Abstract group 7680.ez: $C_2^3.(D_4\times S_5)$

• Label: 7680.ez
• Order: \( 2^{9} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 8
• $\card{\Aut(G)}$: \( 2^{18} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $5$

Group information

Description:	$C_2^3.(D_4\times S_5)$
Order:	\(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^{10}.A_4.C_2^3.S_5$, of order \(11796480\)\(\medspace = 2^{18} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 7, $A_5$
Derived length:	$2$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	10	12	20
Elements	1	1039	20	2544	24	1580	936	960	576	7680
Conjugacy classes	1	65	1	46	1	43	21	12	6	196
Divisions	1	65	1	46	1	43	21	12	6	196
Autjugacy classes	1	12	1	7	1	7	4	2	1	36

Dimension	1	2	4	5	6	8	10	12
Irr. complex chars.	32	24	32	32	16	24	24	12	196
Irr. rational chars.	32	24	32	32	16	24	24	12	196

Minimal presentations

Permutation degree:	$15$
Transitive degree:	not computed
Rank:	$5$
Inequivalent generating 5-tuples:	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 $15$ $\langle(6,13)(7,12)(8,9)(10,15)(11,14), (1,2), (6,10)(7,11), (6,14,7,15)(8,9)(10,12,11,13) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_2^3\times S_5)$ . $D_4$ (24)	$S_5$ . $(C_2^3:D_4)$ (8)	$(C_2^3:D_4)$ . $S_5$	$C_2^3$ . $(D_4\times S_5)$ (12)	all 65
Aut. group:	$\Aut(C_2\times C_8\times S_5)$	$\Aut(C_2\times C_8:S_5)$	$\Aut(C_2^2:\GL(2,5))$	$\Aut(C_4:\GL(2,5))$	all 9

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 661 normal subgroups (18 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 $C_2^3$
Commutator:	a subgroup isomorphic to $C_2^2\times A_5$
Frattini:	a subgroup isomorphic to $C_2^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4^2$

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 15 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 7680: $C_2^3.(D_4\times S_5)$

Order 3840: $S_5\times C_2^2\wr C_2$ x 16, $C_2^2.(D_4\times S_5)$ x 3, $C_2^3:C_4\times S_5$ x 3, $C_2^2\times D_4\times S_5$ x 3, $(C_2^2\times D_4).S_5$ x 3, $C_2^5\times S_5$, $C_2^5.S_5$, $C_2^3.(D_4\times A_5)$

Order 1920: $C_2\times D_4\times S_5$ x 48, $C_2^4\times S_5$ x 39, $C_2.(D_4\times S_5)$ x 24, $(C_2\times D_4).S_5$ x 24, $C_2^2:C_4\times S_5$ x 24, $A_5\times C_2^2\wr C_2$ x 8, $C_2^4.S_5$ x 8, $C_2^4:S_5$ x 6, $(C_2^2\times C_4):S_5$ x 6, $C_2^2\times D_4\times A_5$ x 3, $C_2^2\times C_4:S_5$ x 3, $C_4\times C_2^2\times S_5$ x 3, $C_2^3:C_4\times A_5$ x 3, $C_2^4.S_5$ x 3, $C_2^5\times A_5$

Order 1536: $C_2^7:D_6$

Order 1280: $C_2^3:D_4\times F_5$

Order 960: $C_2^3\times S_5$ x 353, $D_4\times S_5$ x 96, $C_2^3:S_5$ x 48, $(C_2\times C_4):S_5$ x 24, $C_2\times D_4\times A_5$ x 24, $C_2\times C_4:S_5$ x 24, $C_2\times C_4\times S_5$ x 24, $C_2^4\times A_5$ x 20, $C_2^2:C_4\times A_5$ x 12, $C_2^3.S_5$ x 12, $C_4\times C_2^2\times A_5$ x 3, $C_2^3.S_5$ x 3

Order 768: $C_2^6:D_6$ x 16, $\GL(2,\mathbb{Z}/4):C_2^3$ x 3, $C_2^5:D_{12}$ x 3, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^6.D_6$ x 3, $S_4\times C_2^5$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^7:C_6$, $C_2^6:D_6$

Order 640: $C_2^2\wr C_2\times F_5$ x 16, $D_{10}.C_2^5$ x 3, $(D_4\times D_{10}):C_4$ x 3, $D_{10}.(C_4\times D_4)$ x 3, $C_2\times D_{10}:C_4^2$ x 3, $F_5\times C_2^5$, $C_2^5:D_{10}$, $C_2^5:F_5$

Order 512: $C_2^3:D_4^2$

Order 480: $C_2^2\times S_5$ x 82, $C_2^3\times A_5$ x 21, $C_2\times C_4\times A_5$ x 6, $C_2^2.S_5$ x 6

Order 240: $C_2\times S_5$ x 40, $C_2^2\times A_5$ x 19

Order 120: $S_5$ x 8, $C_2\times A_5$ x 7

Order 64: $C_2^3:D_4$

Order 60: $A_5$

Order 32: $C_2^2\wr C_2$ x 8, $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^5$

Order 16: $C_2^2:C_4$ x 12, $C_2\times D_4$ x 12, $C_2^4$ x 8, $C_2^2\times C_4$ x 3

Order 8: $C_2^3$ x 21, $C_2\times C_4$ x 6

Order 4: $C_2^2$ x 19

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 7680: $C_2^3.(D_4\times S_5)$ ($C_1$)

Order 3840: $S_5\times C_2^2\wr C_2$ ($C_2$) x 16, $C_2^2.(D_4\times S_5)$ ($C_2$) x 3, $C_2^3:C_4\times S_5$ ($C_2$) x 3, $C_2^2\times D_4\times S_5$ ($C_2$) x 3, $(C_2^2\times D_4).S_5$ ($C_2$) x 3, $C_2^5\times S_5$ ($C_2$), $C_2^5.S_5$ ($C_2$), $C_2^3.(D_4\times A_5)$ ($C_2$)

Order 1920: $C_2\times D_4\times S_5$ ($C_2^2$) x 24, $C_2.(D_4\times S_5)$ ($C_2^2$) x 24, $(C_2\times D_4).S_5$ ($C_2^2$) x 24, $C_2^2:C_4\times S_5$ ($C_2^2$) x 24, $C_2^4\times S_5$ ($C_2^2$) x 15, $A_5\times C_2^2\wr C_2$ ($C_2^2$) x 8, $C_2^4.S_5$ ($C_2^2$) x 8, $C_2^4:S_5$ ($C_2^2$) x 6, $(C_2^2\times C_4):S_5$ ($C_2^2$) x 6, $C_2^2\times D_4\times A_5$ ($C_2^2$) x 3, $C_2^2\times C_4:S_5$ ($C_2^2$) x 3, $C_4\times C_2^2\times S_5$ ($C_2^2$) x 3, $C_2^3:C_4\times A_5$ ($C_2^2$) x 3, $C_2^4.S_5$ ($C_2^2$) x 3, $C_2^5\times A_5$ ($C_2^2$)

Order 960: $C_2^3\times S_5$ ($C_2^3$) x 33, $(C_2\times C_4):S_5$ ($C_2^3$) x 24, $C_2^3\times S_5$ ($D_4$) x 24, $C_2^3:S_5$ ($C_2^3$) x 24, $C_2^2:C_4\times A_5$ ($C_2^3$) x 12, $C_2^3.S_5$ ($C_2^3$) x 12, $C_2\times D_4\times A_5$ ($C_2^3$) x 12, $C_2\times C_4:S_5$ ($C_2^3$) x 12, $C_2\times C_4\times S_5$ ($C_2^3$) x 12, $C_2^4\times A_5$ ($C_2^3$) x 8, $C_4\times C_2^2\times A_5$ ($C_2^3$) x 3, $C_2^3.S_5$ ($C_2^3$) x 3

Order 480: $C_2^2\times S_5$ ($C_2\times D_4$) x 72, $C_2^3\times A_5$ ($C_2\times D_4$) x 12, $C_2^2\times S_5$ ($C_2^4$) x 10, $C_2^3\times A_5$ ($C_2^4$) x 9, $C_2\times C_4\times A_5$ ($C_2^4$) x 6, $C_2^2.S_5$ ($C_2^4$) x 6

Order 240: $C_2\times S_5$ ($C_2^2\times D_4$) x 24, $C_2^2\times A_5$ ($C_2^2\times D_4$) x 18, $C_2\times S_5$ ($C_2^2\wr C_2$) x 16, $C_2^2\times A_5$ ($C_2^5$)

Order 120: $S_5$ ($C_2^3:D_4$) x 8, $C_2\times A_5$ ($C_2^3:D_4$) x 4, $C_2\times A_5$ ($D_4\times C_2^3$) x 3

Order 64: $C_2^3:D_4$ ($S_5$)

Order 60: $A_5$ ($C_2^4:D_4$)

Order 32: $C_2^2\wr C_2$ ($C_2\times S_5$) x 8, $C_2^2\times D_4$ ($C_2\times S_5$) x 3, $C_2^3:C_4$ ($C_2\times S_5$) x 3, $C_2^5$ ($C_2\times S_5$)

Order 16: $C_2^2:C_4$ ($C_2^2\times S_5$) x 12, $C_2\times D_4$ ($C_2^2\times S_5$) x 12, $C_2^4$ ($C_2^2\times S_5$) x 8, $C_2^2\times C_4$ ($C_2^2\times S_5$) x 3

Order 8: $C_2^3$ ($D_4\times S_5$) x 12, $C_2^3$ ($C_2^3\times S_5$) x 9, $C_2\times C_4$ ($C_2^3\times S_5$) x 6

Order 4: $C_2^2$ ($C_2\times D_4\times S_5$) x 18, $C_2^2$ ($C_2^4\times S_5$)

Order 2: $C_2$ ($S_5\times C_2^2\wr C_2$) x 4, $C_2$ ($C_2^2\times D_4\times S_5$) x 3

Order 1: $C_1$ ($C_2^3.(D_4\times S_5)$)

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 7 larger groups in the database.

This group is a maximal quotient of 2 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 $196 \times 196$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.