Abstract group 20000.cv: $C_5^4:(C_4^2:C_2)$

• Label: 20000.cv
• Order: \( 2^{5} \cdot 5^{4} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $24$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$C_5^4:(C_4^2:C_2)$
Order:	\(20000\)\(\medspace = 2^{5} \cdot 5^{4} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_5^4.C_4.C_2^5.C_2^4$, of order \(1280000\)\(\medspace = 2^{11} \cdot 5^{4} \)
Composition factors:	$C_2$ x 5, $C_5$ x 4
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	20
Elements	1	1351	10200	624	3024	4800	20000
Conjugacy classes	1	5	14	48	60	24	152
Divisions	1	5	8	30	38	8	90
Autjugacy classes	1	4	4	8	10	2	29

Dimension	1	2	4	8	16	32
Irr. complex chars.	16	4	0	72	60	0	152
Irr. rational chars.	8	4	2	16	32	28	90

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$40$
Rank:	$3$
Inequivalent generating triples:	$2044224$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{4}=c^{5}=d^{10}=e^{5}=f^{5}=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,4,2,6,7)(5,9)(8,10)(15,18,20,23,16,19,17,22,21,24), (1,2,4,7)(3,5,8,10) \!\cdots\! \rangle$
Transitive group:	40T10773	40T10789	40T10880	40T10882	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_5^4.C_4^2)$ $\,\rtimes\,$ $C_2$	$C_5^4$ $\,\rtimes\,$ $(C_4^2:C_2)$	$(C_5^4:(C_2\times C_4))$ $\,\rtimes\,$ $C_4$	$((C_5^3\times C_{10}).Q_8)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(C_5^4:D_4:C_2)$	$(C_5^4:C_2^3)$ . $C_2^2$ (5)	$(C_5^4:C_2^2)$ . $C_2^3$	$(C_5^4:C_2)$ . $(D_4:C_2)$	all 10
Aut. group:	$\Aut(C_{12}.(S_3\times D_{12}))$	$\Aut(D_4^2:S_3^2)$	$\Aut(C_2\wr S_4\times D_6)$	$\Aut(C_3\times C_4^2:\GL(2,\mathbb{Z}/4))$	all 23

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2} \times C_{20}$
Commutator length:	$1$

Subgroups

There are 202324 subgroups in 1722 conjugacy classes, 36 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_5^4:D_4:C_2$
Commutator:	$G' \simeq$ $C_5^4:C_2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_5^4:D_4:C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^3\times C_{10}$	$G/\operatorname{Fit} \simeq$ $D_4:C_2$
Radical:	$R \simeq$ $C_5^4:(C_4^2:C_2)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^3\times C_{10}$	$G/\operatorname{soc} \simeq$ $D_4:C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2$
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 20000: $C_5^4:(C_4^2:C_2)$

Order 10000: $(C_5^3\times C_{10}).D_4$ x 2, $C_5^4.C_4^2$ x 2, $(C_5^3\times C_{10}).D_4$, $C_5^4:(C_2^2\times C_4)$, $(C_5^3\times C_{10}).Q_8$

Order 5000: $C_5^4:(C_2\times C_4)$ x 4, $C_5^4:(C_2\times C_4)$ x 4, $C_5^4:(C_2\times C_4)$ x 2, $C_5^4:C_2^3$

Order 2500: $C_5^4:C_4$ x 6, $C_5^4:C_2^2$ x 3, $C_5^3:C_{20}$ x 2, $C_5^4:C_2^2$, $C_5^2:C_{10}^2$

Order 2000: $C_5^3:C_4^2$ x 4, $(C_5\times D_{10}):F_5$ x 2

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

Order 1000: $C_2\times C_5^3:C_4$ x 18, $C_5^3:(C_2\times C_4)$ x 8, $C_{10}.D_5^2$ x 8, $C_{10}:D_5^2$ x 4

Order 800: $D_5^2.(C_2\times C_4)$ x 2

Order 625: $C_5^4$

Order 500: $C_5^3:C_4$ x 36, $C_5^2:D_{10}$ x 30, $C_5:D_5^2$ x 16, $C_5^2:C_{20}$ x 8, $C_5^2:C_{20}$ x 8, $C_5^2:D_{10}$ x 4, $D_{10}\times C_5^2$ x 4

Order 400: $D_5^2:C_4$ x 8, $C_5^2:C_4^2$ x 6, $(C_5\times C_{10}).Q_8$ x 4, $(C_5\times C_{10}).D_4$ x 4, $D_{10}:F_5$ x 3, $C_{20}:F_5$ x 2, $C_{10}^2:C_4$

Order 250: $C_5^3:C_2$ x 60, $C_5^2\times C_{10}$ x 30, $C_5^2:C_{10}$ x 8, $D_5\times C_5^2$ x 8

Order 200: $C_{10}:F_5$ x 55, $C_{10}.D_{10}$ x 26, $D_5\times D_{10}$ x 13, $D_5:F_5$ x 12, $C_{10}:F_5$ x 5, $C_{20}:D_5$ x 2, $C_{10}:D_{10}$

Order 125: $C_5^3$ x 30

Order 100: $C_5:D_{10}$ x 139, $C_5:F_5$ x 90, $D_5^2$ x 46, $C_5:C_{20}$ x 40, $C_5\times D_{10}$ x 20, $C_5:F_5$ x 10, $C_5\times C_{20}$ x 2, $C_5^2:C_4$ x 2, $C_{10}^2$

Order 80: $C_4\times F_5$ x 4, $C_2^2\times F_5$ x 2

Order 50: $C_5:D_5$ x 270, $C_5\times C_{10}$ x 136, $C_5\times D_5$ x 40

Order 40: $C_2\times F_5$ x 26, $C_4\times D_5$ x 8, $C_2\times D_{10}$ x 4

Order 32: $C_4^2:C_2$

Order 25: $C_5^2$ x 134

Order 20: $D_{10}$ x 50, $F_5$ x 36, $C_{20}$ x 8, $C_5:C_4$ x 8, $C_2\times C_{10}$ x 4

Order 16: $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_4^2$ x 2, $C_2^2\times C_4$

Order 10: $D_5$ x 68, $C_{10}$ x 38

Order 8: $C_2\times C_4$ x 10, $C_2^3$

Order 5: $C_5$ x 30

Order 4: $C_4$ x 8, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 20000: $C_5^4:(C_4^2:C_2)$

Order 10000: $(C_5^3\times C_{10}).D_4$, $(C_5^3\times C_{10}).D_4$, $C_5^4.C_4^2$, $C_5^4:(C_2^2\times C_4)$, $(C_5^3\times C_{10}).Q_8$

Order 5000: $C_5^4:(C_2\times C_4)$ x 3, $C_5^4:C_2^3$, $C_5^4:(C_2\times C_4)$, $C_5^4:(C_2\times C_4)$

Order 2500: $C_5^4:C_4$ x 3, $C_5^4:C_2^2$ x 2, $C_5^4:C_2^2$, $C_5^2:C_{10}^2$, $C_5^3:C_{20}$

Order 2000: $(C_5\times D_{10}):F_5$, $C_5^3:C_4^2$

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

Order 1000: $C_2\times C_5^3:C_4$ x 6, $C_{10}:D_5^2$ x 2, $C_{10}.D_5^2$ x 2, $C_5^3:(C_2\times C_4)$

Order 800: $D_5^2.(C_2\times C_4)$

Order 625: $C_5^4$

Order 500: $C_5^2:D_{10}$ x 8, $C_5^3:C_4$ x 6, $C_5:D_5^2$ x 4, $C_5^2:D_{10}$ x 2, $D_{10}\times C_5^2$ x 2, $C_5^2:C_{20}$ x 2, $C_5^2:C_{20}$ x 2

Order 400: $D_{10}:F_5$ x 2, $(C_5\times C_{10}).Q_8$ x 2, $(C_5\times C_{10}).D_4$ x 2, $D_5^2:C_4$ x 2, $C_5^2:C_4^2$ x 2, $C_{10}^2:C_4$, $C_{20}:F_5$

Order 250: $C_5^3:C_2$ x 16, $C_5^2\times C_{10}$ x 8, $C_5^2:C_{10}$ x 2, $D_5\times C_5^2$ x 2

Order 200: $C_{10}:F_5$ x 19, $C_{10}.D_{10}$ x 6, $D_5\times D_{10}$ x 5, $C_{10}:F_5$ x 3, $D_5:F_5$ x 2, $C_{10}:D_{10}$, $C_{20}:D_5$

Order 125: $C_5^3$ x 8

Order 100: $C_5:D_{10}$ x 38, $C_5:F_5$ x 18, $D_5^2$ x 10, $C_5:C_{20}$ x 7, $C_5\times D_{10}$ x 6, $C_5:F_5$ x 3, $C_5\times C_{20}$, $C_5^2:C_4$, $C_{10}^2$

Order 80: $C_2^2\times F_5$, $C_4\times F_5$

Order 50: $C_5:D_5$ x 71, $C_5\times C_{10}$ x 36, $C_5\times D_5$ x 6

Order 40: $C_2\times F_5$ x 7, $C_4\times D_5$ x 2, $C_2\times D_{10}$ x 2

Order 32: $C_4^2:C_2$

Order 25: $C_5^2$ x 35

Order 20: $D_{10}$ x 14, $F_5$ x 6, $C_2\times C_{10}$ x 2, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $C_4:C_4$ x 2, $C_2^2:C_4$, $C_4^2$, $C_2^2\times C_4$

Order 10: $D_5$ x 18, $C_{10}$ x 10

Order 8: $C_2\times C_4$ x 5, $C_2^3$

Order 5: $C_5$ x 8

Order 4: $C_2^2$ x 4, $C_4$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 20000: $C_5^4:(C_4^2:C_2)$ ($C_1$)

Order 10000: $(C_5^3\times C_{10}).D_4$ ($C_2$) x 2, $C_5^4.C_4^2$ ($C_2$) x 2, $(C_5^3\times C_{10}).D_4$ ($C_2$), $C_5^4:(C_2^2\times C_4)$ ($C_2$), $(C_5^3\times C_{10}).Q_8$ ($C_2$)

Order 5000: $C_5^4:(C_2\times C_4)$ ($C_2^2$) x 4, $C_5^4:(C_2\times C_4)$ ($C_4$) x 4, $C_5^4:(C_2\times C_4)$ ($C_2^2$) x 2, $C_5^4:C_2^3$ ($C_2^2$)

Order 2500: $C_5^4:C_4$ ($C_2\times C_4$) x 4, $C_5^4:C_2^2$ ($C_2\times C_4$) x 2, $C_5^4:C_2^2$ ($C_2^3$)

Order 1250: $C_5^4:C_2$ ($D_4:C_2$), $C_5^4:C_2$ ($C_2^2\times C_4$), $C_5^3\times C_{10}$ ($D_4:C_2$)

Order 625: $C_5^4$ ($C_4^2:C_2$)

Order 50: $C_5\times C_{10}$ ($D_5^2.C_2^2$) x 2

Order 25: $C_5^2$ ($D_5^2.(C_2\times C_4)$) x 2

Order 2: $C_2$ ($C_5^4:D_4:C_2$)

Order 1: $C_1$ ($C_5^4:(C_4^2:C_2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 20000: $C_5^4:(C_4^2:C_2)$ ($C_1$)

Order 10000: $(C_5^3\times C_{10}).D_4$ ($C_2$), $(C_5^3\times C_{10}).D_4$ ($C_2$), $C_5^4.C_4^2$ ($C_2$), $C_5^4:(C_2^2\times C_4)$ ($C_2$), $(C_5^3\times C_{10}).Q_8$ ($C_2$)

Order 5000: $C_5^4:(C_2\times C_4)$ ($C_2^2$) x 3, $C_5^4:C_2^3$ ($C_2^2$), $C_5^4:(C_2\times C_4)$ ($C_2^2$), $C_5^4:(C_2\times C_4)$ ($C_4$)

Order 2500: $C_5^4:C_4$ ($C_2\times C_4$) x 2, $C_5^4:C_2^2$ ($C_2\times C_4$), $C_5^4:C_2^2$ ($C_2^3$)

Order 1250: $C_5^4:C_2$ ($D_4:C_2$), $C_5^4:C_2$ ($C_2^2\times C_4$), $C_5^3\times C_{10}$ ($D_4:C_2$)

Order 625: $C_5^4$ ($C_4^2:C_2$)

Order 50: $C_5\times C_{10}$ ($D_5^2.C_2^2$)

Order 25: $C_5^2$ ($D_5^2.(C_2\times C_4)$)

Order 2: $C_2$ ($C_5^4:D_4:C_2$)

Order 1: $C_1$ ($C_5^4:(C_4^2:C_2)$)

Series

Derived series

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4.C_4^2$

$\rhd$

$C_5^4.C_4^2$

$\rhd$

$C_5^4.C_4^2$

$\rhd$

$C_5^4:(C_2\times C_4)$

$\rhd$

$C_5^4:(C_2\times C_4)$

$\rhd$

$C_5^4:(C_2\times C_4)$

$\rhd$

$C_5^4:C_2^2$

$\rhd$

$C_5^4:C_2^2$

$\rhd$

$C_5^4:C_2^2$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_5^2$

$\rhd$

$C_5^2$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:(C_4^2:C_2)$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4:C_2$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 35 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $152 \times 152$ 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 $90 \times 90$ rational character table.