Abstract group 3265920.a: $\PSU(4,3)$

• Label: 3265920.a
• Order: \( 2^{7} \cdot 3^{6} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $112$
• Trans deg.: $112$
• Rank: $2$

Group information

Description:	$\PSU(4,3)$
Order:	\(3265920\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 5 \cdot 7 \)
Exponent:	\(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$\PGammaU(4,3)$, of order \(26127360\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \)
Composition factors:	$\PSU(4,3)$
Derived length:	$0$

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

Group statistics

Order	1	2	3	4	5	6	7	8	9	12
Elements	1	2835	47600	238140	653184	226800	933120	408240	483840	272160	3265920
Conjugacy classes	1	1	4	2	1	3	2	1	4	1	20
Divisions	1	1	4	2	1	3	1	1	2	1	17
Autjugacy classes	1	1	3	2	1	2	1	1	1	1	14

Dimension	1	21	35	90	140	189	210	280	315	420	560	640	729	896	1280
Irr. complex chars.	1	1	2	1	1	1	1	4	2	1	1	2	1	1	0	20
Irr. rational chars.	1	1	2	1	1	1	1	0	2	1	3	0	1	1	1	17

Minimal presentations

Permutation degree:	$112$
Transitive degree:	$112$
Rank:	$2$
Inequivalent generating pairs:	$390255$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	21	21	21
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSU(4,3)$, $\POmegaMinus(6,3)$
Permutation group:	Degree $112$ $\langle(1,3,7,12,20,32,45,64,89)(2,5,11,18,30,43,62,87,106)(4,9,6,13,21,33,46,65,91) \!\cdots\! \rangle$
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 equivalence classes (represented by square brackets) of matrices in $\SU(4,3)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_{3} \times C_{12}$
Commutator length:	$1$

Subgroups

There are 10009764 subgroups in 381 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSU(4,3)$
Commutator:	$G' \simeq$ $\PSU(4,3)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSU(4,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSU(4,3)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSU(4,3)$
Socle:	$\operatorname{soc} \simeq$ $\PSU(4,3)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 3265920: $\PSU(4,3)$

Order 29160: $C_3^4:A_6$

Order 25920: $\SU(4,2)$ x 2

Order 20160: $\PSL(3,4)$ x 2

Order 11664: $\He_3:(C_3:S_3).S_4$

Order 6048: $\SU(3,3)$

Order 5832: $\He_3.\ASL(2,3)$

Order 5760: $C_2^4:A_6$ x 2

Order 4860: $C_3^4:A_5$ x 2

Order 3888: $C_3^3:C_3^2.Q_{16}$

Order 2916: $C_3^4:C_3^2:C_4$

Order 2520: $A_7$ x 4

Order 1944: $C_3^4:S_4$ x 2, $\He_3.\PSU(3,2)$, $C_3^3:\PSU(3,2)$, $\He_3.F_9$

Order 1458: $C_3^4:C_3:S_3$

Order 1152: $C_2^3:A_4.D_6$

Order 972: $C_3\wr A_4$ x 2, $C_3^3:C_3^2:C_4$, $C_3^3:C_3^2:C_4$

Order 960: $C_2^4:A_5$ x 2, $C_2^4:A_5$ x 2

Order 810: $C_3^4:D_5$

Order 729: $C_3^4:C_3^2$

Order 720: $A_6.C_2$ x 2, $S_6$

Order 648: $C_3^3:S_4$ x 6, $\Unitary(3,2)$ x 2, $C_3^4:D_4$

Order 576: $A_4^2:C_4$ x 2, $\OmegaPlus(4,3):C_2$ x 2, $\SL(2,3).S_4$

Order 486: $C_3^4:S_3$ x 2, $(C_3\times \He_3):S_3$

Order 405: $C_3^4:C_5$

Order 384: $C_2^4:S_4$ x 2, $C_4^2:S_4$

Order 360: $A_6$ x 11

Order 324: $C_3^3:A_4$ x 6, $C_3\wr C_2^2$ x 2, $C_3^4:C_4$

Order 288: $\PSOPlus(4,3)$ x 2, $\OmegaPlus(4,3)$

Order 243: $C_3^2.C_3^3$ x 2, $C_3^4:C_3$ x 2, $C_3^3:C_3^2$

Order 216: $\SU(3,2)$ x 2, $C_3^3:Q_8$ x 2, $S_3^2:S_3$ x 2, $\He_3:C_8$

Order 192: $C_2\wr A_4$ x 2, $C_2^3:S_4$ x 2, $Q_8.S_4$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$, $C_4^2:A_4$

Order 168: $\PSL(2,7)$ x 5

Order 162: $C_3\wr S_3$ x 6, $C_3^3:S_3$ x 3, $C_3^2\wr C_2$

Order 160: $C_2^4:D_5$ x 2

Order 144: $A_4^2$ x 2, $C_6.S_4$

Order 128: $C_4^2:D_4$

Order 120: $S_5$ x 2

Order 108: $C_3^3:C_4$ x 9, $\He_3:C_4$ x 3, $C_3:S_3^2$ x 2, $C_3\times S_3^2$ x 2, $C_3^2:C_{12}$ x 2

Order 96: $C_2^2:S_4$ x 2, $Q_8:A_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $\Unitary(2,3)$, $C_4^2:S_3$, $\SL(2,3):C_2^2$, $C_4.S_4$

Order 81: $C_3.\He_3$ x 6, $C_3\wr C_3$ x 6, $C_3\times \He_3$ x 3, $C_9:C_3^2$ x 2, $C_3^4$

Order 80: $C_2^4:C_5$ x 2

Order 72: $C_3:S_4$ x 2, $\PSU(3,2)$ x 2, $\SOPlus(4,2)$, $C_3\times \SL(2,3)$, $D_6:S_3$

Order 64: $C_2\wr C_4$ x 2, $C_2\wr C_2^2$ x 2, $C_4^2:C_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 60: $A_5$ x 8

Order 54: $C_3^2:S_3$ x 9, $C_3^2:C_6$ x 7, $S_3\times C_3^2$ x 2

Order 48: $C_2.S_4$ x 3, $C_2^2:A_4$ x 2, $C_2^2\times A_4$ x 2, $C_2\times S_4$ x 2, $A_4:C_4$ x 2, $C_4^2:C_3$ x 2, $\SL(2,3):C_2$, $C_3:Q_{16}$

Order 36: $C_3^2:C_4$ x 9, $C_6\times S_3$ x 2, $C_3\times A_4$ x 2, $S_3^2$ x 2, $C_3^2:C_4$

Order 32: $C_2^3:C_4$ x 2, $C_4^2:C_2$ x 2, $C_2^2\wr C_2$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $Q_{16}:C_2$, $C_4\wr C_2$

Order 27: $\He_3$ x 9, $C_3^3$ x 7, $C_9:C_3$ x 6, $C_3\times C_9$ x 2

Order 24: $S_4$ x 7, $\SL(2,3)$ x 3, $C_3:D_4$ x 2, $C_2\times A_4$ x 2, $C_3:Q_8$, $C_3\times Q_8$, $C_3:C_8$

Order 21: $C_7:C_3$

Order 20: $F_5$

Order 18: $C_3\times S_3$ x 7, $C_3:S_3$ x 5, $C_3\times C_6$

Order 16: $C_2^2:C_4$ x 4, $C_4^2$ x 2, $C_2^4$ x 2, $C_2\times D_4$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 12: $A_4$ x 7, $C_3:C_4$ x 3, $C_2\times C_6$ x 2, $D_6$ x 2, $C_{12}$

Order 10: $D_5$

Order 9: $C_3^2$ x 9, $C_9$ x 2

Order 8: $C_2^3$ x 2, $Q_8$ x 2, $D_4$ x 2, $C_2\times C_4$ x 2, $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 3, $S_3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3265920: $\PSU(4,3)$

Order 29160: $C_3^4:A_6$

Order 25920: $\SU(4,2)$

Order 20160: $\PSL(3,4)$

Order 11664: $\He_3:(C_3:S_3).S_4$

Order 6048: $\SU(3,3)$

Order 5832: $\He_3.\ASL(2,3)$

Order 5760: $C_2^4:A_6$

Order 4860: $C_3^4:A_5$

Order 3888: $C_3^3:C_3^2.Q_{16}$

Order 2916: $C_3^4:C_3^2:C_4$

Order 2520: $A_7$

Order 1944: $C_3^4:S_4$, $\He_3.\PSU(3,2)$, $C_3^3:\PSU(3,2)$, $\He_3.F_9$

Order 1458: $C_3^4:C_3:S_3$

Order 1152: $C_2^3:A_4.D_6$

Order 972: $C_3\wr A_4$, $C_3^3:C_3^2:C_4$, $C_3^3:C_3^2:C_4$

Order 960: $C_2^4:A_5$, $C_2^4:A_5$

Order 810: $C_3^4:D_5$

Order 729: $C_3^4:C_3^2$

Order 720: $A_6.C_2$, $S_6$

Order 648: $C_3^3:S_4$ x 2, $C_3^4:D_4$, $\Unitary(3,2)$

Order 576: $A_4^2:C_4$, $\SL(2,3).S_4$, $\OmegaPlus(4,3):C_2$

Order 486: $(C_3\times \He_3):S_3$, $C_3^4:S_3$

Order 405: $C_3^4:C_5$

Order 384: $C_2^4:S_4$, $C_4^2:S_4$

Order 360: $A_6$ x 4

Order 324: $C_3^3:A_4$ x 2, $C_3\wr C_2^2$, $C_3^4:C_4$

Order 288: $\OmegaPlus(4,3)$, $\PSOPlus(4,3)$

Order 243: $C_3^3:C_3^2$, $C_3^2.C_3^3$, $C_3^4:C_3$

Order 216: $\SU(3,2)$, $\He_3:C_8$, $C_3^3:Q_8$, $S_3^2:S_3$

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$, $C_2\wr A_4$, $C_2^3:S_4$, $Q_8.S_4$, $C_4^2:A_4$

Order 168: $\PSL(2,7)$ x 2

Order 162: $C_3^3:S_3$ x 2, $C_3\wr S_3$ x 2, $C_3^2\wr C_2$

Order 160: $C_2^4:D_5$

Order 144: $A_4^2$, $C_6.S_4$

Order 128: $C_4^2:D_4$

Order 120: $S_5$

Order 108: $C_3^3:C_4$ x 4, $\He_3:C_4$ x 2, $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^2:C_{12}$

Order 96: $\Unitary(2,3)$, $C_4^2:S_3$, $C_2^2:S_4$, $Q_8:A_4$, $\SL(2,3):C_2^2$, $\GL(2,\mathbb{Z}/4)$, $C_4.S_4$

Order 81: $C_3.\He_3$ x 2, $C_3\wr C_3$ x 2, $C_3\times \He_3$ x 2, $C_3^4$, $C_9:C_3^2$

Order 80: $C_2^4:C_5$

Order 72: $C_3:S_4$, $\PSU(3,2)$, $\SOPlus(4,2)$, $C_3\times \SL(2,3)$, $D_6:S_3$

Order 64: $C_4^2:C_4$, $C_2\wr C_4$, $C_4^2:C_2^2$, $C_2\wr C_2^2$, $C_4^2:C_2^2$

Order 60: $A_5$ x 3

Order 54: $C_3^2:S_3$ x 4, $C_3^2:C_6$ x 4, $S_3\times C_3^2$

Order 48: $C_4^2:C_3$ x 2, $C_2.S_4$ x 2, $C_2^2:A_4$, $C_2^2\times A_4$, $C_2\times S_4$, $\SL(2,3):C_2$, $A_4:C_4$, $C_3:Q_{16}$

Order 36: $C_3^2:C_4$ x 4, $C_3^2:C_4$, $C_6\times S_3$, $C_3\times A_4$, $S_3^2$

Order 32: $C_4^2:C_2$ x 2, $\OD_{16}:C_2$, $C_2^3:C_4$, $D_4:C_2^2$, $Q_{16}:C_2$, $C_2^2\wr C_2$, $C_4\wr C_2$

Order 27: $C_3^3$ x 4, $\He_3$ x 4, $C_9:C_3$ x 2, $C_3\times C_9$

Order 24: $S_4$ x 3, $\SL(2,3)$ x 2, $C_3:D_4$, $C_3:Q_8$, $C_2\times A_4$, $C_3\times Q_8$, $C_3:C_8$

Order 21: $C_7:C_3$

Order 20: $F_5$

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 3, $C_3\times C_6$

Order 16: $C_2^2:C_4$ x 2, $C_4^2$ x 2, $C_2\times D_4$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $C_2^4$, $D_4:C_2$, $C_2\times Q_8$

Order 12: $A_4$ x 3, $C_3:C_4$ x 2, $C_2\times C_6$, $D_6$, $C_{12}$

Order 10: $D_5$

Order 9: $C_3^2$ x 6, $C_9$

Order 8: $Q_8$ x 2, $D_4$ x 2, $C_2\times C_4$ x 2, $C_2^3$, $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 2, $S_3$ x 2

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3265920: $\PSU(4,3)$ ($C_1$)

Order 1: $C_1$ ($\PSU(4,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3265920: $\PSU(4,3)$ ($C_1$)

Order 1: $C_1$ ($\PSU(4,3)$)

Series

Derived series	$\PSU(4,3)$
Chief series	$\PSU(4,3)$	$\rhd$	$C_1$
Lower central series	$\PSU(4,3)$
Upper central series	$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $20 \times 20$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.