Abstract group 230400.co: $(C_2\times S_5^2).C_2^3$

• Label: 230400.co
• Order: \( 2^{10} \cdot 3^{2} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{2} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $16$
• Trans deg.: $80$
• Rank: $4$

Group information

Description:	$(C_2\times S_5^2).C_2^3$
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, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	30	60
Elements	1	5087	440	44960	624	40840	28800	35088	45760	960	13440	10560	3840	230400
Conjugacy classes	1	35	2	36	2	44	4	28	22	1	6	7	2	190
Divisions	1	35	2	36	2	44	4	28	22	1	6	7	2	190
Autjugacy classes	1	17	2	14	2	21	1	12	10	1	3	4	1	89

Dimension	1	2	8	10	12	16	20	24	25	32	36	40	48	50	60	80	96	120
Irr. complex chars.	16	12	16	16	8	20	4	2	16	12	16	16	8	12	8	4	2	2	190
Irr. rational chars.	16	12	16	16	8	20	4	2	16	12	16	16	8	12	8	4	2	2	190

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$80$
Rank:	$4$
Inequivalent generating quadruples:	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 $16$ $\langle(1,7,3,4)(2,5,10,6,8,9)(13,14), (1,7,8,5)(2,4,3,6)(9,10)(11,14)(12,13), (1,7,3,6,10,9,8,4,2,5) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^3$ . $(S_5\wr C_2)$	$(C_2.S_5^2)$ . $C_2^3$ (2)	$(C_2\times S_5^2)$ . $C_2^3$ (2)	$(A_5^2:C_2^3)$ . $D_4$ (8)	all 33
Aut. group:	$\Aut(A_5^2:(C_2\times D_4))$	$\Aut((C_2\times A_5^2).C_2^3)$	$\Aut(A_5^2:\OD_{16})$	$\Aut(A_5^2:(C_2\times C_8))$	all 9

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{8}$
Commutator length:	$1$

Subgroups

There are 113 normal subgroups (19 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2^2$
Commutator:	a subgroup isomorphic to $A_5^2:C_2^2$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 100 (order at least 2304) are shown, as well as all normal subgroups of any index.

Order 230400: $(C_2\times S_5^2).C_2^3$

Order 115200: $C_2^2.A_5^2.D_4$ x 8, $C_2^2.A_5^2.D_4$ x 2, $C_2^2\times S_5\wr C_2$ x 2, $C_2^3\times A_5^2.C_2^2$, $(C_2^3\times A_5).S_5.C_2$, $C_2^3.A_5^2.C_4$

Order 57600: $A_5^2:C_2^4$ x 17, $S_5^2:C_2^2$ x 16, $\POPlus(4,5):C_4$ x 8, $(C_2^2\times A_5).S_5.C_2$ x 6, $(C_2^2\times A_5^2):C_4$ x 4, $(C_2^2\times A_5).S_5.C_2$ x 2, $A_5^2:C_4\times C_2^2$ x 2, $C_2^3.A_5^2.C_2$, $(C_2^3\times A_5).S_5$, $(C_2^3\times A_5).A_5.C_2$, $C_2^2\times S_5^2$

Order 28800: $A_5^2:C_2^3$ x 62, $S_5\wr C_2$ x 16, $A_5^2:C_2^3$ x 14, $A_5^2:C_2^3$ x 10, $C_2\times A_5^2:C_4$ x 8, $C_2\times S_5^2$ x 7, $A_5^2:D_4$ x 4, $A_5^2:D_4$ x 4, $C_2.S_5^2$ x 3, $(C_2^3\times A_5).A_5$, $(C_2^2\times A_5).A_5.C_2$, $A_5^2:C_2^3$

Order 23040: $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$

Order 19200: $(C_2\times C_2^2:F_5).S_5$

Order 14400: $\POPlus(4,5)$ x 36, $A_5^2:C_2^2$ x 28, $A_5^2:C_2^2$ x 18, $S_5^2$ x 6, $C_2^2\times A_5^2$ x 5, $A_5^2:C_2^2$ x 4, $A_5^2:C_4$ x 4, $A_5^2:C_4$ x 2

Order 11520: $\GL(2,\mathbb{Z}/4):S_5$ x 8, $D_4\times D_6\times S_5$, $C_2^2.(S_4\times S_5)$, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times S_4):S_5$, $(C_2^3\times A_4).S_5$, $C_2^3\times A_4:S_5$, $C_2^2\times S_4\times S_5$, $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$

Order 9600: $D_{10}:(C_4\times S_5)$ x 8, $(C_2^3\times D_5).S_5$ x 2, $(C_2^2\times F_5).S_5$ x 2, $D_{10}.C_2^2:S_5$, $D_{10}.(C_2^3\times A_5)$, $C_2^2\times F_5\times S_5$

Order 9216: $C_2^7:\SOPlus(4,2)$

Order 7680: $A_5:(C_2\times D_4^2)$

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

Order 6400: $C_2\times D_{10}^2.D_4$

Order 5760: $S_3\times D_4\times S_5$ x 16, $C_2^2\times A_4:S_5$ x 10, $C_2\times S_4\times S_5$ x 8, $A_4\times C_2^2:S_5$ x 4, $A_4:(C_4\times S_5)$ x 4, $(C_2\times S_4):S_5$ x 4, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $A_5\times \GL(2,\mathbb{Z}/4)$ x 4, $S_3\times C_2^3:S_5$ 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, $C_2^2.S_4\times A_5$, $A_4\times C_2^2.S_5$, $C_2^5:\GL(2,4)$, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $C_6:S_5\times D_4$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $C_2^2\times A_4\times S_5$, $C_2^2\times S_4\times A_5$, $D_4\times D_6\times A_5$

Order 4800: $C_2\times D_{10}.S_5$ x 10, $D_5\times C_2^2:S_5$ x 8, $C_2\times F_5\times S_5$ x 8, $A_5\times D_{10}:C_4$ x 4, $C_{10}.(C_4\times S_5)$ x 4, $(C_{10}\times S_5):C_4$ x 4, $A_5:C_4\times F_5$ x 4, $C_{10}.C_2^2:S_5$ x 2, $(C_2^2\times D_5).S_5$ x 2, $C_2^2\times F_5\times A_5$ x 2, $(C_2^3\times D_5).A_5$, $(C_2^2\times C_{10}).S_5$, $C_2\times D_{10}\times S_5$

Order 4608: $C_2^6:\SOPlus(4,2)$ x 8, $C_2^6.\SOPlus(4,2)$ x 2, $S_4^2:C_2^3$ x 2, $C_2^3:S_4^2$, $C_2^5:D_6^2$, $C_2^3:(A_4^2:C_4)$

Order 3840: $A_5.D_4^2$ x 16, $(C_2^2\times S_5):D_4$ x 2, $C_2^3:(C_4\times S_5)$ x 2, $C_2^2.(D_4\times S_5)$ x 2, $(C_2^2\times D_4):S_5$ x 2, $C_2^5:S_5$ x 2, $C_2^2\times D_4\times S_5$ x 2, $(C_2^2\times D_4):S_5$, $C_2^3.(D_4\times A_5)$, $C_2\times \GL(2,\mathbb{Z}/4):F_5$, $(C_2^2\times D_4).S_5$

Order 3600: $A_5^2$

Order 3200: $D_{10}^2.D_4$ x 8, $F_5^2:C_2^3$ x 4, $D_{10}^2.C_2^3$, $D_{10}^2.C_2^3$, $C_2\times C_{10}^2:\OD_{16}$

Order 2880: $\GL(2,4):C_2^4$ x 39, $C_2\times A_4:S_5$ x 18, $C_3:(D_4\times S_5)$ x 16, $S_3\times C_2^2:S_5$ x 16, $C_3:D_4\times S_5$ x 16, $D_{12}:S_5$ 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, $S_4\times S_5$ x 8, $C_2^4:\GL(2,4)$ x 5, $C_2\times S_4\times A_5$ x 4, $C_2\times A_4\times S_5$ x 4, $\GL(2,4):C_2^4$ x 2, $C_6:D_4\times A_5$ x 2, $A_4:C_4\times A_5$ 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, $A_4\times A_5:C_4$ x 2, $C_2\times D_{12}\times A_5$, $C_4\times D_6\times A_5$, $A_5:C_4\times D_6$, $C_6\times D_4\times A_5$, $C_2\times C_{12}:S_5$, $C_6:(C_4\times S_5)$, $C_6:C_4\times S_5$, $C_2\times C_{12}:S_5$, $C_2\times C_{12}\times S_5$, $C_2\times C_{12}:S_5$

Order 2400: $D_{10}.S_5$ x 18, $C_2\times D_{10}\times A_5$ x 10, $D_{10}\times S_5$ x 8, $C_2\times F_5\times A_5$ x 8, $F_5\times S_5$ x 8, $D_{10}.S_5$ x 4, $D_{10}:S_5$ x 4, $D_{10}:S_5$ x 4, $C_5\times C_2^2:S_5$ x 4, $(C_2\times C_{10}):S_5$ x 4, $C_{10}.A_5:C_4$, $C_5.(C_2^3\times A_5)$, $(C_2\times C_{10}).S_5$, $C_2\times C_{10}:S_5$, $C_2\times C_{10}\times S_5$

Order 2304: $C_2^4:D_6^2$ x 17, $S_4^2:C_2^2$ x 16, $\POPlus(4,3):C_4$ x 8, $\GL(2,\mathbb{Z}/4):S_4$ x 6, $(C_2^2\times A_4^2):C_4$ x 4, $A_4^2:C_4\times C_2^2$ x 2, $D_6^2.C_2^4$, $C_2^2.S_4^2$, $A_4^2:C_2^4$, $A_4^2:C_2^4$, $C_2^4:D_6^2$, $C_2^2.S_4^2$, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$, $C_2^2\times S_4^2$

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

not computed

Normal subgroups (quotient in parentheses)

Order 230400: $(C_2\times S_5^2).C_2^3$ ($C_1$)

Order 115200: $C_2^2.A_5^2.D_4$ ($C_2$) x 8, $C_2^2.A_5^2.D_4$ ($C_2$) x 2, $C_2^2\times S_5\wr C_2$ ($C_2$) x 2, $C_2^3\times A_5^2.C_2^2$ ($C_2$), $(C_2^3\times A_5).S_5.C_2$ ($C_2$), $C_2^3.A_5^2.C_4$ ($C_2$)

Order 57600: $S_5^2:C_2^2$ ($C_2^2$) x 8, $\POPlus(4,5):C_4$ ($C_2^2$) x 8, $A_5^2:C_2^4$ ($C_2^2$) x 6, $(C_2^2\times A_5^2):C_4$ ($C_2^2$) x 4, $(C_2^2\times A_5).S_5.C_2$ ($C_2^2$) x 4, $A_5^2:C_4\times C_2^2$ ($C_2^2$) x 2, $(C_2^3\times A_5).S_5$ ($C_2^2$), $(C_2^2\times A_5).S_5.C_2$ ($C_2^2$), $C_2^2\times S_5^2$ ($C_2^2$)

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

Order 14400: $\POPlus(4,5)$ ($C_2\times D_4$) x 8, $A_5^2:C_2^2$ ($C_2\times D_4$) x 7, $C_2^2\times A_5^2$ ($C_2\times D_4$) x 3, $A_5^2:C_2^2$ ($C_2^4$)

Order 7200: $\PSOPlus(4,5)$ ($C_2^2\times D_4$) x 2, $\PSOPlus(4,5)$ ($C_2^2\wr C_2$) x 2, $C_2\times A_5^2$ ($C_2^2\wr C_2$) x 2, $C_2\times A_5^2$ ($C_2^2\times D_4$)

Order 3600: $A_5^2$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($S_5\wr C_2$)

Order 4: $C_2^2$ ($S_5^2:C_2^2$) x 3

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

Order 1: $C_1$ ($(C_2\times S_5^2).C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $190 \times 190$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.