Abstract group 5663616.a: $\SU(3,7)$

• Label: 5663616.a
• Order: \( 2^{7} \cdot 3 \cdot 7^{3} \cdot 43 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \cdot 43 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 7^{3} \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $344$
• Trans deg.: $344$
• Rank: $2$

Group information

Description:	$\SU(3,7)$
Order:	\(5663616\)\(\medspace = 2^{7} \cdot 3 \cdot 7^{3} \cdot 43 \)
Exponent:	\(14448\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 43 \)
Automorphism group:	$\PGammaU(3,7)$, of order \(11327232\)\(\medspace = 2^{8} \cdot 3 \cdot 7^{3} \cdot 43 \)
Composition factors:	$\SU(3,7)$
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	14	16	24	28	43	48	56
Elements	1	2107	117992	92708	117992	117648	539392	235984	101136	471968	471968	202272	1843968	943936	404544	5663616
Conjugacy classes	1	1	1	3	1	2	10	2	1	4	4	2	14	8	4	58
Divisions	1	1	1	2	1	2	4	1	1	1	1	1	1	1	1	20
Autjugacy classes	1	1	1	2	1	2	6	1	1	2	2	1	7	4	2	34

Dimension	1	42	43	86	172	258	301	343	344	384	516	602	688	1204	1376	2752	5376
Irr. complex chars.	1	1	7	0	0	7	7	1	20	14	0	0	0	0	0	0	0	58
Irr. rational chars.	1	1	1	1	1	1	1	1	2	0	3	1	1	1	2	1	1	20

Minimal presentations

Permutation degree:	$344$
Transitive degree:	$344$
Rank:	$2$
Inequivalent generating pairs:	$2822079$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	42	43	43
Arbitrary	42	43	43

Constructions

Groups of Lie type:	$\SU(3,7)$, $\PSU(3,7)$, $\PU(3,7)$
Permutation group:	Degree $344$ $\langle(2,4,8,20,18,7,17,43,109,42,108,230,223,185,202,253,127,51,125,99,38,97,213,116,45,115,143,95,37,14,35,88,195,112,138,57,22,56,135,264,333,298,176,80,101,39,15,6) \!\cdots\! \rangle$
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 $\SU(3,7)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 3639825 subgroups in 95 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\SU(3,7)$
Commutator:	$G' \simeq$ $\SU(3,7)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\SU(3,7)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\SU(3,7)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\SU(3,7)$
Socle:	$\operatorname{soc} \simeq$ $\SU(3,7)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $\He_7$
43-Sylow subgroup:	$P_{ 43 } \simeq$ $C_{43}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 5663616: $\SU(3,7)$

Order 16464: $\He_7:C_{48}$

Order 8232: $\He_7:C_{24}$

Order 5488: $\He_7:C_{16}$

Order 4116: $\He_7:C_{12}$

Order 2744: $\He_7:C_8$

Order 2688: $\Unitary(2,7)$

Order 2058: $C_7^2:F_7$

Order 1372: $\He_7:C_4$

Order 1344: $\SL(2,7):C_4$

Order 1029: $\He_7:C_3$

Order 686: $C_7^2:D_7$

Order 672: $\SL(2,7):C_2$

Order 384: $C_8^2:S_3$

Order 343: $\He_7$

Order 336: $\PGL(2,7)$, $\SL(2,7)$, $C_7:C_{48}$

Order 294: $C_7:F_7$

Order 192: $\GL(2,3):C_4$, $C_8^2:C_3$

Order 168: $\PSL(2,7)$, $C_7:C_{24}$

Order 147: $C_7^2:C_3$

Order 129: $C_{43}:C_3$

Order 128: $C_8\wr C_2$

Order 112: $C_7:C_{16}$

Order 98: $C_7\times D_7$

Order 96: $C_8.A_4$, $\Unitary(2,3)$, $C_4^2:S_3$, $C_{48}:C_2$, $\GL(2,3):C_2$

Order 84: $C_7:C_{12}$

Order 64: $C_8.C_8$, $C_8^2$, $D_8:C_4$

Order 56: $C_{56}$

Order 49: $C_7^2$

Order 48: $S_3\times C_8$, $\SL(2,3):C_2$, $C_4^2:C_3$, $\GL(2,3)$, $C_2.S_4$, $C_{48}$, $C_3:C_{16}$

Order 43: $C_{43}$

Order 42: $C_7:C_6$, $F_7$

Order 32: $D_8:C_2$, $\OD_{16}:C_2$, $C_4\times C_8$, $\OD_{32}$, $C_8.C_4$, $C_4\wr C_2$

Order 28: $C_{28}$

Order 24: $C_4\times S_3$, $\SL(2,3)$, $C_{24}$, $S_4$, $C_3:C_8$

Order 21: $C_7:C_3$ x 2

Order 16: $C_2\times C_8$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4^2$, $D_4:C_2$, $C_{16}$

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

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

Order 8: $C_8$ x 4, $Q_8$, $D_4$, $C_2\times C_4$

Order 7: $C_7$ x 2

Order 6: $C_6$, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5663616: $\SU(3,7)$

Order 16464: $\He_7:C_{48}$

Order 8232: $\He_7:C_{24}$

Order 5488: $\He_7:C_{16}$

Order 4116: $\He_7:C_{12}$

Order 2744: $\He_7:C_8$

Order 2688: $\Unitary(2,7)$

Order 2058: $C_7^2:F_7$

Order 1372: $\He_7:C_4$

Order 1344: $\SL(2,7):C_4$

Order 1029: $\He_7:C_3$

Order 686: $C_7^2:D_7$

Order 672: $\SL(2,7):C_2$

Order 384: $C_8^2:S_3$

Order 343: $\He_7$

Order 336: $\PGL(2,7)$, $\SL(2,7)$, $C_7:C_{48}$

Order 294: $C_7:F_7$

Order 192: $\GL(2,3):C_4$, $C_8^2:C_3$

Order 168: $\PSL(2,7)$, $C_7:C_{24}$

Order 147: $C_7^2:C_3$

Order 129: $C_{43}:C_3$

Order 128: $C_8\wr C_2$

Order 112: $C_7:C_{16}$

Order 98: $C_7\times D_7$

Order 96: $C_8.A_4$, $\Unitary(2,3)$, $C_4^2:S_3$, $C_{48}:C_2$, $\GL(2,3):C_2$

Order 84: $C_7:C_{12}$

Order 64: $C_8.C_8$, $C_8^2$, $D_8:C_4$

Order 56: $C_{56}$

Order 49: $C_7^2$

Order 48: $S_3\times C_8$, $\SL(2,3):C_2$, $C_4^2:C_3$, $\GL(2,3)$, $C_2.S_4$, $C_{48}$, $C_3:C_{16}$

Order 43: $C_{43}$

Order 42: $C_7:C_6$, $F_7$

Order 32: $D_8:C_2$, $\OD_{16}:C_2$, $C_4\times C_8$, $\OD_{32}$, $C_8.C_4$, $C_4\wr C_2$

Order 28: $C_{28}$

Order 24: $C_4\times S_3$, $\SL(2,3)$, $C_{24}$, $S_4$, $C_3:C_8$

Order 21: $C_7:C_3$ x 2

Order 16: $C_2\times C_8$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4^2$, $D_4:C_2$, $C_{16}$

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

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

Order 8: $C_8$ x 4, $Q_8$, $D_4$, $C_2\times C_4$

Order 7: $C_7$ x 2

Order 6: $C_6$, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5663616: $\SU(3,7)$ ($C_1$)

Order 1: $C_1$ ($\SU(3,7)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5663616: $\SU(3,7)$ ($C_1$)

Order 1: $C_1$ ($\SU(3,7)$)

Series

Derived series	$\SU(3,7)$
Chief series	$\SU(3,7)$	$\rhd$	$C_1$
Lower central series	$\SU(3,7)$
Upper central series	$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $58 \times 58$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.