Abstract group 40960.yn: $C_2^5.C_2^8:C_5$

• Label: 40960.yn
• Order: \( 2^{13} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{20} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $40$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$C_2^5.C_2^8:C_5$
Order:	\(40960\)\(\medspace = 2^{13} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_{2316}:C_{16}$, of order \(5242880\)\(\medspace = 2^{20} \cdot 5 \)
Composition factors:	$C_2$ x 13, $C_5$
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	4	5	10
Elements	1	671	7520	16384	16384	40960
Conjugacy classes	1	12	67	4	4	88
Divisions	1	12	63	1	1	78
Autjugacy classes	1	11	47	4	4	67

Minimal presentations

Permutation degree:	$40$
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	20	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid f^{4}=g^{4}=h^{2}=i^{4}=j^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(1,28,40,20,15,2,27,39,19,16)(3,26,37,18,13,4,25,38,17,14)(5,30,34,24,10,6,29,33,23,9) \!\cdots\! \rangle$
Transitive group:	40T17177	40T17193	40T17357	40T17534	all 6
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:	$C_2^5$ . $(C_2^8:C_5)$	$C_2^4$ . $(C_2^5.C_2^4.C_5)$	$(C_2^6.C_2^3)$ . $(C_2^4:C_5)$ (2)	$(C_2^5.C_2^4)$ . $(C_2^4:C_5)$ (6)	all 11

Elements of the group are displayed as permutations of degree 40.

Homology

Abelianization:	$C_{5} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 23 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4.C_2^4.C_2^4.C_5$
Commutator:	$G' \simeq$ $C_2^5.C_2^6.C_2^2$	$G/G' \simeq$ $C_5$
Frattini:	$\Phi \simeq$ $C_2^5$	$G/\Phi \simeq$ $C_2^8:C_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5.C_2^6.C_2^2$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_2^5.C_2^8:C_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2^8:C_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^6.C_2^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 16 (order at least 2560) are shown, as well as all normal subgroups of any index.

Order 40960: $C_2^5.C_2^8:C_5$

Order 8192: $C_2^5.C_2^6.C_2^2$

Order 4096: $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$

Order 2560: $C_2^5.C_2^4:C_5$ x 6, $C_2^4.C_2\wr C_5$ x 2, $C_2^5.C_2^4:C_5$ x 2, $C_2^5.C_2^4:C_5$ x 2, $C_2^4.C_2\wr C_5$ x 2, $C_2^4.C_2\wr C_5$, $C_2.C_4^4:C_5$, $C_2^5.C_2^4:C_5$

Order 512: $C_2^5.C_2^4$ x 6, $C_2\times C_2^4.C_2^4$ x 2, $C_2\times C_2^5.C_2^3$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2\times C_2^4.C_2^4$, $C_2^5.C_2^4$, $C_2^5.C_2^4$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 16 (order at least 2560) are shown, as well as all normal subgroups of any index.

Order 40960: $C_2^5.C_2^8:C_5$

Order 8192: $C_2^5.C_2^6.C_2^2$

Order 4096: $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$

Order 2560: $C_2^5.C_2^4:C_5$ x 6, $C_2^4.C_2\wr C_5$ x 2, $C_2^5.C_2^4:C_5$ x 2, $C_2^5.C_2^4:C_5$ x 2, $C_2^4.C_2\wr C_5$ x 2, $C_2^4.C_2\wr C_5$, $C_2.C_4^4:C_5$, $C_2^5.C_2^4:C_5$

Order 512: $C_2^5.C_2^4$ x 6, $C_2\times C_2^4.C_2^4$ x 2, $C_2\times C_2^5.C_2^3$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2\times C_2^4.C_2^4$, $C_2^5.C_2^4$, $C_2^5.C_2^4$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 40960: $C_2^5.C_2^8:C_5$ ($C_1$)

Order 8192: $C_2^5.C_2^6.C_2^2$ ($C_5$)

Order 512: $C_2^5.C_2^4$ ($C_2^4:C_5$) x 6, $C_2\times C_2^4.C_2^4$ ($C_2^4:C_5$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2^4:C_5$) x 2, $C_2^5.C_2^4$ ($C_2^4:C_5$) x 2, $C_2^6.C_2^3$ ($C_2^4:C_5$) x 2, $C_2\times C_2^4.C_2^4$ ($C_2^4:C_5$), $C_2^5.C_2^4$ ($C_2^4:C_5$), $C_2^5.C_2^4$ ($C_2^4:C_5$)

Order 32: $C_2^5$ ($C_2^8:C_5$)

Order 16: $C_2^4$ ($C_2^5.C_2^4.C_5$)

Order 2: $C_2$ ($C_2^4.C_2^4.C_2^4.C_5$)

Order 1: $C_1$ ($C_2^5.C_2^8:C_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 40960: $C_2^5.C_2^8:C_5$ ($C_1$)

Order 8192: $C_2^5.C_2^6.C_2^2$ ($C_5$)

Order 512: $C_2^5.C_2^4$ ($C_2^4:C_5$) x 6, $C_2\times C_2^4.C_2^4$ ($C_2^4:C_5$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2^4:C_5$) x 2, $C_2^5.C_2^4$ ($C_2^4:C_5$) x 2, $C_2^6.C_2^3$ ($C_2^4:C_5$) x 2, $C_2\times C_2^4.C_2^4$ ($C_2^4:C_5$), $C_2^5.C_2^4$ ($C_2^4:C_5$), $C_2^5.C_2^4$ ($C_2^4:C_5$)

Order 32: $C_2^5$ ($C_2^8:C_5$)

Order 16: $C_2^4$ ($C_2^5.C_2^4.C_5$)

Order 2: $C_2$ ($C_2^4.C_2^4.C_2^4.C_5$)

Order 1: $C_1$ ($C_2^5.C_2^8:C_5$)

Series

Derived series

$C_2^5.C_2^8:C_5$

$\rhd$

$C_2^5.C_2^6.C_2^2$

$\rhd$

$C_2^5$

$\rhd$

$C_1$

Chief series

$C_2^5.C_2^8:C_5$

$\rhd$

$C_2^5.C_2^6.C_2^2$

$\rhd$

$C_2^6.C_2^3$

$\rhd$

$C_2^5$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^5.C_2^8:C_5$

$\rhd$

$C_2^5.C_2^6.C_2^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

The $88 \times 88$ character table is not available for this group.

Rational character table

The $78 \times 78$ rational character table is not available for this group.