 4, $C_2^3\times S_4^2$, $C_2^3:(A_4^2:C_4)$, $C_2^7.S_3^2$

Order 3840: $C_2^2\times D_4\times S_5$ x 56, $C_2^5\times S_5$ x 2, $C_2^5:S_5$ x 2, $D_4\times C_2^3\times A_5$, $C_4\times C_2^3\times S_5$, $C_2^3\times F_5\times S_4$, $C_2^3\times C_4:S_5$

Order 3600: $A_5^2$

Order 3200: $D_{10}^2.D_4$ x 4, $D_5^2.C_2^5$, $D_{10}^2.C_2^3$, $C_2\times C_{10}^2:\OD_{16}$

Order 2880: $\GL(2,4):C_2^4$ x 560, $S_4\times S_5$ x 64, $\GL(2,4):C_2^4$ x 30, $\GL(2,4):C_2^4$ x 30, $\GL(2,4):C_2^4$ x 30, $C_2\times S_4\times A_5$ x 28, $C_2\times A_4\times S_5$ x 28, $C_2\times A_4:S_5$ x 28, $C_2^4:\GL(2,4)$ x 7, $(C_2^3\times C_6).A_5$

Order 2400: $D_{10}\times S_5$ x 112, $F_5\times S_5$ x 64, $C_2\times F_5\times A_5$ x 28, $D_{10}.S_5$ x 28, $C_2\times C_{10}:S_5$ x 14, $C_2\times C_{10}\times S_5$ x 14, $C_2\times D_{10}\times A_5$ x 14, $C_5.(C_2^3\times A_5)$

Order 2304: $C_2^2\times S_4^2$ x 17, $S_4^2:C_4$ x 8, $A_4^2:C_4\times C_2^2$ x 2, $C_2^6.S_3^2$ x 2, $(C_2^2\times A_4^2):C_4$ x 2, $C_3^2.C_2^6.C_2^2$, $A_4^2:C_2^4$, $C_2^4:D_6^2$, $A_4^2:C_2^4$, $(C_2^2\times A_4^2):C_4$, $C_2^6.S_3^2$

Order 1920: $C_2\times D_4\times S_5$ x 448, $C_2^4\times S_5$ x 75, $C_2^4:S_5$ x 56, $C_2^2\times D_4\times A_5$ x 28, $C_2^2\times C_4:S_5$ x 28, $C_4\times C_2^2\times S_5$ x 28, $C_2^2\times F_5\times S_4$ x 28, $C_2^5\times A_5$ x 2, $C_4\times C_2^3\times A_5$, $C_2^4.S_5$, $C_2^2:C_4\times S_5$, $C_2^2\times D_{10}\times S_4$, $C_2^3\times A_4\times F_5$, $C_2^3\times A_4:F_5$, $C_2^3\times D_6\times F_5$

Order 1600: $C_2^2\times F_5^2$ x 17, $F_5^2:C_4$ x 8, $C_{10}^2:\OD_{16}$ x 6, $D_{10}^2.C_2^2$ x 2, $D_{10}^2.C_2^2$ x 2, $C_{10}^2:\OD_{16}$ x 2, $C_2\times C_{10}^2:C_8$, $D_{10}^2.C_4$, $(C_2\times C_{10}^2).Q_8$, $(C_2\times C_{10}^2).D_4$, $D_{10}^2:C_4$, $C_{10}^2:C_4^2$, $D_{10}^2.C_2^2$, $D_5^2:C_4^2$, $D_{10}^2:C_4$, $D_{10}^2:C_4$

Order 1536: $\GL(2,\mathbb{Z}/4):C_2^4$, $C_2^7.D_6$

Order 1440: $D_6\times S_5$ x 960, $\GL(2,4):C_2^3$ x 140, $\GL(2,4):C_2^3$ x 140, $\GL(2,4):C_2^3$ x 140, $C_2^3\times \GL(2,4)$ x 15, $S_4\times A_5$ x 8, $A_4:S_5$ x 8, $A_4\times S_5$ x 8, $C_2^3:\GL(2,4)$ x 7

Order 1280: $D_{10}.C_2^6$

Order 1200: $D_5\times S_5$ x 64, $D_{10}\times A_5$ x 28, $C_{10}\times S_5$ x 28, $C_{10}:S_5$ x 28, $F_5\times A_5$ x 8, $A_5:F_5$ x 8, $C_2\times C_{10}\times A_5$ x 7

Order 1152: $C_2\times S_4^2$ x 62, $C_2^3:D_6^2$ x 60, $A_4^2:C_2^3$ x 14, $C_2\times C_3^2.(C_2^5.C_2)$ x 12, $A_4^2:C_2^3$ x 10, $C_2\times A_4^2:C_4$ x 6, $C_2^5.S_3^2$ x 4, $C_2\times A_4^2:C_4$ x 2, $(C_2\times C_6^2).C_2^4$, $C_2\times C_6^2.(C_2^2\times C_4)$, $C_2^3\times A_4\times D_6$, $C_2^3\times C_6:S_4$, $C_6\times C_2^3\times S_4$, $C_2^3\times A_4^2$, $C_2^3\times D_6^2$

Order 1024: $C_2^4.C_2^5.C_2$

Order 25: $C_5^2$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 3

Order 2: $C_2$ x 3

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: $(C_2\times S_5^2).D_4$

Order 115200: $C_2^3\times A_5.S_5.C_2$, $C_2^3.\POPlus(4,5)$, $C_2^3.A_5^2.C_4$, $C_2\times S_5^2:C_4$

Order 57600: $C_2^2\times S_5^2$ x 5, $(C_2^3\times A_5).A_5.C_2$, $(C_2^3\times A_5).S_5$, $(C_2^2\times A_5^2):C_4$, $(C_2^2\times A_5^2):C_4$, $S_5^2:C_4$, $(C_2\times A_5^2).D_4$, $A_5^2:(C_2^2\times C_4)$, $A_5^2:C_4\times C_2^2$

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

Order 23040: $C_2^3\times S_5\times S_4$

Order 19200: $(C_2^3\times F_5).S_5$

Order 14400: $A_5^2:C_2^2$ x 7, $S_5^2$ x 4, $C_2^2\times A_5^2$ x 3, $A_5^2:C_2^2$ x 3, $A_5^2:C_4$, $A_5^2:C_4$

Order 11520: $C_2^2\times S_4\times S_5$ x 5, $(C_2^4\times S_3).S_5$, $(C_2^3\times A_4).S_5$, $A_5.(C_2^3\times S_4)$, $C_2^3\times A_4:S_5$

Order 9600: $C_2^2\times F_5\times S_5$ x 5, $(C_2^3\times F_5).A_5$, $(C_2^3\times D_5).S_5$, $(C_2^3\times D_5).S_5$

Order 9216: $(C_2\times S_4^2).D_4$

Order 7680: $C_2^3\times S_5\times D_4$

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

Order 6400: $(C_2^2\times F_5^2):C_4$

Order 5760: $\GL(2,4):C_2^5$ x 14, $C_2\times S_4\times S_5$ x 8, $C_2^2\times A_4:S_5$ x 5, $C_2^2\times A_4\times S_5$ x 3, $C_2^2\times S_4\times A_5$ x 3, $(C_2^3\times C_6).S_5$, $(C_2^3\times C_6).S_5$, $(C_2^4\times S_3).A_5$, $C_2^5:\GL(2,4)$

Order 4800: $C_2\times F_5\times S_5$ x 8, $C_2\times D_{10}\times S_5$ x 6, $C_2\times D_{10}.S_5$ x 5, $C_2^2\times F_5\times A_5$ x 3, $(C_2^2\times C_{10}).S_5$, $(C_2^2\times C_{10}).S_5$, $(C_2^3\times D_5).A_5$

Order 4608: $C_2^3\times S_4^2$, $C_2^3:(A_4^2:C_4)$, $C_2^7.S_3^2$, $C_2\times S_4^2:C_4$

Order 3840: $C_2^2\times D_4\times S_5$ x 10, $C_2^5\times S_5$ x 2, $C_2^5:S_5$ x 2, $D_4\times C_2^3\times A_5$, $C_4\times C_2^3\times S_5$, $C_2^3\times F_5\times S_4$, $C_2^3\times C_4:S_5$

Order 3600: $A_5^2$

Order 3200: $D_5^2.C_2^5$, $D_{10}^2.D_4$, $D_{10}^2.C_2^3$, $C_2\times C_{10}^2:\OD_{16}$

Order 2880: $\GL(2,4):C_2^4$ x 52, $\GL(2,4):C_2^4$ x 8, $\GL(2,4):C_2^4$ x 7, $\GL(2,4):C_2^4$ x 7, $C_2\times A_4:S_5$ x 7, $S_4\times S_5$ x 4, $C_2^4:\GL(2,4)$ x 3, $C_2\times S_4\times A_5$ x 3, $C_2\times A_4\times S_5$ x 3, $(C_2^3\times C_6).A_5$

Order 2400: $D_{10}\times S_5$ x 9, $D_{10}.S_5$ x 7, $C_2\times D_{10}\times A_5$ x 6, $F_5\times S_5$ x 4, $C_2\times C_{10}:S_5$ x 3, $C_2\times C_{10}\times S_5$ x 3, $C_2\times F_5\times A_5$ x 3, $C_5.(C_2^3\times A_5)$

Order 2304: $C_2^2\times S_4^2$ x 5, $C_3^2.C_2^6.C_2^2$, $A_4^2:C_2^4$, $C_2^4:D_6^2$, $A_4^2:C_2^4$, $(C_2^2\times A_4^2):C_4$, $A_4^2:C_4\times C_2^2$, $C_2^6.S_3^2$, $C_2^6.S_3^2$, $S_4^2:C_4$, $(C_2^2\times A_4^2):C_4$

Order 1920: $C_2\times D_4\times S_5$ x 26, $C_2^4\times S_5$ x 19, $C_2^4:S_5$ x 15, $C_2^2\times D_4\times A_5$ x 6, $C_2^2\times C_4:S_5$ x 5, $C_4\times C_2^2\times S_5$ x 5, $C_2^2\times F_5\times S_4$ x 5, $C_2^5\times A_5$ x 2, $C_4\times C_2^3\times A_5$, $C_2^4.S_5$, $C_2^2:C_4\times S_5$, $C_2^2\times D_{10}\times S_4$, $C_2^3\times A_4\times F_5$, $C_2^3\times A_4:F_5$, $C_2^3\times D_6\times F_5$

Order 1600: $C_2^2\times F_5^2$ x 5, $C_{10}^2:\OD_{16}$ x 2, $C_2\times C_{10}^2:C_8$, $D_{10}^2.C_2^2$, $D_{10}^2.C_4$, $F_5^2:C_4$, $(C_2\times C_{10}^2).Q_8$, $(C_2\times C_{10}^2).D_4$, $D_{10}^2:C_4$, $C_{10}^2:C_4^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_5^2:C_4^2$, $D_{10}^2:C_4$, $D_{10}^2:C_4$, $C_{10}^2:\OD_{16}$

Order 1536: $\GL(2,\mathbb{Z}/4):C_2^4$, $C_2^7.D_6$

Order 1440: $D_6\times S_5$ x 66, $\GL(2,4):C_2^3$ x 18, $\GL(2,4):C_2^3$ x 16, $\GL(2,4):C_2^3$ x 16, $C_2^3\times \GL(2,4)$ x 4, $C_2^3:\GL(2,4)$ x 3, $A_4:S_5$ x 3, $S_4\times A_5$, $A_4\times S_5$

Order 1280: $D_{10}.C_2^6$

Order 1200: $D_{10}\times A_5$ x 9, $D_5\times S_5$ x 4, $C_2\times C_{10}\times A_5$ x 3, $C_{10}\times S_5$ x 3, $C_{10}:S_5$ x 3, $A_5:F_5$ x 3, $F_5\times A_5$

Order 1152: $C_2^3:D_6^2$ x 14, $C_2\times S_4^2$ x 8, $C_2\times C_3^2.(C_2^5.C_2)$ x 6, $A_4^2:C_2^3$ x 5, $A_4^2:C_2^3$ x 3, $C_2\times A_4^2:C_4$ x 2, $(C_2\times C_6^2).C_2^4$, $C_2\times C_6^2.(C_2^2\times C_4)$, $C_2^3\times A_4\times D_6$, $C_2^3\times C_6:S_4$, $C_6\times C_2^3\times S_4$, $C_2^3\times A_4^2$, $C_2\times A_4^2:C_4$, $C_2^5.S_3^2$, $C_2^3\times D_6^2$

Order 1024: $C_2^4.C_2^5.C_2$

Order 25: $C_5^2$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 230400: $(C_2\times S_5^2).D_4$ ($C_1$)

Order 115200: $C_2\times S_5^2:C_4$ ($C_2$) x 4, $C_2^3\times A_5.S_5.C_2$ ($C_2$), $C_2^3.\POPlus(4,5)$ ($C_2$), $C_2^3.A_5^2.C_4$ ($C_2$)

Order 57600: $C_2^2\times S_5^2$ ($C_4$) x 4, $C_2^2\times S_5^2$ ($C_2^2$) x 2, $A_5^2:(C_2^2\times C_4)$ ($C_2^2$) x 2, $A_5^2:C_4\times C_2^2$ ($C_2^2$) x 2, $(C_2^3\times A_5).S_5$ ($C_2^2$)

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

Order 14400: $S_5^2$ ($C_2^2:C_4$) x 8, $A_5^2:C_2^2$ ($C_2\times D_4$) x 5, $C_2^2\times A_5^2$ ($C_2^2:C_4$) x 2, $A_5^2:C_2^2$ ($C_2^2:C_4$) x 2, $C_2^2\times A_5^2$ ($C_2\times D_4$), $A_5^2:C_2^2$ ($C_2^2\times C_4$)

Order 7200: $\PSOPlus(4,5)$ ($C_2^2\wr C_2$) x 2, $\PSOPlus(4,5)$ ($C_2^3:C_4$) x 2, $C_2\times A_5^2$ ($C_2^2\wr C_2$) x 2, $C_2\times A_5^2$ ($C_2^3:C_4$)

Order 3600: $A_5^2$ ($C_2^4:C_4$)

Order 8: $C_2^3$ ($A_5^2.D_4$)

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

Order 2: $C_2$ ($S_5^2:D_4$) x 2, $C_2$ ($C_2\times S_5^2:C_4$)

Order 1: $C_1$ ($(C_2\times S_5^2).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 230400: $(C_2\times S_5^2).D_4$ ($C_1$)

Order 115200: $C_2^3\times A_5.S_5.C_2$ ($C_2$), $C_2^3.\POPlus(4,5)$ ($C_2$), $C_2^3.A_5^2.C_4$ ($C_2$), $C_2\times S_5^2:C_4$ ($C_2$)

Order 57600: $(C_2^3\times A_5).S_5$ ($C_2^2$), $C_2^2\times S_5^2$ ($C_2^2$), $C_2^2\times S_5^2$ ($C_4$), $A_5^2:(C_2^2\times C_4)$ ($C_2^2$), $A_5^2:C_4\times C_2^2$ ($C_2^2$)

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

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

Order 7200: $\PSOPlus(4,5)$ ($C_2^2\wr C_2$), $\PSOPlus(4,5)$ ($C_2^3:C_4$), $C_2\times A_5^2$ ($C_2^2\wr C_2$), $C_2\times A_5^2$ ($C_2^3:C_4$)

Order 3600: $A_5^2$ ($C_2^4:C_4$)

Order 8: $C_2^3$ ($A_5^2.D_4$)

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

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

Order 1: $C_1$ ($(C_2\times S_5^2).D_4$)

Series

Derived series

$(C_2\times S_5^2).D_4$

$\rhd$

$A_5^2:C_2^2$

$\rhd$

$A_5^2$

Chief series

$(C_2\times S_5^2).D_4$

$\rhd$

$C_2^3\times A_5.S_5.C_2$

$\rhd$

$(C_2^3\times A_5).S_5$

$\rhd$

$(C_2^3\times A_5).A_5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2\times S_5^2).D_4$

$\rhd$

$A_5^2:C_2^2$

$\rhd$

$A_5^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $238 \times 238$ 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 $224 \times 224$ rational character table (warning: may be slow to load).