Abstract group 81920.cud: $C_2^5.C_2^8:C_{10}$

• Label: 81920.cud
• Order: \( 2^{14} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{21} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_2^5.C_2^8:C_{10}$
Order:	\(81920\)\(\medspace = 2^{14} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_3^2:C_4\times A_6$, of order \(10485760\)\(\medspace = 2^{21} \cdot 5 \)
Composition factors:	$C_2$ x 14, $C_5$
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	543	15840	16384	49152	81920
Conjugacy classes	1	15	180	4	12	212
Divisions	1	15	163	1	3	183
Autjugacy classes	1	11	102	4	8	126

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
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:	${\langle a, b, c, d, e, f, g, h, i, j \mid a^{10}=e^{4}=g^{4}=h^{4}=i^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(1,28,38,17,16,2,27,37,18,15)(3,25,39,19,14,4,26,40,20,13)(5,30,35,22,9,6,29,36,21,10) \!\cdots\! \rangle$
Transitive group:	40T23676	40T25010	40T29480	40T29621	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:	$C_2^6$ . $(C_2^8:C_5)$	$C_2^5$ . $(C_2^8:C_{10})$	$C_4^4$ . $(C_2.C_2\wr C_5)$	$(C_2^5.C_2^8:C_5)$ . $C_2$	all 16

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

Homology

Abelianization:	$C_{10} \simeq C_{2} \times C_{5}$
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$2$

Subgroups

There are 48 normal subgroups (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4.C_2^4.C_2^4.C_{10}$
Commutator:	$G' \simeq$ $C_2^5.C_2^6.C_2^2$	$G/G' \simeq$ $C_{10}$
Frattini:	$\Phi \simeq$ $C_2^5$	$G/\Phi \simeq$ $C_2^8:C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6.C_2^6.C_2^2$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_2^5.C_2^8:C_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2^8:C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^6.C_2^2$
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 5120) are shown, as well as all normal subgroups of any index.

Order 81920: $C_2^5.C_2^8:C_{10}$

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

Order 16384: $C_2^6.C_2^6.C_2^2$

Order 8192: $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$, $C_2^4.Q_8.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^4.D_4.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^4.Q_8.C_2^6$, $C_2^4.D_4.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^4.D_4.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.Q_8.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$

Order 5120: $C_2^5.C_2\wr C_5$ x 2, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2\times C_4^4:C_{10}$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$

Order 1024: $C_2^6.C_2^4$ x 6, $C_2^6.C_2^4$ x 5, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$, $C_2^6.C_2^4$

Order 512: $C_2^5.C_2^4$ x 6, $C_2^5.C_2^4$ x 4, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_4^4.C_2$ x 2, $C_2^5.C_2^4$ x 2, $C_2\times C_4^4$

Order 256: $C_4^4$

Order 64: $C_2^6$

Order 32: $C_2^5$ x 3

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 5120) are shown, as well as all normal subgroups of any index.

Order 81920: $C_2^5.C_2^8:C_{10}$

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

Order 16384: $C_2^6.C_2^6.C_2^2$

Order 8192: $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2^2$ x 2, $C_2^5.C_2^6.C_2^2$, $C_2^4.Q_8.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^4.D_4.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^4.Q_8.C_2^6$, $C_2^4.D_4.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^4.D_4.C_2^6$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.D_4.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^4.Q_8.C_2^6$, $C_2^6.C_2^6.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^2$

Order 5120: $C_2^5.C_2\wr C_5$ x 2, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$, $C_2\times C_4^4:C_{10}$, $C_2^5.C_2\wr C_5$, $C_2^5.C_2\wr C_5$

Order 1024: $C_2^6.C_2^4$ x 6, $C_2^6.C_2^4$ x 5, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$, $C_2^6.C_2^4$

Order 512: $C_2^5.C_2^4$ x 6, $C_2^5.C_2^4$ x 4, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_4^4.C_2$, $C_2\times C_4^4$

Order 256: $C_4^4$

Order 64: $C_2^6$

Order 32: $C_2^5$ x 2

Order 16: $C_2^4$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 81920: $C_2^5.C_2^8:C_{10}$ ($C_1$)

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

Order 16384: $C_2^6.C_2^6.C_2^2$ ($C_5$)

Order 8192: $C_2^5.C_2^6.C_2^2$ ($C_{10}$)

Order 1024: $C_2^6.C_2^4$ ($C_2^4:C_5$) x 6, $C_2^6.C_2^4$ ($C_2^4:C_5$) x 5, $C_2^6.C_2^4$ ($C_2^4:C_5$) x 2, $C_2^6.C_2^4$ ($C_2^4:C_5$) x 2, $C_2^6.C_2^4$ ($C_2^4:C_5$), $C_2^6.C_2^4$ ($C_2^4:C_5$)

Order 512: $C_2^5.C_2^4$ ($C_2\wr C_5$) x 6, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 4, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 2, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 2, $C_4^4.C_2$ ($C_2.C_2^4:C_5$) x 2, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 2, $C_2\times C_4^4$ ($C_2\wr C_5$)

Order 256: $C_4^4$ ($C_2.C_2\wr C_5$)

Order 64: $C_2^6$ ($C_2^8:C_5$)

Order 32: $C_2^5$ ($C_4.C_2^3.C_2^4.C_5$) x 2, $C_2^5$ ($C_2^8:C_{10}$)

Order 16: $C_2^4$ ($C_4.C_2^3.C_2^4.C_{10}$)

Order 2: $C_2$ ($C_2^4.C_2^4.C_2^4.C_{10}$)

Order 1: $C_1$ ($C_2^5.C_2^8:C_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 81920: $C_2^5.C_2^8:C_{10}$ ($C_1$)

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

Order 16384: $C_2^6.C_2^6.C_2^2$ ($C_5$)

Order 8192: $C_2^5.C_2^6.C_2^2$ ($C_{10}$)

Order 1024: $C_2^6.C_2^4$ ($C_2^4:C_5$) x 6, $C_2^6.C_2^4$ ($C_2^4:C_5$) x 5, $C_2^6.C_2^4$ ($C_2^4:C_5$) x 2, $C_2^6.C_2^4$ ($C_2^4:C_5$) x 2, $C_2^6.C_2^4$ ($C_2^4:C_5$), $C_2^6.C_2^4$ ($C_2^4:C_5$)

Order 512: $C_2^5.C_2^4$ ($C_2\wr C_5$) x 6, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 4, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 2, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 2, $C_2^5.C_2^4$ ($C_2\wr C_5$) x 2, $C_4^4.C_2$ ($C_2.C_2^4:C_5$), $C_2\times C_4^4$ ($C_2\wr C_5$)

Order 256: $C_4^4$ ($C_2.C_2\wr C_5$)

Order 64: $C_2^6$ ($C_2^8:C_5$)

Order 32: $C_2^5$ ($C_4.C_2^3.C_2^4.C_5$), $C_2^5$ ($C_2^8:C_{10}$)

Order 16: $C_2^4$ ($C_4.C_2^3.C_2^4.C_{10}$)

Order 2: $C_2$ ($C_2^4.C_2^4.C_2^4.C_{10}$)

Order 1: $C_1$ ($C_2^5.C_2^8:C_{10}$)

Series

Derived series

$C_2^5.C_2^8:C_{10}$

$\rhd$

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

$\rhd$

$C_2^5$

$\rhd$

$C_1$

Chief series

$C_2^5.C_2^8:C_{10}$

$\rhd$

$C_2^5.C_2^8:C_5$

$\rhd$

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

$\rhd$

$C_2^5.C_2^4$

$\rhd$

$C_2^5$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^5.C_2^8:C_{10}$

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $212 \times 212$ character table is not available for this group.

Rational character table

The $183 \times 183$ rational character table is not available for this group.