Abstract group 29484000000.c: $\POMinus(6,5)$

• Label: 29484000000.c
• Order: \( 2^{8} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $756$
• Trans deg.: $756$
• Rank: $2$

Group information

Description:	$\POMinus(6,5)$
Order:	\(29484000000\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
Exponent:	\(32760\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Automorphism group:	Group of order \(58968000000\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
Composition factors:	$C_2$, $\PSU(4,5)$
Derived length:	$1$

This group is nonabelian, almost simple, and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	13	15	20	21	24	26	30	52	60	63
Elements	1	1433775	7192250	256410000	244140624	911478750	468000000	819000000	468000000	2123175600	3091725000	850500000	782964000	2751840000	936000000	2866500000	4252500000	3652740000	1701000000	491400000	2808000000	29484000000
Conjugacy classes	1	5	3	6	6	13	1	2	1	16	8	3	6	6	2	7	9	11	3	1	6	116
Divisions	1	5	3	6	6	13	1	2	1	16	8	1	6	6	1	4	3	11	1	1	1	97
Autjugacy classes	1	4	3	5	4	10	1	2	1	9	7	3	5	4	2	6	6	7	3	1	6	90

Minimal presentations

Permutation degree:	$756$
Transitive degree:	$756$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	105	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PGOMinus(6,5)$
Permutation group:	Degree $756$ $\langle(7,14,16,36)(10,21,23,55)(12,26,34,84)(13,28,68,162)(15,32,37,90)(17,40,42,103) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\PSU(4,5)$ . $C_2$				more information

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\GOMinus(6,5)$.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 3 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\PSU(4,5)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^4.C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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 33696 (order at least 875000) are shown, as well as all normal subgroups of any index.

Order 29484000000: $\POMinus(6,5)$

Order 14742000000: $\PSU(4,5)$

Order 39000000: $C_5^4.C_4.\PSL(2,25).C_2$

Order 19500000: $C_5^4.C_2.\PSL(2,25).C_2$ x 2, $C_5^4.C_4.\PSL(2,25)$

Order 18720000: $C_2.\PSp(4,5).C_2$ x 2

Order 9750000: $C_5^4.\PSL(2,25).C_2$ x 2, $C_5^4.C_2.\PSL(2,25)$

Order 9360000: $\SO(5,5)$ x 4, $C_2\times \PSp(4,5)$ x 2

Order 9000000: $C_5.C_5^4.D_6.C_2.S_5$

Order 4875000: $C_5^4.\PSL(2,25)$

Order 4680000: $\PSp(4,5)$ x 2

Order 4500000: $C_5.C_5^4.C_{12}.S_5$ x 2, $C_5.C_5^4.D_6.C_2.A_5$

Order 3000000: $\He_5.C_5:D_5.C_2^2.S_5$

Order 2268000: $C_3.\PSU(3,5).S_3$

Order 2250000: $C_5.C_5^4.D_6.A_5$, $C_5.C_5^4.C_{12}.A_5$, $C_5.C_5^4.C_6.C_2.A_5$

Order 1800000: $C_5.C_5^4.Q_8.(C_6\times S_3).C_2$

Order 1500000: $\He_5.C_5^2.C_4.S_5$ x 2, $\He_5.C_5^2.C_4.C_2.A_5$, $C_5^3.C_5^3.(C_4^2\times S_3)$

Order 1134000: $C_3.\PSU(3,5).C_3$

Order 1125000: $C_5.C_5^4.C_6.A_5$

Order 900000: $C_5.C_5^4.Q_8.C_3^2.C_4$, $C_5.C_5^4.Q_8.C_3.C_{12}$, $C_5.C_5^4.C_6.A_4.C_2^2$, $C_5.C_5^4.C_3.C_{12}.C_2^3$

Order 172800: $D_6.A_5^2.C_2^2$

Order 124800: $D_4.\PSL(2,25).C_2$

Order 51840: $\SO(5,3)$ x 2

Order 31200: $\PSL(2,25).C_2^2$

Order 28800: $C_2\times S_5^2$

Order 15625: $C_5^4.C_5^2$

Order 5184: $S_3^3:S_4$

Order 5040: $S_7$ x 2

Order 256: $D_4^2:C_2^2$

Order 81: $C_3\wr C_3$

Order 13: $C_{13}$

Order 7: $C_7$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 29484000000: $\POMinus(6,5)$ ($C_1$)

Order 14742000000: $\PSU(4,5)$ ($C_2$)

Order 1: $C_1$ ($\POMinus(6,5)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $116 \times 116$ character table is not available for this group.

Rational character table

The $97 \times 97$ rational character table is not available for this group.