Abstract group 9904359628800.b: $\PSOPlus(8,3)$

• Label: 990...800.b
• Order: \( 2^{13} \cdot 3^{12} \cdot 5^{2} \cdot 7 \cdot 13 \)
• 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
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{12} \cdot 5^{2} \cdot 7 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $1080$
• Trans deg.: $1080$
• Rank: $2$

Group information

Description:	$\PSOPlus(8,3)$
Order:	\(9904359628800\)\(\medspace = 2^{13} \cdot 3^{12} \cdot 5^{2} \cdot 7 \cdot 13 \)
Exponent:	\(32760\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Automorphism group:	Group of order \(39617438515200\)\(\medspace = 2^{15} \cdot 3^{12} \cdot 5^{2} \cdot 7 \cdot 13 \)
Composition factors:	$C_2$, $\POmegaPlus(8,3)$
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	14	15	18	20	24	26	28	30	36	40
Elements	1	25455951	515322080	22916902320	8253633024	221190147360	176863564800	625900504320	281914214400	696056385024	1055480025600	761873817600	530590694400	660290641920	458535168000	990435962880	550242201600	761873817600	707454259200	220096880640	183414067200	990435962880	9904359628800
Conjugacy classes	1	6	9	9	3	29	1	8	8	7	24	2	3	4	6	4	6	2	4	2	2	4	144
Divisions	1	6	9	9	3	29	1	8	8	7	24	1	3	4	6	4	6	1	2	2	2	2	138
Autjugacy classes	1	5	6	8	2	18	1	6	4	5	16	2	2	2	3	3	3	2	3	1	1	2	96

Minimal presentations

Permutation degree:	$1080$
Transitive degree:	$1080$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	260	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSOPlus(8,3)$
Permutation group:	Degree $1080$ $\langle(1,4)(2,6)(3,9)(5,13)(7,16)(8,19)(10,23)(11,25)(12,28)(14,32)(15,35)(17,38) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\POmegaPlus(8,3)$ . $C_2$				more information

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SOPlus(8,3)$.

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 $\POmegaPlus(8,3)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2\wr D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5.C_3^3.C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
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 40320 (order at least 245643840) are shown, as well as all normal subgroups of any index.

Order 9904359628800: $\PSOPlus(8,3)$

Order 4952179814400: $\POmegaPlus(8,3)$

Order 9170703360: $\SO(7,3)$ x 2

Order 8843178240: $C_3^6.\PSL(4,3).C_2$ x 2, $C_3^6.C_2.\PSL(4,3)$

Order 4585351680: $\Omega(7,3)$ x 4

Order 4421589120: $C_3^6.\PSL(4,3)$ x 3

Order 272097792: $C_3.C_3^8.Q_8.C_2^4.C_3^2.D_6$

Order 52254720: $C_2.\PSU(4,3).D_4$ x 2, $C_4.\PSU(4,3).C_2^2$

Order 24261120: $\PSL(4,3).C_2^2$ x 2

Order 2580480: $C_2^6.S_8$ x 2

Order 1244160: $S_4.\SU(4,2).C_2$ x 2

Order 1036800: $A_6^2.C_2^3$ x 2, $A_6^2.C_2^3$

Order 663552: $C_2.C_2^8.C_3^4.C_2^3.C_2$

Order 531441: $C_3^5.C_3^3.C_3^4$

Order 8192: $C_2^6.C_2\wr D_4$

Order 25: $C_5^2$

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 9904359628800: $\PSOPlus(8,3)$ ($C_1$)

Order 4952179814400: $\POmegaPlus(8,3)$ ($C_2$)

Order 1: $C_1$ ($\PSOPlus(8,3)$)

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

Supergroups

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

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

Character theory

Complex character table

The $144 \times 144$ character table is not available for this group.

Rational character table

The $138 \times 138$ rational character table is not available for this group.