Abstract group 10000.em: $D_5^4$

• Label: 10000.em
• Order: \( 2^{4} \cdot 5^{4} \)
• Exponent: \( 2 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \cdot 5^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$D_5^4$
Order:	\(10000\)\(\medspace = 2^{4} \cdot 5^{4} \)
Exponent:	\(10\)\(\medspace = 2 \cdot 5 \)
Automorphism group:	$C_5^4:C_4\wr S_4$, of order \(3840000\)\(\medspace = 2^{11} \cdot 3 \cdot 5^{4} \)
Composition factors:	$C_2$ x 4, $C_5$ x 4
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	5	10
Elements	1	1295	624	8080	10000
Conjugacy classes	1	15	80	160	256
Divisions	1	15	40	80	136
Autjugacy classes	1	4	4	6	15

Dimension	1	2	4	8	16	32
Irr. complex chars.	16	64	96	64	16	0	256
Irr. rational chars.	16	0	32	48	32	8	136

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	$775757640$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	32
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{10}=c^{10}=d^{10}=e^{5}=[a,d]=[b,e]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(11,13,14,19,18,12,16,20,17,15), (1,2,4,5,8,7,10,3,6,9)(11,14,18,16,17) \!\cdots\! \rangle$
Transitive group:	40T5805				more information
Direct product:	$D_5$ ${}^4$
Semidirect product:	$C_5^4$ $\,\rtimes\,$ $C_2^4$	$C_5^2$ $\,\rtimes\,$ $D_{10}^2$	$(C_5:D_5^3)$ $\,\rtimes\,$ $C_2$	$(C_5\times D_5^3)$ $\,\rtimes\,$ $C_2$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_6^2.D_4^2)$	$\Aut(C_2\times C_4^2.\GL(2,\mathbb{Z}/4))$	$\Aut((C_2\times C_4^2).\GL(2,\mathbb{Z}/4))$	$\Aut(C_2^8.A_4)$	all 17

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 134816 subgroups in 2396 conjugacy classes, 170 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_5^4$
Commutator:	$G' \simeq$ $C_5^4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_5^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^4$	$G/\operatorname{Fit} \simeq$ $C_2^4$
Radical:	$R \simeq$ $D_5^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^4$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 10000: $D_5^4$

Order 5000: $D_5\times C_5:D_5\times D_5$ x 6, $C_5\times D_5^3$ x 4, $C_5:D_5^3$ x 4, $C_5:D_5^3$

Order 2500: $C_5\times D_5\times C_5:D_5$ x 12, $C_5^4:C_2^2$ x 6, $C_5^2\times D_5^2$ x 6, $C_5^2:D_5\times D_5$ x 4, $C_5\wr C_2^2$ x 4, $C_5^4:C_2^2$ x 3

Order 2000: $C_2\times D_5^3$ x 4

Order 1250: $C_5^2\wr C_2$ x 6, $C_5^3:C_{10}$ x 4, $D_5\times C_5^3$ x 4, $C_5^4:C_2$

Order 1000: $D_5^3$ x 44, $C_{10}:D_5^2$ x 12, $C_{10}\times D_5^2$ x 12, $C_{10}:D_5^2$ x 4

Order 625: $C_5^4$

Order 500: $C_5:D_5^2$ x 100, $C_5\times D_5^2$ x 84, $C_5:D_5^2$ x 28, $C_5^2:D_{10}$ x 12, $D_{10}\times C_5^2$ x 12, $C_5^2:D_{10}$ x 4

Order 400: $D_{10}^2$ x 6

Order 250: $C_5^2:C_{10}$ x 76, $D_5\times C_5^2$ x 76, $C_5^3:C_2$ x 44, $C_5^2\times C_{10}$ x 4

Order 200: $D_5\times D_{10}$ x 96, $C_{10}\times D_{10}$ x 12, $C_{10}:D_{10}$ x 6

Order 125: $C_5^3$ x 40

Order 100: $D_5^2$ x 220, $C_5\times D_{10}$ x 120, $C_5:D_{10}$ x 76, $C_{10}^2$ x 6

Order 80: $C_2^2\times D_{10}$ x 4

Order 50: $C_5:D_5$ x 204, $C_5\times D_5$ x 200, $C_5\times C_{10}$ x 58

Order 40: $C_2\times D_{10}$ x 68, $C_2^2\times C_{10}$ x 4

Order 25: $C_5^2$ x 146

Order 20: $D_{10}$ x 200, $C_2\times C_{10}$ x 40

Order 16: $C_2^4$

Order 10: $D_5$ x 120, $C_{10}$ x 80

Order 8: $C_2^3$ x 15

Order 5: $C_5$ x 40

Order 4: $C_2^2$ x 35

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 10000: $D_5^4$

Order 5000: $D_5\times C_5:D_5\times D_5$, $C_5\times D_5^3$, $C_5:D_5^3$, $C_5:D_5^3$

Order 2500: $C_5^2:D_5\times D_5$, $C_5^4:C_2^2$, $C_5^4:C_2^2$, $C_5\times D_5\times C_5:D_5$, $C_5^2\times D_5^2$, $C_5\wr C_2^2$

Order 2000: $C_2\times D_5^3$

Order 1250: $C_5^3:C_{10}$, $C_5^4:C_2$, $C_5^2\wr C_2$, $D_5\times C_5^3$

Order 1000: $D_5^3$ x 5, $C_{10}:D_5^2$, $C_{10}:D_5^2$, $C_{10}\times D_5^2$

Order 625: $C_5^4$

Order 500: $C_5:D_5^2$ x 7, $C_5\times D_5^2$ x 5, $C_5:D_5^2$ x 3, $C_5^2:D_{10}$, $C_5^2:D_{10}$, $D_{10}\times C_5^2$

Order 400: $D_{10}^2$

Order 250: $C_5^3:C_2$ x 5, $C_5^2:C_{10}$ x 5, $D_5\times C_5^2$ x 5, $C_5^2\times C_{10}$

Order 200: $D_5\times D_{10}$ x 7, $C_{10}:D_{10}$, $C_{10}\times D_{10}$

Order 125: $C_5^3$ x 4

Order 100: $D_5^2$ x 13, $C_5:D_{10}$ x 6, $C_5\times D_{10}$ x 6, $C_{10}^2$

Order 80: $C_2^2\times D_{10}$

Order 50: $C_5:D_5$ x 11, $C_5\times D_5$ x 10, $C_5\times C_{10}$ x 4

Order 40: $C_2\times D_{10}$ x 7, $C_2^2\times C_{10}$

Order 25: $C_5^2$ x 7

Order 20: $D_{10}$ x 14, $C_2\times C_{10}$ x 4

Order 16: $C_2^4$

Order 10: $D_5$ x 10, $C_{10}$ x 6

Order 8: $C_2^3$ x 4

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 6

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 10000: $D_5^4$ ($C_1$)

Order 5000: $D_5\times C_5:D_5\times D_5$ ($C_2$) x 6, $C_5\times D_5^3$ ($C_2$) x 4, $C_5:D_5^3$ ($C_2$) x 4, $C_5:D_5^3$ ($C_2$)

Order 2500: $C_5\times D_5\times C_5:D_5$ ($C_2^2$) x 12, $C_5^4:C_2^2$ ($C_2^2$) x 6, $C_5^2\times D_5^2$ ($C_2^2$) x 6, $C_5^2:D_5\times D_5$ ($C_2^2$) x 4, $C_5\wr C_2^2$ ($C_2^2$) x 4, $C_5^4:C_2^2$ ($C_2^2$) x 3

Order 1250: $C_5^2\wr C_2$ ($C_2^3$) x 6, $C_5^3:C_{10}$ ($C_2^3$) x 4, $D_5\times C_5^3$ ($C_2^3$) x 4, $C_5^4:C_2$ ($C_2^3$)

Order 1000: $D_5^3$ ($D_5$) x 4

Order 625: $C_5^4$ ($C_2^4$)

Order 500: $C_5:D_5^2$ ($D_{10}$) x 12, $C_5\times D_5^2$ ($D_{10}$) x 12, $C_5:D_5^2$ ($D_{10}$) x 4

Order 250: $C_5^2:C_{10}$ ($C_2\times D_{10}$) x 12, $D_5\times C_5^2$ ($C_2\times D_{10}$) x 12, $C_5^3:C_2$ ($C_2\times D_{10}$) x 4

Order 125: $C_5^3$ ($C_2^2\times D_{10}$) x 4

Order 100: $D_5^2$ ($D_5^2$) x 6

Order 50: $C_5\times D_5$ ($D_5\times D_{10}$) x 12, $C_5:D_5$ ($D_5\times D_{10}$) x 6

Order 25: $C_5^2$ ($D_{10}^2$) x 6

Order 10: $D_5$ ($D_5^3$) x 4

Order 5: $C_5$ ($C_2\times D_5^3$) x 4

Order 1: $C_1$ ($D_5^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 10000: $D_5^4$ ($C_1$)

Order 5000: $D_5\times C_5:D_5\times D_5$ ($C_2$), $C_5\times D_5^3$ ($C_2$), $C_5:D_5^3$ ($C_2$), $C_5:D_5^3$ ($C_2$)

Order 2500: $C_5^2:D_5\times D_5$ ($C_2^2$), $C_5^4:C_2^2$ ($C_2^2$), $C_5^4:C_2^2$ ($C_2^2$), $C_5\times D_5\times C_5:D_5$ ($C_2^2$), $C_5^2\times D_5^2$ ($C_2^2$), $C_5\wr C_2^2$ ($C_2^2$)

Order 1250: $C_5^3:C_{10}$ ($C_2^3$), $C_5^4:C_2$ ($C_2^3$), $C_5^2\wr C_2$ ($C_2^3$), $D_5\times C_5^3$ ($C_2^3$)

Order 1000: $D_5^3$ ($D_5$)

Order 625: $C_5^4$ ($C_2^4$)

Order 500: $C_5:D_5^2$ ($D_{10}$), $C_5:D_5^2$ ($D_{10}$), $C_5\times D_5^2$ ($D_{10}$)

Order 250: $C_5^3:C_2$ ($C_2\times D_{10}$), $C_5^2:C_{10}$ ($C_2\times D_{10}$), $D_5\times C_5^2$ ($C_2\times D_{10}$)

Order 125: $C_5^3$ ($C_2^2\times D_{10}$)

Order 100: $D_5^2$ ($D_5^2$)

Order 50: $C_5:D_5$ ($D_5\times D_{10}$), $C_5\times D_5$ ($D_5\times D_{10}$)

Order 25: $C_5^2$ ($D_{10}^2$)

Order 10: $D_5$ ($D_5^3$)

Order 5: $C_5$ ($C_2\times D_5^3$)

Order 1: $C_1$ ($D_5^4$)

Series

Derived series

$D_5^4$

$\rhd$

$D_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$D_5^4$

$\rhd$

$D_5^4$

$\rhd$

$D_5\times C_5:D_5\times D_5$

$\rhd$

$D_5\times C_5:D_5\times D_5$

$\rhd$

$C_5^2\times D_5^2$

$\rhd$

$C_5^2\times D_5^2$

$\rhd$

$C_5\times D_5^2$

$\rhd$

$C_5\times D_5^2$

$\rhd$

$D_5\times C_5^2$

$\rhd$

$D_5\times C_5^2$

$\rhd$

$C_5\times D_5$

$\rhd$

$C_5\times D_5$

$\rhd$

$C_5^2$

$\rhd$

$C_5^2$

$\rhd$

$C_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$D_5^4$

$\rhd$

$D_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

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

Character theory

Complex character table

See the $256 \times 256$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $136 \times 136$ rational character table (warning: may be slow to load).