Abstract group 40000.op: $(\SO(3,7)\times S_4^2).C_2^2$

• Label: 40000.op
• Order: \( 2^{6} \cdot 5^{4} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 5^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $28$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$(\SO(3,7)\times S_4^2).C_2^2$
Order:	\(40000\)\(\medspace = 2^{6} \cdot 5^{4} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_5^4.C_2.C_2^6.C_2^5$, of order \(2560000\)\(\medspace = 2^{12} \cdot 5^{4} \)
Composition factors:	$C_2$ x 6, $C_5$ 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	1451	25300	624	5424	7200	40000
Conjugacy classes	1	5	19	27	39	18	109
Divisions	1	5	15	21	29	9	80
Autjugacy classes	1	4	7	8	10	4	34

Dimension	1	2	4	16	32	64
Irr. complex chars.	16	8	1	60	24	0	109
Irr. rational chars.	16	0	5	34	13	12	80

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	$307617186240$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{4}=c^{4}=d^{10}=e^{5}=f^{5}=g^{5}=[d,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(1,3,10,16,6,15,20,8,13,17)(2,7,4,12,11,9,19,18,5,14)(21,24)(22,27)(23,28) \!\cdots\! \rangle$
Transitive group:	40T15272				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_5^4$ $\,\rtimes\,$ $(C_4^2.C_2^2)$	$(C_5^3:C_{20}:Q_8)$ $\,\rtimes\,$ $C_2$	$(C_5^4:(C_4\times Q_8))$ $\,\rtimes\,$ $C_2$	$(C_5^4.(C_4\times Q_8))$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_5^4.C_4^2)$ . $C_2^2$ (3)	$(C_2\times C_5^4:Q_8)$ . $C_2^2$ (2)	$(C_5^4:(C_4\times D_4))$ . $C_2$	$(C_5^4:C_4)$ . $(D_4:C_2)$ (2)	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:	$C_{2}^{3} \times C_{10}$
Commutator length:	$1$

Subgroups

There are 336830 subgroups in 1848 conjugacy classes, 77 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_5^4:C_2.C_2^4$
Commutator:	$G' \simeq$ $C_2\times C_5^4:C_2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_5^4:C_2.C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^3\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4.C_2^3$
Radical:	$R \simeq$ $(\SO(3,7)\times S_4^2).C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^3\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_4.C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^2$
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 40000: $(\SO(3,7)\times S_4^2).C_2^2$

Order 20000: $C_5^4.C_4^2:C_2$ x 4, $C_5^4:(C_4\times Q_8)$ x 2, $C_5^2:C_{10}^2:Q_8$ x 2, $(\SO(3,7)\times S_4^2).C_2$ x 2, $C_5^4:(C_4^2:C_2)$ x 2, $C_5^3:C_{20}:Q_8$, $C_5^4:(C_4\times D_4)$, $C_5^4.(C_4\times Q_8)$

Order 10000: $C_5^4.C_4^2$ x 7, $(C_5^3\times C_{10}).Q_8$ x 7, $(C_5^3\times C_{10}).D_4$ x 6, $C_5^2:C_{10}^2:C_4$ x 4, $(C_5^3\times C_{10}).D_4$ x 3, $C_2\times C_5^4:Q_8$ x 3, $C_5^4:(C_2^2\times C_4)$ x 2, $C_5^3:C_{20}:C_4$ x 2, $\SO(3,7)\times S_4^2$

Order 5000: $C_5^4:Q_8$ x 6, $C_5^4:(C_2\times C_4)$ x 5, $C_2\times C_5^4:C_4$ x 5, $C_5^4:(C_2\times C_4)$ x 4, $C_5^4:(C_2\times C_4)$ x 3, $C_2\times C_5^4:C_2^2$ x 2, $C_5^4:D_4$ x 2

Order 2500: $C_5^4:C_4$ x 12, $C_5^4:C_2^2$ x 4, $C_5^3:C_{20}$ x 3, $C_5^2:C_{10}^2$ x 2, $C_2\times C_5^4:C_2$

Order 2000: $C_5^3:C_4^2$ x 7, $(C_5\times D_{10}):F_5$ x 4, $(C_5\times D_{10}):F_5$ x 2, $C_5:C_{20}:F_5$ 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 30, $C_{10}.D_5^2$ x 9, $C_5^3:(C_2\times C_4)$ x 8, $C_{10}:D_5^2$ x 6

Order 800: $(D_5\times D_{10}).C_2^2$ x 4, $(D_5\times D_{10}).C_2^2$ x 2, $D_5^2.(C_2\times C_4)$ x 2, $C_{10}^2:Q_8$ x 2, $C_4\times C_5^2:Q_8$ x 2, $C_5^2:(C_4\times Q_8)$, $C_5^2:(C_4\times D_4)$, $(C_5\times C_{20}):Q_8$

Order 625: $C_5^4$

Order 500: $C_5^3:C_4$ x 48, $C_5^2:D_{10}$ x 21, $C_5:D_5^2$ x 16, $C_5^2:C_{20}$ x 9, $C_5^2:C_{20}$ x 9, $C_5^2:D_{10}$ x 6, $D_{10}\times C_5^2$ x 6

Order 400: $(C_5\times C_{10}).Q_8$ x 21, $D_5^2:C_4$ x 18, $C_5^2:C_4^2$ x 14, $C_2\times C_5^2:Q_8$ x 9, $(C_5\times C_{10}).D_4$ x 9, $D_{10}:F_5$ x 8, $C_{20}:F_5$ x 7, $D_{10}:F_5$ x 4, $C_{10}^2:C_4$ x 4, $C_{10}.(C_2\times F_5)$ x 4, $D_5^2:C_2^2$ x 3, $C_{10}^2:C_4$ x 2, $C_{20}:F_5$ x 2

Order 250: $C_5^3:C_2$ x 42, $C_5^2\times C_{10}$ x 21, $C_5^2:C_{10}$ x 8, $D_5\times C_5^2$ x 8

Order 200: $C_{10}:F_5$ x 94, $C_{10}.D_{10}$ x 27, $C_5^2:Q_8$ x 24, $D_5\times D_{10}$ x 18, $D_5:F_5$ x 12, $C_{10}:F_5$ x 10, $D_5\wr C_2$ x 8, $C_{20}:D_5$ x 3, $C_{10}:D_{10}$ x 2

Order 125: $C_5^3$ x 21

Order 100: $C_5:F_5$ x 138, $C_5:D_{10}$ x 92, $D_5^2$ x 52, $C_5:C_{20}$ x 36, $C_5\times D_{10}$ x 24, $C_5:F_5$ x 14, $C_5\times C_{20}$ x 3, $C_5^2:C_4$ x 3, $C_{10}^2$ x 2

Order 80: $C_4\times F_5$ x 7, $D_{10}:C_4$ x 4, $C_2^2\times F_5$ x 2, $C_{20}:C_4$ x 2

Order 64: $C_4^2.C_2^2$

Order 50: $C_5:D_5$ x 174, $C_5\times C_{10}$ x 88, $C_5\times D_5$ x 40

Order 40: $C_2\times F_5$ x 38, $C_4\times D_5$ x 9, $C_2\times D_{10}$ x 6

Order 32: $C_4^2:C_2$ x 4, $C_4\times Q_8$ x 3, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_4^2:C_2$ x 2, $C_4:Q_8$, $C_4\times D_4$

Order 25: $C_5^2$ x 86

Order 20: $F_5$ x 48, $D_{10}$ x 43, $C_{20}$ x 9, $C_5:C_4$ x 9, $C_2\times C_{10}$ x 6

Order 16: $C_4:C_4$ x 12, $C_2^2:C_4$ x 10, $C_4^2$ x 7, $C_2\times Q_8$ x 3, $C_2^2\times C_4$ x 2, $C_2\times D_4$

Order 10: $D_5$ x 50, $C_{10}$ x 29

Order 8: $C_2\times C_4$ x 17, $Q_8$ x 6, $C_2^3$ x 2, $D_4$ x 2

Order 5: $C_5$ x 21

Order 4: $C_4$ x 15, $C_2^2$ x 7

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 40000: $(\SO(3,7)\times S_4^2).C_2^2$

Order 20000: $C_5^4:(C_4\times Q_8)$, $C_5^2:C_{10}^2:Q_8$, $C_5^3:C_{20}:Q_8$, $C_5^4.C_4^2:C_2$, $C_5^4:(C_4\times D_4)$, $(\SO(3,7)\times S_4^2).C_2$, $C_5^4.(C_4\times Q_8)$, $C_5^4:(C_4^2:C_2)$

Order 10000: $C_5^4.C_4^2$ x 3, $(C_5^3\times C_{10}).Q_8$ x 3, $(C_5^3\times C_{10}).D_4$ x 2, $(C_5^3\times C_{10}).D_4$ x 2, $C_2\times C_5^4:Q_8$ x 2, $\SO(3,7)\times S_4^2$, $C_5^2:C_{10}^2:C_4$, $C_5^4:(C_2^2\times C_4)$, $C_5^3:C_{20}:C_4$

Order 5000: $C_2\times C_5^4:C_4$ x 3, $C_5^4:(C_2\times C_4)$ x 2, $C_5^4:(C_2\times C_4)$ x 2, $C_5^4:Q_8$ x 2, $C_2\times C_5^4:C_2^2$, $C_5^4:D_4$, $C_5^4:(C_2\times C_4)$

Order 2500: $C_5^4:C_4$ x 5, $C_5^4:C_2^2$ x 2, $C_5^3:C_{20}$ x 2, $C_2\times C_5^4:C_2$, $C_5^2:C_{10}^2$

Order 2000: $C_5^3:C_4^2$ x 3, $(C_5\times D_{10}):F_5$, $C_5:C_{20}:F_5$, $(C_5\times D_{10}):F_5$

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

Order 800: $(D_5\times D_{10}).C_2^2$, $(D_5\times D_{10}).C_2^2$, $C_5^2:(C_4\times Q_8)$, $C_5^2:(C_4\times D_4)$, $D_5^2.(C_2\times C_4)$, $C_{10}^2:Q_8$, $(C_5\times C_{20}):Q_8$, $C_4\times C_5^2:Q_8$

Order 625: $C_5^4$

Order 500: $C_5^3:C_4$ x 11, $C_5^2:D_{10}$ x 8, $C_5:D_5^2$ x 4, $C_5^2:C_{20}$ x 4, $C_5^2:C_{20}$ x 4, $C_5^2:D_{10}$ x 2, $D_{10}\times C_5^2$ x 2

Order 400: $(C_5\times C_{10}).Q_8$ x 8, $C_5^2:C_4^2$ x 6, $C_2\times C_5^2:Q_8$ x 5, $D_5^2:C_4$ x 5, $(C_5\times C_{10}).D_4$ x 4, $C_{20}:F_5$ x 3, $D_5^2:C_2^2$ x 2, $D_{10}:F_5$ x 2, $D_{10}:F_5$ x 2, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $C_{20}:F_5$, $C_{10}.(C_2\times F_5)$

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 34, $C_{10}.D_{10}$ x 11, $D_5\times D_{10}$ x 6, $C_{10}:F_5$ x 5, $C_5^2:Q_8$ x 5, $D_5\wr C_2$ x 2, $D_5:F_5$ x 2, $C_{20}:D_5$ x 2, $C_{10}:D_{10}$

Order 125: $C_5^3$ x 8

Order 100: $C_5:D_{10}$ x 38, $C_5:F_5$ x 33, $C_5:C_{20}$ x 13, $D_5^2$ x 12, $C_5\times D_{10}$ x 7, $C_5:F_5$ x 5, $C_5\times C_{20}$ x 2, $C_5^2:C_4$ x 2, $C_{10}^2$

Order 80: $C_4\times F_5$ x 3, $C_2^2\times F_5$, $D_{10}:C_4$, $C_{20}:C_4$

Order 64: $C_4^2.C_2^2$

Order 50: $C_5:D_5$ x 71, $C_5\times C_{10}$ x 36, $C_5\times D_5$ x 7

Order 40: $C_2\times F_5$ x 12, $C_4\times D_5$ x 4, $C_2\times D_{10}$ x 2

Order 32: $C_4\times Q_8$ x 2, $C_4:Q_8$, $C_4^2:C_2$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_4\times D_4$, $C_4^2:C_2$

Order 25: $C_5^2$ x 35

Order 20: $D_{10}$ x 14, $F_5$ x 11, $C_{20}$ x 4, $C_5:C_4$ x 4, $C_2\times C_{10}$ x 2

Order 16: $C_4:C_4$ x 6, $C_2^2:C_4$ x 3, $C_4^2$ x 3, $C_2\times Q_8$ x 2, $C_2\times D_4$, $C_2^2\times C_4$

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

Order 8: $C_2\times C_4$ x 8, $Q_8$ x 2, $C_2^3$, $D_4$

Order 5: $C_5$ x 8

Order 4: $C_4$ x 7, $C_2^2$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 40000: $(\SO(3,7)\times S_4^2).C_2^2$ ($C_1$)

Order 20000: $C_5^4.C_4^2:C_2$ ($C_2$) x 4, $C_5^4:(C_4\times Q_8)$ ($C_2$) x 2, $C_5^2:C_{10}^2:Q_8$ ($C_2$) x 2, $(\SO(3,7)\times S_4^2).C_2$ ($C_2$) x 2, $C_5^4:(C_4^2:C_2)$ ($C_2$) x 2, $C_5^3:C_{20}:Q_8$ ($C_2$), $C_5^4:(C_4\times D_4)$ ($C_2$), $C_5^4.(C_4\times Q_8)$ ($C_2$)

Order 10000: $C_5^4.C_4^2$ ($C_2^2$) x 7, $(C_5^3\times C_{10}).Q_8$ ($C_2^2$) x 7, $(C_5^3\times C_{10}).D_4$ ($C_2^2$) x 6, $C_5^2:C_{10}^2:C_4$ ($C_2^2$) x 4, $(C_5^3\times C_{10}).D_4$ ($C_2^2$) x 3, $C_2\times C_5^4:Q_8$ ($C_2^2$) x 3, $C_5^4:(C_2^2\times C_4)$ ($C_2^2$) x 2, $C_5^3:C_{20}:C_4$ ($C_2^2$) x 2, $\SO(3,7)\times S_4^2$ ($C_2^2$)

Order 5000: $C_5^4:(C_2\times C_4)$ ($C_2^3$) x 5, $C_2\times C_5^4:C_4$ ($C_2^3$) x 5, $C_5^4:(C_2\times C_4)$ ($C_2^3$) x 3, $C_2\times C_5^4:C_2^2$ ($C_2^3$) x 2

Order 2500: $C_5^4:C_4$ ($D_4:C_2$) x 4, $C_2\times C_5^4:C_2$ ($C_2^4$)

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

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

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

Order 1: $C_1$ ($(\SO(3,7)\times S_4^2).C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 40000: $(\SO(3,7)\times S_4^2).C_2^2$ ($C_1$)

Order 20000: $C_5^4:(C_4\times Q_8)$ ($C_2$), $C_5^2:C_{10}^2:Q_8$ ($C_2$), $C_5^3:C_{20}:Q_8$ ($C_2$), $C_5^4.C_4^2:C_2$ ($C_2$), $C_5^4:(C_4\times D_4)$ ($C_2$), $(\SO(3,7)\times S_4^2).C_2$ ($C_2$), $C_5^4.(C_4\times Q_8)$ ($C_2$), $C_5^4:(C_4^2:C_2)$ ($C_2$)

Order 10000: $C_5^4.C_4^2$ ($C_2^2$) x 3, $(C_5^3\times C_{10}).Q_8$ ($C_2^2$) x 3, $(C_5^3\times C_{10}).D_4$ ($C_2^2$) x 2, $(C_5^3\times C_{10}).D_4$ ($C_2^2$) x 2, $C_2\times C_5^4:Q_8$ ($C_2^2$) x 2, $\SO(3,7)\times S_4^2$ ($C_2^2$), $C_5^2:C_{10}^2:C_4$ ($C_2^2$), $C_5^4:(C_2^2\times C_4)$ ($C_2^2$), $C_5^3:C_{20}:C_4$ ($C_2^2$)

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

Order 2500: $C_5^4:C_4$ ($D_4:C_2$) x 2, $C_2\times C_5^4:C_2$ ($C_2^4$)

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

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

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

Order 1: $C_1$ ($(\SO(3,7)\times S_4^2).C_2^2$)

Series

Derived series

$(\SO(3,7)\times S_4^2).C_2^2$

$\rhd$

$(\SO(3,7)\times S_4^2).C_2^2$

$\rhd$

$C_2\times C_5^4:C_2$

$\rhd$

$C_2\times C_5^4:C_2$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(\SO(3,7)\times S_4^2).C_2^2$

$\rhd$

$(\SO(3,7)\times S_4^2).C_2^2$

$\rhd$

$C_5^4.C_4^2:C_2$

$\rhd$

$C_5^4.C_4^2:C_2$

$\rhd$

$C_5^3:C_{20}:C_4$

$\rhd$

$C_5^3:C_{20}:C_4$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_2\times C_5^4:C_2$

$\rhd$

$C_2\times C_5^4:C_2$

$\rhd$

$C_5^3\times C_{10}$

$\rhd$

$C_5^3\times C_{10}$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(\SO(3,7)\times S_4^2).C_2^2$

$\rhd$

$(\SO(3,7)\times S_4^2).C_2^2$

$\rhd$

$C_2\times C_5^4:C_2$

$\rhd$

$C_2\times C_5^4:C_2$

$\rhd$

$C_5^4$

$\rhd$

$C_5^4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

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

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

Character theory

Complex character table

See the $109 \times 109$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $80 \times 80$ rational character table.