Abstract group 811273008.a: $\SU(3,13)$

• Label: 811273008.a
• Order: \( 2^{4} \cdot 3 \cdot 7^{2} \cdot 13^{3} \cdot 157 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 157 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 7^{2} \cdot 13^{3} \cdot 157 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $2198$
• Trans deg.: $2198$
• Rank: $2$

Group information

Description:	$\SU(3,13)$
Order:	\(811273008\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{2} \cdot 13^{3} \cdot 157 \)
Exponent:	\(342888\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 157 \)
Automorphism group:	$\PGammaU(3,13)$, of order \(1622546016\)\(\medspace = 2^{5} \cdot 3 \cdot 7^{2} \cdot 13^{3} \cdot 157 \)
Composition factors:	$\SU(3,13)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	6	7	8	12	13	14	21	24	26	28	42	56	84	91	157	168	182
Elements	1	26533	4829006	4829006	4829006	20854938	9658012	9658012	4826808	87081306	28974036	19316024	4457544	28974036	28974036	57948072	57948072	26745264	268701888	115896144	26745264	811273008
Conjugacy classes	1	1	1	1	1	11	2	2	2	27	6	4	1	6	6	12	12	6	52	24	6	184
Divisions	1	1	1	1	1	3	1	1	2	5	1	1	1	1	1	1	1	1	1	1	1	28
Autjugacy classes	1	1	1	1	1	7	1	2	2	15	3	2	1	3	3	6	6	3	26	12	3	100

Dimension	1	156	157	942	1884	2041	2197	2198	2352	3768	4396	5652	8792	11304	12246	13188	26376	52752	122304
Irr. complex chars.	1	1	13	0	26	13	1	77	52	0	0	0	0	0	0	0	0	0	0	184
Irr. rational chars.	1	1	1	2	0	1	1	3	0	1	2	2	1	3	2	3	2	1	1	28

Minimal presentations

Permutation degree:	$2198$
Transitive degree:	$2198$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	156	157	157
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\SU(3,13)$, $\PSU(3,13)$, $\PU(3,13)$
Permutation group:	Degree $2198$ $\langle(1,2,5,12,33,92,259,687,491,180,152,54,20,53,148,413,621,1392,1895,1880,1799,2184,2183,2180,1747,899,532,196,531,1194,1362,2070,1373,1104,1023,448,1087,1905,2186,1833,1805,956,380,136,378,952,976,1345,735,282,350,125,45,19,7,3) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SU(3,13)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 335083691 subgroups in 121 conjugacy classes, 2 normal, 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 $\SU(3,13)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $\He_{13}$
157-Sylow subgroup:	$P_{ 157 } \simeq$ $C_{157}$

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 811273008 (order at least 1) are shown, as well as all normal subgroups of any index.

Order 811273008: $\SU(3,13)$

Order 369096: $\He_{13}.C_{168}$

Order 184548: $\He_{13}:C_{84}$

Order 123032: $\He_{13}:C_{56}$

Order 92274: $\He_{13}:C_{42}$

Order 61516: $\He_{13}:C_{28}$

Order 52728: $\He_{13}:C_{24}$

Order 46137: $\He_{13}:C_{21}$

Order 30758: $\He_{13}:C_{14}$

Order 30576: $\Unitary(2,13)$

Order 26364: $\He_{13}:C_{12}$

Order 17576: $\He_{13}:C_8$

Order 15379: $\He_{13}:C_7$

Order 15288: $C_7\times \SL(2,13)$

Order 13182: $\He_{13}:C_6$

Order 8788: $\He_{13}:C_4$

Order 6591: $\He_{13}:C_3$

Order 4394: $\He_{13}:C_2$

Order 4368: $\SL(2,13):C_2$

Order 2197: $\He_{13}$

Order 2184: $C_{13}:C_{168}$, $\PGL(2,13)$, $\SL(2,13)$

Order 2028: $C_{13}^2:C_{12}$

Order 1176: $C_7^2:S_4$

Order 1092: $\PSL(2,13)$, $C_{13}:C_{84}$

Order 1014: $C_{13}^2:C_6$

Order 728: $C_{13}:C_{56}$

Order 676: $C_{13}^2:C_4$

Order 588: $C_{14}^2:C_3$

Order 546: $C_{13}:C_{42}$

Order 507: $C_{13}^2:C_3$

Order 471: $C_{157}:C_3$

Order 392: $C_{14}\wr C_2$

Order 364: $C_{13}:C_{28}$

Order 338: $C_{13}\times D_{13}$

Order 336: $C_{168}:C_2$, $C_7\times \GL(2,3)$

Order 312: $C_{13}:C_{24}$

Order 294: $C_7^2:S_3$

Order 273: $C_{13}:C_{21}$

Order 196: $C_7:C_{28}$, $C_{14}^2$, $C_7\times D_{14}$

Order 182: $C_{182}$

Order 169: $C_{13}^2$

Order 168: $C_{168}$, $\PSL(2,7)$, $C_7\times D_{12}$, $C_{21}:Q_8$, $C_7\times \SL(2,3)$

Order 157: $C_{157}$

Order 156: $F_{13}$, $C_{13}:C_{12}$

Order 147: $C_7^2:C_3$

Order 112: $C_7\times \SD_{16}$

Order 104: $C_{13}:C_8$

Order 98: $C_7\times C_{14}$, $C_7\times D_7$

Order 91: $C_{91}$

Order 84: $C_{84}$, $C_3:C_{28}$, $S_3\times C_{14}$, $C_7:A_4$

Order 78: $C_{13}:C_6$, $C_{13}:C_6$

Order 56: $C_7\times D_4$, $C_7:D_4$, $C_{56}$, $C_7\times Q_8$

Order 52: $C_{13}:C_4$, $C_{13}:C_4$

Order 49: $C_7^2$

Order 48: $C_{24}:C_2$, $\GL(2,3)$

Order 42: $C_{42}$, $S_3\times C_7$

Order 39: $C_{13}:C_3$ x 2

Order 28: $C_2\times C_{14}$ x 3, $D_{14}$, $C_{28}$, $C_7:C_4$

Order 26: $C_{26}$, $D_{13}$

Order 24: $D_{12}$, $C_3:Q_8$, $\SL(2,3)$, $C_{24}$, $S_4$

Order 21: $C_{21}$, $C_7:C_3$

Order 16: $\SD_{16}$

Order 14: $C_{14}$ x 5, $D_7$

Order 13: $C_{13}$ x 2

Order 12: $D_6$, $A_4$, $C_{12}$, $C_3:C_4$

Order 8: $Q_8$, $D_4$, $C_8$

Order 7: $C_7$ x 3

Order 6: $C_6$, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 811273008: $\SU(3,13)$ ($C_1$)

Order 1: $C_1$ ($\SU(3,13)$)

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

See the $184 \times 184$ 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 $28 \times 28$ rational character table.