C2^2xGL(2,3) - GroupNames

G = C₂²×GL₂(𝔽₃) order 192 = 2⁶·3

Direct product of C₂² and GL₂(𝔽₃)

Aliases: C₂²×GL₂(𝔽₃), C₂³.₂₁S₄, SL₂(𝔽₃)⋊₁C₂³, (C₂×Q₈)⋊₂D₆, Q₈⋊(C₂²×S₃), C₂.₉(C₂²×S₄), (C₂²×Q₈)⋊₃S₃, C₂².₂₆(C₂×S₄), (C₂×SL₂(𝔽₃))⋊₄C₂², (C₂²×SL₂(𝔽₃))⋊₅C₂, SmallGroup(192,1475)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — Q₈ — SL₂(𝔽₃) — C₂²×GL₂(𝔽₃)

Chief series

C₁ — C₂ — Q₈ — SL₂(𝔽₃) — GL₂(𝔽₃) — C₂×GL₂(𝔽₃) — C₂²×GL₂(𝔽₃)

Lower central

SL₂(𝔽₃) — C₂²×GL₂(𝔽₃)

Upper central

C₁ — C₂³

Generators and relations for C₂²×GL₂(𝔽₃)
G = < a,b,c,d,e,f | a²=b²=c⁴=e³=f²=1, d²=c², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd^-1=fdf=c^-1, ece^-1=cd, fcf=c²d, ede^-1=c, fef=e^-1 >

Subgroups: 875 in 213 conjugacy classes, 37 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, D₆, C₂×C₆, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, SL₂(𝔽₃), C₂²×S₃, C₂²×C₆, C₂²×C₈, C₂×SD₁₆, C₂²×D₄, C₂²×Q₈, GL₂(𝔽₃), C₂×SL₂(𝔽₃), S₃×C₂³, C₂²×SD₁₆, C₂×GL₂(𝔽₃), C₂²×SL₂(𝔽₃), C₂²×GL₂(𝔽₃)
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, S₄, C₂²×S₃, GL₂(𝔽₃), C₂×S₄, C₂×GL₂(𝔽₃), C₂²×S₄, C₂²×GL₂(𝔽₃)

Smallest permutation representation of C₂²×GL₂(𝔽₃)
►On 32 points

Generators in S₃₂

(1 22)(2 23)(3 24)(4 21)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)

(1 14)(2 15)(3 16)(4 13)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)

(1 11 3 9)(2 10 4 12)(5 30 7 32)(6 29 8 31)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)

(2 11 10)(4 9 12)(5 8 29)(6 31 7)(13 19 18)(15 17 20)(21 27 26)(23 25 28)

(1 14)(2 19)(3 16)(4 17)(5 21)(6 28)(7 23)(8 26)(9 15)(10 18)(11 13)(12 20)(22 32)(24 30)(25 31)(27 29)

G:=sub<Sym(32)| (1,22)(2,23)(3,24)(4,21)(5,17)(6,18)(7,19)(8,20)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30), (1,14)(2,15)(3,16)(4,13)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,11,3,9)(2,10,4,12)(5,30,7,32)(6,29,8,31)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27), (2,11,10)(4,9,12)(5,8,29)(6,31,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28), (1,14)(2,19)(3,16)(4,17)(5,21)(6,28)(7,23)(8,26)(9,15)(10,18)(11,13)(12,20)(22,32)(24,30)(25,31)(27,29)>;

G:=Group( (1,22)(2,23)(3,24)(4,21)(5,17)(6,18)(7,19)(8,20)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30), (1,14)(2,15)(3,16)(4,13)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,11,3,9)(2,10,4,12)(5,30,7,32)(6,29,8,31)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27), (2,11,10)(4,9,12)(5,8,29)(6,31,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28), (1,14)(2,19)(3,16)(4,17)(5,21)(6,28)(7,23)(8,26)(9,15)(10,18)(11,13)(12,20)(22,32)(24,30)(25,31)(27,29) );

G=PermutationGroup([[(1,22),(2,23),(3,24),(4,21),(5,17),(6,18),(7,19),(8,20),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30)], [(1,14),(2,15),(3,16),(4,13),(5,25),(6,26),(7,27),(8,28),(9,19),(10,20),(11,17),(12,18),(21,31),(22,32),(23,29),(24,30)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,11,3,9),(2,10,4,12),(5,30,7,32),(6,29,8,31),(13,18,15,20),(14,17,16,19),(21,26,23,28),(22,25,24,27)], [(2,11,10),(4,9,12),(5,8,29),(6,31,7),(13,19,18),(15,17,20),(21,27,26),(23,25,28)], [(1,14),(2,19),(3,16),(4,17),(5,21),(6,28),(7,23),(8,26),(9,15),(10,18),(11,13),(12,20),(22,32),(24,30),(25,31),(27,29)]])

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	3	4A	4B	4C	4D	6A	···	6G	8A	···	8H
order	1	2	···	2	2	2	2	2	3	4	4	4	4	6	···	6	8	···	8
size	1	1	···	1	12	12	12	12	8	6	6	6	6	8	···	8	6	···	6

32 irreducible representations

dim	1	1	1	2	2	2	3	3	4
type	+	+	+	+	+		+	+	+
image	C₁	C₂	C₂	S₃	D₆	GL₂(𝔽₃)	S₄	C₂×S₄	GL₂(𝔽₃)
kernel	C₂²×GL₂(𝔽₃)	C₂×GL₂(𝔽₃)	C₂²×SL₂(𝔽₃)	C₂²×Q₈	C₂×Q₈	C₂²	C₂³	C₂²	C₂²
# reps	1	6	1	1	3	8	2	6	4

Matrix representation of C₂²×GL₂(𝔽₃) ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	53	52
0	0	0	0	33	20

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	53	40
0	0	0	0	21	20

0	72	0	0	0	0
1	72	0	0	0	0
0	0	0	72	0	0
0	0	1	72	0	0
0	0	0	0	0	72
0	0	0	0	1	72

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	72	0	0
0	0	72	0	0	0
0	0	0	0	0	72
0	0	0	0	72	0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,33,0,0,0,0,52,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,21,0,0,0,0,40,20],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

C₂²×GL₂(𝔽₃) in GAP, Magma, Sage, TeX

C_2^2\times {\rm GL}_2({\mathbb F}_3)

% in TeX

G:=Group("C2^2xGL(2,3)");

// GroupNames label

G:=SmallGroup(192,1475);

// by ID

G=gap.SmallGroup(192,1475);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,451,1684,655,172,1013,404,285,124]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^3=f^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*d*f=c^-1,e*c*e^-1=c*d,f*c*f=c^2*d,e*d*e^-1=c,f*e*f=e^-1>;

// generators/relations

G = C22×GL2(𝔽3) order 192 = 26·3

Direct product of C22 and GL2(𝔽3)

G = C₂²×GL₂(𝔽₃) order 192 = 2⁶·3

Direct product of C₂² and GL₂(𝔽₃)