Abstract group 61440.e: $C_2^8.(C_2\times S_5)$

• Label: 61440.e
• Order: \( 2^{12} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $20$
• Trans deg.: $20$
• Rank: $2$

Group information

Description:	$C_2^8.(C_2\times S_5)$
Order:	\(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^8.(C_2\times S_5)$, of order \(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 10, $A_5$
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	8	10	12
Elements	1	1551	320	15600	6144	8640	11520	6144	11520	61440
Conjugacy classes	1	17	1	29	1	9	4	1	6	69
Divisions	1	17	1	29	1	9	4	1	6	69
Autjugacy classes	1	17	1	29	1	9	4	1	6	69

Dimension	1	4	5	6	10	15	20	30	40	60	80
Irr. complex chars.	4	4	8	2	12	4	13	6	6	9	1	69
Irr. rational chars.	4	4	8	2	12	4	13	6	6	9	1	69

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$20$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	10	10	10
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $20$ $\langle(1,2)(3,6,7,9,4,5,8,10)(11,13,17,19)(12,14,18,20), (1,18,7,13,6,15,3,11,10,19)(2,17,8,14,5,16,4,12,9,20)\rangle$
Transitive group:	20T667	20T675	20T681	20T692	all 26
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^8.S_5)$ . $C_2$	$(C_2^8.S_5)$ . $C_2$	$C_2^8$ . $(C_2\times S_5)$	$(C_2^4\wr C_2)$ . $S_5$	all 8
Aut. group:	$\Aut(C_4.C_2^4)$	$\Aut(C_2^5.C_2^4:A_5)$	$\Aut(C_2^5.C_2^4:A_5)$	$\Aut(C_2^5.C_2^4:A_5)$	all 7

Elements of the group are displayed as permutations of degree 20.

Homology

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

Subgroups

There are 10 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_2^8.A_5$
Frattini:	a subgroup isomorphic to $C_2^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.D_4^2$

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 192 (order at least 320) are shown, as well as all normal subgroups of any index.

Order 61440: $C_2^8.(C_2\times S_5)$

Order 30720: $C_2^8.S_5$, $C_2^8:(C_2\times A_5)$, $C_2^8.S_5$

Order 15360: $C_2^8.A_5$

Order 12288: $C_2^6.(D_4\times S_4)$

Order 10240: $C_2^8.(C_2\times F_5)$

Order 6144: $C_2^5.(D_4\times S_4)$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^7.(C_2\times S_4)$, $C_2^8.S_4$, $C_2^6.(C_4\times S_4)$, $C_2^7.(C_2\times S_4)$, $C_2\wr C_2^3:C_3$

Order 5120: $C_2^8.F_5$ x 2, $C_2^8.D_{10}$

Order 4096: $C_2^6.D_4^2$

Order 3840: $C_2\wr S_5$

Order 3072: $C_2^5:(C_4\times S_4)$ x 2, $C_2^2\wr S_3:D_4$ x 2, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^7:S_4$ x 2, $C_2^7.S_4$ x 2, $C_2^8:D_6$ x 2, $C_2^8.D_6$, $C_2^7:(C_4\times S_3)$, $C_2^7:D_{12}$, $C_2\wr C_6.C_2^3$, $C_2^5:\GL(2,\mathbb{Z}/4)$, $C_2^4.C_2^4:S_3.C_2$, $C_2^7.C_6.C_2^2$, $C_2^7:S_4$, $C_2^3\wr C_2:S_4$, $C_2^7.S_4$, $C_2^6:(C_2\times S_4)$, $C_2^2\wr A_4$, $C_2^6.(C_4\times A_4)$, $C_2^7:(C_2\times A_4)$

Order 2560: $C_2^8:D_5$ x 2, $C_2^8:C_{10}$

Order 2048: $C_2\wr D_4$ x 2, $C_2^7.(C_2\times D_4)$ x 2, $(C_2^2\times D_4^2).D_4$ x 2, $C_2^2\wr D_4$ x 2, $C_2^5.D_4^2$, $C_2^7.(C_2\times D_4)$, $C_2\wr (C_2\times C_4)$, $C_2^7.(C_2\times D_4)$, $C_2\wr C_2^3$, $C_2^7.(C_2\times D_4)$, $C_2^7.(C_2\times D_4)$

Order 1920: $C_2^4:S_5$ x 3, $C_2\wr A_5$

Order 1536: $C_2^7:D_6$ x 5, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:D_{12}$ x 4, $C_2^7.D_6$ x 4, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^7:D_6$ x 3, $C_2^7:D_6$ x 2, $C_2^6.S_4$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:D_{12}$ x 2, $C_2^7.D_6$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr D_6:C_2$ x 2, $C_2\wr D_6:C_2$ x 2, $C_2^5:(C_2\times S_4)$ x 2, $D_4\times C_2^3:S_4$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^7:A_4$ x 2, $C_2^8:C_6$ x 2, $C_2^5:(C_4\times A_4)$ x 2, $C_2^6:S_4$, $C_2^7:D_6$, $C_2^6:S_4$, $C_2^6:S_4$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2^5:(C_2\times S_4)$, $C_2\wr C_2^2\times S_4$, $C_2\wr D_6:C_2$, $C_2\wr D_6:C_2$, $C_2^5:\GL(2,3)$, $C_2^8:C_6$, $(D_4\times C_2^5):C_6$, $C_2^6:(C_2\times A_4)$, $C_2^7:C_{12}$

Order 1280: $C_2^8:C_5$

Order 1024: $C_2^7.C_2^3$ x 11, $C_2^7.D_4$ x 4, $C_2^7.D_4$ x 4, $C_2^7.D_4$ x 4, $C_2^2\wr C_2^2$ x 4, $C_2^7.D_4$ x 4, $C_2^7:D_4$ x 4, $C_2^6.C_2^4$ x 3, $C_2^7.C_2^3$ x 3, $C_2^6.C_2^4$ x 2, $C_2^7:D_4$ x 2, $C_2^2\wr C_4$ x 2, $C_2^4.D_4^2$ x 2, $C_2^6.C_4^2$ x 2, $C_2^7.D_4$ x 2, $C_2^5:C_4\wr C_2$ x 2, $C_2^3\wr C_2:D_4$ x 2, $C_2^4.D_4^2$ x 2, $C_2^3\wr C_2:D_4$ x 2, $C_2^6.(C_2\times D_4)$ x 2, $C_2^7:D_4$ x 2, $C_2^7.(C_2\times C_4)$, $C_2^6.(C_2\times D_4)$, $C_2^6.(C_2\times D_4)$, $C_2^6.(C_2\times D_4)$, $(C_2\times D_4^2):D_4$, $C_2^7:(C_2\times C_4)$

Order 960: $C_2^4:A_5$ x 2

Order 768: $C_2\wr D_6$ x 26, $C_2^5:S_4$ x 13, $C_2^6:D_6$ x 10, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 9, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^6.D_6$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^5:D_{12}$ x 5, $C_2^5:S_4$ x 5, $C_2^6.D_6$ x 5, $C_2^7:C_6$ x 5, $C_2^6:D_6$ x 4, $C_2^6.D_6$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:C_{12}$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5:S_4$ x 3, $C_2^5:S_4$ x 3, $C_2^5:D_{12}$ x 3, $(C_2^2\times D_4):S_4$ x 3, $(C_2^2\times Q_8):S_4$ x 3, $C_2^3:C_4\times S_4$ x 3, $C_2^5:D_{12}$ x 2, $C_2^6.D_6$ x 2, $C_2^6.D_6$ x 2, $(C_2^2\times D_4):S_4$ x 2, $C_2^5:S_4$ x 2, $C_2^4:(C_2\times S_4)$ x 2, $(C_2^3\times C_4):S_4$ x 2, $(C_2^3\times C_4):S_4$ x 2, $(C_2^3\times C_4):S_4$ x 2, $C_4\times C_2^3:S_4$ x 2, $C_2^4:\GL(2,3)$ x 2, $C_2^7:C_6$ x 2, $C_2^6:C_{12}$ x 2, $C_2\wr C_6:C_2$ x 2, $D_4\times C_2^3:A_4$ x 2, $C_2^4:(C_2\times S_4)$, $C_2^4.(C_2\times S_4)$, $C_2^6.D_6$, $\GL(2,\mathbb{Z}/4):C_2^3$, $(C_2^2\times D_4):S_4$, $(C_2^3\times C_4):S_4$, $(C_2^2\times D_4):S_4$, $(C_2^3\times C_4):S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times Q_8):S_4$, $(C_2^2\times D_4):S_4$, $C_2^4.\GL(2,3)$, $C_2^8.C_3$, $C_2^6:A_4$, $(D_4\times C_2^3):A_4$, $A_4\times C_2\wr C_2^2$, $C_2\wr C_6:C_2$, $C_2^5:\SL(2,3)$

Order 640: $C_2\wr F_5$

Order 512: $C_2^6.D_4$ x 17, $C_2^6:D_4$ x 12, $C_2^7:C_2^2$ x 9, $C_2^4.C_2^5$ x 8, $C_2^6:D_4$ x 7, $C_2^4.C_2^5$ x 6, $C_2^6.D_4$ x 6, $C_2^6.D_4$ x 6, $C_2^6.D_4$ x 6, $D_4^2:D_4$ x 6, $C_2^5.C_2^4$ x 6, $C_2^7.C_2^2$ x 5, $C_2^4.C_2^5$ x 4, $C_2^3:D_4^2$ x 4, $C_2^6:C_2^3$ x 4, $C_2^4\wr C_2$ x 4, $C_2^5.(C_2\times D_4)$ x 4, $C_2^7.C_2^2$ x 4, $C_2^4.C_2^5$ x 4, $C_2^6.D_4$ x 4, $C_2^6:D_4$ x 4, $C_2^6:(C_2\times C_4)$ x 4, $C_2^4.(C_4\times D_4)$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $D_4^2:D_4$ x 4, $D_4^2:D_4$ x 4, $C_2^7.C_4$ x 4, $C_2^7:C_4$ x 4, $C_2^6.D_4$ x 4, $C_2^7.C_2^2$ x 4, $C_2^6:D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.C_2^3$ x 4, $C_2^6.C_2^3$ x 4, $C_2^4.C_2^5$ x 3, $C_2^4.C_2^5$ x 3, $C_2^6:D_4$ x 3, $C_2^6.C_2^3$ x 3, $C_2^5.C_2^4$ x 3, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5:C_2^4$ x 2, $C_2^3:D_4^2$ x 2, $D_4^2:C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5:\SD_{16}$ x 2, $C_2^5:D_8$ x 2, $C_2^5:\OD_{16}$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6:D_4$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^6.D_4$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^3\wr C_2:C_4$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^4.C_2^5$ x 2, $D_4^2:D_4$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $(C_2\times D_4^2):C_4$ x 2, $C_2^6:(C_2\times C_4)$ x 2, $C_2^7:C_4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^4.C_2^5$, $(C_2^2\times C_4^2):C_2^3$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $(C_2^2\times C_4^2):Q_8$, $C_2^6:Q_8$, $(C_2^2\times C_4^2):D_4$, $C_2^6.(C_2\times C_4)$, $C_2^5.(C_2\times D_4)$, $C_2^6.C_2^3$, $D_4^2.C_2^3$, $C_2^6:D_4$

Order 384: $C_2^2\wr S_3$ x 55, $C_2\wr S_3$ x 18, $C_2^4:S_4$ x 12, $C_2^5.D_6$ x 12, $\GL(2,\mathbb{Z}/4):C_2^2$ x 12, $C_2^4:S_4$ x 12, $C_2\wr S_4$ x 11, $C_2\wr C_6$ x 10, $C_2^4:D_{12}$ x 10, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 9, $C_2^4:D_{12}$ x 9, $C_2^5.D_6$ x 9, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^5:A_4$ x 7, $C_2^4.S_4$ x 7, $\GL(2,\mathbb{Z}/4):C_2^2$ x 7, $C_2^5.D_6$ x 6, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^5:C_{12}$ x 5, $C_2^4:D_{12}$ x 5, $C_2^6:C_6$ x 5, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:C_6$ x 4, $C_2^5:A_4$ x 3, $C_2^5:C_{12}$ x 3, $C_2^4.S_4$ x 3, $C_2^4:S_4$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $C_2^5:A_4$ x 3, $C_2^3:\GL(2,3)$ x 3, $C_2^4:\SL(2,3)$ x 2, $C_2^4:(C_2\times A_4)$ x 2, $C_4\times C_2^3:A_4$ x 2, $C_2^4.S_4$ x 2, $(C_2^2\times C_4):S_4$ x 2, $C_2^4.S_4$ x 2, $C_2^4:S_4$ x 2, $C_2^5.D_6$ x 2, $C_2^4\times S_4$ x 2, $C_2^5:C_{12}$ x 2, $C_2^4.S_4$ x 2, $C_2^4:S_4$ x 2, $C_2^4:\SL(2,3)$, $(C_2^2\times D_4):A_4$, $(C_2^3\times C_4):A_4$, $(C_2^2\times Q_8):A_4$, $(C_2^2\times D_4):A_4$, $(C_2^2\times C_4).S_4$, $(C_2^2\times C_4):S_4$, $C_2^4.S_4$, $C_2^4:(C_2\times A_4)$, $A_4\times D_4:C_2^2$, $Q_8.(C_2\times S_4)$, $(C_2^2\times C_4):S_4$, $Q_8.(C_2\times S_4)$, $(C_2^2\times C_4):S_4$, $C_2^5.D_6$, $C_2^4:S_4$, $C_2\wr C_2^2\times S_3$

Order 320: $C_2^4:F_5$ x 4, $C_2\wr D_5$

Order 256: $C_2^8$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 61440: $C_2^8.(C_2\times S_5)$ ($C_1$)

Order 30720: $C_2^8.S_5$ ($C_2$), $C_2^8:(C_2\times A_5)$ ($C_2$), $C_2^8.S_5$ ($C_2$)

Order 15360: $C_2^8.A_5$ ($C_2^2$)

Order 512: $C_2^4\wr C_2$ ($S_5$)

Order 256: $C_2^8$ ($C_2\times S_5$)

Order 32: $C_2^5$ ($C_2^4:S_5$)

Order 16: $C_2^4$ ($C_2\wr S_5$)

Order 1: $C_1$ ($C_2^8.(C_2\times S_5)$)

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 9 larger groups in the database.

This group is a maximal quotient of 15 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 $69 \times 69$ rational character table. Alternatively, you may search for characters of this group with desired properties.