Abstract group $\PSL(3,4)\wr C_2$

• Label: 812851200.b
• Order: \( 2^{13} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\mathrm{Aut}(G)}$: \( 2^{15} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $42$
• Trans deg.: $42$
• Rank: not computed

Group information

Description:	$\PSL(3,4)\wr C_2$
Order:	\(812851200\)\(\medspace = 2^{13} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(9754214400\)\(\medspace = 2^{15} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \) (generators)
Outer automorphisms:	$D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \)
Composition factors:	$C_2$, $\PSL(3,4)$ x 2
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	20	21	28	35
Elements	1	120015	5022080	23027760	65044224	46569600	33189120	76204800	167650560	16934400	119750400	36126720	60963840	25804800	43545600	92897280	812851200
Conjugacy classes	1	3	2	13	5	2	5	3	4	3	4	2	6	2	6	4	65
Divisions	1	3	2	13	3	2	3	3	2	3	2	1	3	1	3	1	46
Autjugacy classes	1	3	2	5	3	2	3	1	2	1	2	1	1	1	1	1	30

Dimension	1	40	70	90	126	128	180	252	400	1225	1400	1800	2025	2450	2520	2560	3150	3600	3969	4050	4096	4410	4480	5040	5670	5760	6300	7938	8064	8820	11520	16128	22680
Irr. complex chars.	2	1	3	2	2	1	0	0	2	6	3	2	4	3	2	1	6	0	4	1	2	6	3	0	4	2	0	1	2	0	0	0	0	65
Irr. rational chars.	2	1	3	0	0	1	1	1	2	6	3	0	0	3	0	1	0	1	0	3	2	0	3	1	0	0	3	3	0	3	1	1	1	46

Minimal Presentations

Permutation degree:	$42$
Transitive degree:	$42$
Rank:	not computed
Inequivalent generating tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	40	40	40
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $42$ $\langle(1,15,21,11,12)(2,7,14,6,8)(3,4,5,10,16)(9,17,20,18,19)(22,31,39,29,32) \!\cdots\! \rangle$
Transitive group:	42T5012				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Non-split product:	$PSL(3,4)^2$ . $C_2$				more information

Elements of the group are displayed as permutations of degree 42.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{4} \times C_{12}$
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:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^6.C_2$

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 subgroup of index up to 259200 (order at least 3136) are shown, as well as all normal subgroups of any index.

Order 812851200: $\PSL(3,4)\wr C_2$

Order 406425600: $PSL(3,4)^2$

Order 19353600: $C_2^4.A_5\times {\rm PSL}(3,4)$ x 2

Order 7257600: $A_6\times {\rm PSL}(3,4)$ x 3

Order 3870720: $C_4^2:A_4.{\rm PSL}(3,4)$

Order 3386880: $PSL(2,7)\times {\rm PSL}(3,4)$ x 3

Order 3225600: $PSL(3,4)\times C_2^4:D_5$ x 2

Order 1935360: $C_2^2:S_4.{\rm PSL}(3,4)$ x 2

Order 1843200: $C_2^8.\SOPlus(4,4)$ x 2

Order 1612800: $PSL(3,4)\times C_2^4:C_5$, $C_2^4:C_5.{\rm PSL}(3,4)$

Order 1451520: $PSU(3,2)\times {\rm PSL}(3,4)$

Order 1290240: $C_2^2{\rm wrC}_2:C_2.{\rm PSL}(3,4)$

Order 1209600: $PSL(3,4)\times A_5$ x 7

Order 967680: $C_4^2:C_3.{\rm PSL}(3,4)$ x 3, $C_2^4:C_3.{\rm PSL}(3,4)$ x 2

Order 921600: $C_2^8.A_5.A_5$ x 3

Order 725760: $PSL(3,4)\times C_3:S_3.C_2$ x 2, $C_3:S_3.C_2\times {\rm PSL}(3,4)$

Order 645120: $C_4.D_4.{\rm PSL}(3,4)$ x 3, $PSL(3,4)\times C_2^2{\rm wrC}_2$, $C_2^2{\rm wrC}_2.{\rm PSL}(3,4)$

Order 483840: $PSL(3,4)\times S_4$ x 6

Order 423360: $PSL(3,4)\times C_7:C_3$

Order 362880: $PSL(3,4)\times C_3:S_3$

Order 345600: $C_2^4.A_5\times A_6$ x 6

Order 322560: $PSL(3,4)\times C_2^2:C_4$ x 6, $C_4^2\times {\rm PSL}(3,4)$ x 3, $C_2\times {\rm PSL}(3,4)\times D_4$ x 3, $C_2^4\times {\rm PSL}(3,4)$ x 2, $C_2\times {\rm PSL}(3,4)\times Q_8$

Order 259200: $A_6\wr C_2$ x 3

Order 241920: $PSL(3,4)\times A_4$ x 4, $A_4\times {\rm PSL}(3,4)$ x 3

Order 201600: $PSL(3,4)\times D_5$

Order 184320: $C_4^2:A_4.C_2^4.A_5$ x 2

Order 181440: $C_3^2\times {\rm PSL}(3,4)$

Order 161280: $C_2^4.A_5\times {\rm PSL}(2,7)$ x 6, $PSL(3,4)\times D_4$ x 3, $C_2\times C_4\times {\rm PSL}(3,4)$ x 3, $C_2^3\times {\rm PSL}(3,4)$ x 2, $PSL(3,4)\times Q_8$

Order 153600: $C_2^4:D_5.C_2^4.A_5$ x 2, $C_2^4:D_5.C_2.A_5$ x 2

Order 141120: $C_7\times {\rm PSL}(3,4)$

Order 129600: $A_6^2$ x 6

Order 120960: $PSL(3,4)\times S_3$

Order 100800: $C_5\times {\rm PSL}(3,4)$

Order 92160: $C_2^2:S_4.C_2^4.A_5$ x 3, $C_2^4:C_3.C_2^5.A_5$

Order 80640: $C_2^2\times {\rm PSL}(3,4)$ x 7, $C_4\times {\rm PSL}(3,4)$ x 3

Order 76800: $C_2^4:C_5.C_2^4.A_5$ x 4

Order 73728: $C_2^8.A_4{\rm wrC}_2$

Order 69120: $C_4^2:A_4.A_6$ x 3, $C_3:S_3.C_2^6.A_5$ x 2

Order 61440: $C_2^4.C_2^6.A_5$, $(C_2\times D_4).C_2^6.A_5$

Order 60480: $A_6\times \GL(3,2)$ x 9, $C_3\times {\rm PSL}(3,4)$

Order 57600: $C_2^4.A_5.A_5$ x 14, $C_2^4:D_5.A_6$ x 6

Order 56448: $\GL(3,2)\wr C_2$ x 3

Order 51200: $C_2^8.D_5\wr C_2$ x 2

Order 46080: $C_4^2:C_3.C_2^4.A_5$ x 6, $C_2^4:C_3.C_2^4.A_5$ x 4

Order 40320: $\PSL(3,4):C_2$ x 3, $\PSL(3,4).C_2$ x 3, $C_2\times \PSL(3,4)$ x 2, $\PSL(3,4):C_2$

Order 36864: $C_2^8.A_4^2$

Order 34560: $C_3:S_3.C_2^5.A_5$ x 6, $C_2^2:S_4\times A_6$ x 6

Order 32256: $C_4^2:A_4.{\rm PSL}(2,7)$ x 3

Order 30720: $C_4.D_4.C_2^4.A_5$ x 2, $C_2^7.C_2^2.A_5$ x 2, $C_2^6.(D_5\times C_2^4:C_3)$ x 2, $C_2^3.C_2^6.A_5$ x 2, $C_2^8:(C_2\times A_5)$ x 2, $C_2^8:S_5$ x 2, $C_4.D_4.C_2.A_5$, $C_2^6.C_2^3.A_5$, $C_2^4.C_2^5.A_5$, $C_2^2:C_4.C_2^5.A_5$

Order 28800: $C_2^4:C_5.A_6$ x 5, $A_6\times C_2^4:C_5$

Order 28224: $\GL(3,2)^2$ x 6

Order 26880: $C_2^4:D_5.{\rm PSL}(2,7)$ x 6

Order 25920: $A_6\times \PSU(3,2)$ x 3

Order 25600: $C_2^8.D_5^2$ x 3, $C_2^8.C_5:F_5$ x 2, $C_2^8.D_5^2$ x 2

Order 24576: $C_2^8.C_2^4.S_3$, $C_2^8.C_2^4.C_6$

Order 23040: $S_4.C_2^4.A_5$ x 6, $S_4.C_2.A_5$ x 3, $C_2^2{\rm wrC}_2:C_2.A_6$ x 3, $A_4.C_2^5.A_5$ x 3

Order 21600: $A_5\times A_6$ x 21

Order 20160: $\PSL(3,4)$ x 11, $(C_2^4\times C_7:C_3).A_5$ x 2

Order 18432: $C_2^6.(C_2^3\times C_6).C_6$ x 2, $C_2^8.\SOPlus(4,2)$ x 2

Order 17280: $C_4^2:C_3.A_6$ x 9, $C_2^2:A_4\times A_6$ x 6, $(C_2^4\times C_3:S_3).A_5$ x 2

Order 16128: $C_2^2:S_4\times \GL(3,2)$ x 6

Order 15360: $C_2^3.C_2^5.A_5$ x 6, $(C_2^4\times C_4^2).A_5$ x 6, $C_2^8.A_5$ x 6, $C_2^8.C_5.D_6$ x 4, $C_2^7.C_2.A_5$ x 4, $C_2^8.A_5$ x 3, $D_4.C_2^5.A_5$ x 2, $C_4.C_2^6.A_5$ x 2, $C_2^7.C_2^4.A_5$ x 2, $C_2^6.C_2^2.A_5$ x 2, $C_2^2.C_2^6.A_5$ x 2, $C_4^2.C_2^6.C_{15}$, $C_2^8.(C_5\times A_4)$, $C_2^2\times C_2^6.A_5$

Order 13824: $C_6^2.C_2^5.C_6.C_2$

Order 13440: $C_2^4:C_5.{\rm PSL}(2,7)$ x 6

Order 12960: $C_3^2:C_4\times A_6$ x 9

Order 12800: $C_2^4:C_5.C_2^4.C_{10}$ x 4, $C_2^8.C_5.D_5$ x 3, $C_2^8.(C_5\times D_5)$ x 2

Order 12288: $C_2^8.C_2^4.C_3$ x 2, $C_2^6.A_4.C_2^4$

Order 12096: $PSU(3,2).{\rm PSL}(2,7)$ x 3

Order 11520: $(C_2^4\times A_4).A_5$ x 14, $C_4.D_4.A_6$ x 9, $C_4^2:A_4.A_5$ x 7, $C_2^2\wr C_2\times A_6$ x 6, $C_2^4:D_5\times {\rm PSU}(3,2)$ x 2

Order 10752: $C_2^2{\rm wrC}_2:C_2.{\rm PSL}(2,7)$ x 3

Order 10368: $\PSU(3,2)\wr C_2$

Order 10240: $C_2^4:C_5.C_2^5.C_2^2$, $C_2^4:C_5.C_2^4.C_2^3$

Order 10080: $A_5\times \GL(3,2)$ x 21

Order 9600: $A_5\times C_2^4:D_5$ x 14, $(C_2^4\times D_5).A_5$ x 2

Order 9216: $C_2^6.A_4^2$ x 3, $(C_2^2:S_4)^2$ x 3, $C_2^8.S_3^2$ x 2, $C_2^4.A_4^2:C_4$ x 2, $C_4^4.A_4.C_3$, $C_2^8.A_4.C_3$

Order 8640: $S_4\times A_6$ x 18, $(C_2^2\times C_6^2).A_5$ x 2

Order 8192: $C_2^6.C_2^6.C_2$

Order 8064: $C_4^2:C_3.{\rm PSL}(2,7)$ x 9, $C_2^2:A_4\times \GL(3,2)$ x 6

Order 7680: $C_2^4:C_5.C_4^2.C_6$ x 6, $(C_2^5\times C_4).A_5$ x 6, $D_4.C_2^4.A_5$ x 4, $C_2^8.D_{15}$ x 4, $C_2^7.A_5$ x 4, $C_2^4:C_5.C_2^4.C_6$ x 4, $C_2^4:C_3.C_2^4.C_{10}$ x 4, $C_2.C_2^6.A_5$ x 2, $Q_8.C_2.A_5$, $C_4.C_2^5.A_5$

Order 7560: $C_7:C_3\times A_6$ x 3

Order 7200: $\SOPlus(4,4)$ x 7

Order 6912: $C_6^2.C_2^4.C_{12}$ x 3, $C_6^2.(Q_8\times S_4)$ x 2

Order 6720: $(C_2^3\times C_{14}).A_5$ x 2

Order 6480: $C_3:S_3\times A_6$ x 3

Order 6400: $C_2^8.C_5^2$ x 2, $(C_2^4:C_5)^2$

Order 6144: $C_2^6.C_2^4.S_3$ x 4, $C_2^6.C_2^4.C_6$ x 4, $C_2^6.A_4.C_2^3$ x 3, $C_2^4:C_3.C_2^4.C_2^3$ x 2, $C_2^8.S_4$, $C_2^8.A_4.C_2$, $C_2^5.(D_4\times A_4).C_2$, $C_2^4.C_2^4.C_6.C_2^2$

Order 6048: $C_3^2:C_4\times \GL(3,2)$ x 9

Order 5760: $C_2^2:C_4\times A_6$ x 18, $A_5\times C_2^2:S_4$ x 14, $C_4^2\times A_6$ x 9, $C_2\times D_4\times A_6$ x 9, $C_3^2.C_2^4:C_5.Q_8$ x 6, $C_2^4:D_5\times C_3:S_3.C_2$ x 6, $C_2^4\times A_6$ x 6, $C_2\times A_6\times Q_8$ x 3, $C_2^4:C_5\times {\rm PSU}(3,2)$ x 2, $(C_2^4\times S_3).A_5$ x 2

Order 5376: $C_4.D_4.{\rm PSL}(2,7)$ x 9, $C_2^2\wr C_2\times \GL(3,2)$ x 6

Order 5184: $C_3^4:D_4.D_4$ x 3, $C_3^4:C_4^2.C_4$ x 3, $C_3^4.Q_8^2$

Order 5120: $C_2^4.(D_5\times C_4.D_4)$ x 6, $C_2^4:C_5.C_2^3.C_2^3$ x 4, $C_2^8.D_{10}$ x 4, $C_2^4:C_5.C_2^4.C_2^2$ x 3, $C_2^4.C_2^2:D_{20}.C_2$ x 3, $C_2^6.C_{10}.C_2^3$ x 2, $C_2^8.D_{10}$ x 2, $C_2^8.F_5$ x 2

Order 4800: $C_2^4.(C_5\times A_5)$ x 14, $(C_2^3\times C_{10}).A_5$ x 2

Order 4608: $C_2^4:C_3.C_4^2.C_6$ x 6, $(C_2^4\times A_4).A_4.C_2$ x 6, $C_2^4:(A_4\times S_4)$ x 4, $C_2^4:\PSOPlus(4,3)$ x 3, $C_4^4:(C_3\times S_3)$ x 3, $C_2^8:(C_3\times S_3)$ x 2, $C_2^2{\rm wrC}_2:C_2\times {\rm PSU}(3,2)$

Order 4320: $A_4\times A_6$ x 21, $A_5\times \PSU(3,2)$ x 7

Order 4096: $C_2^5.C_2^6.C_2$ x 8, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^6$

Order 4032: $S_4\times \GL(3,2)$ x 18, $(C_2^3\times C_{14}).A_4.C_3$

Order 3840: $C_2^6.A_5$ x 13, $C_2^4:D_5\times S_4$ x 12, $C_2^2{\rm wrC}_2:C_2.A_5$ x 7, $(C_2^4\times C_4).A_5$ x 6, $C_4^2.C_2^4:C_5.C_3$ x 5, $C_2^8.C_{15}$ x 4, $C_2^6.(C_5\times A_4)$, $C_2^2\times C_2^4.A_5$

Order 3600: $A_5^2$ x 28, $D_5\times A_6$ x 3

Order 3528: $C_7:C_3\times \GL(3,2)$ x 3

Order 3456: $C_6^2.A_4:Q_8$ x 6, $C_6^2:(C_4\times S_4)$ x 6, $C_6^2.(Q_8\times A_4)$ x 3, $(C_2^4\times C_3:S_3).C_6.C_2$ x 2, $C_6^2.C_2^4.C_6$

Order 3360: $C_2^4.C_{35}.C_6$ x 2

Order 3240: $C_3^2\times A_6$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 812851200: $\PSL(3,4)\wr C_2$ ($C_1$)

Order 406425600: $PSL(3,4)^2$ ($C_2$)

Order 1: $C_1$ ($\PSL(3,4)\wr C_2$)

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 4 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 $65 \times 65$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.