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G = GL2(F3):C4order 192 = 26·3

1st semidirect product of GL2(F3) and C4 acting via C4/C2=C2

non-abelian, soluble

Aliases: GL2(F3):1C4, C2.6(C4xS4), (C2xC4).3S4, (C4xQ8):3S3, Q8.3(C4xS3), Q8:Dic3:5C2, (C2xQ8).10D6, C22.12(C2xS4), C2.1(C4.3S4), (C4xSL2(F3)):1C2, SL2(F3):2(C2xC4), C2.2(Q8.D6), (C2xGL2(F3)).1C2, (C2xSL2(F3)).10C22, SmallGroup(192,953)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — GL2(F3):C4
C1C2Q8SL2(F3)C2xSL2(F3)C2xGL2(F3) — GL2(F3):C4
SL2(F3) — GL2(F3):C4
C1C22C2xC4

Generators and relations for GL2(F3):C4
 G = < a,b,c,d,e | a4=c3=d2=e4=1, b2=a2, bab-1=dbd=a-1, cac-1=ab, dad=a2b, ae=ea, cbc-1=a, be=eb, dcd=c-1, ce=ec, ede-1=a2d >

Subgroups: 327 in 77 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, Dic3, C12, D6, C2xC6, C42, C22:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, SL2(F3), C2xDic3, C2xC12, C22xS3, C8:C4, D4:C4, Q8:C4, C2.D8, C4xD4, C4xQ8, C2xSD16, D6:C4, GL2(F3), C2xSL2(F3), SD16:C4, Q8:Dic3, C4xSL2(F3), C2xGL2(F3), GL2(F3):C4
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, C4xS3, S4, C2xS4, C4xS4, Q8.D6, C4.3S4, GL2(F3):C4

Character table of GL2(F3):C4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C8A8B8C8D12A12B12C12D
 size 1111121282266661212888121212128888
ρ111111111111111111111111111    trivial
ρ21111-1-11111111-1-1111-1-1-1-11111    linear of order 2
ρ31111111-1-11-11-1-1-1111-1-111-1-1-1-1    linear of order 2
ρ41111-1-11-1-11-11-11111111-1-1-1-1-1-1    linear of order 2
ρ511-1-11-11-ii-1-i1ii-i-11-1i-i1-1ii-i-i    linear of order 4
ρ611-1-1-111-ii-1-i1i-ii-11-1-ii-11ii-i-i    linear of order 4
ρ711-1-11-11i-i-1i1-i-ii-11-1-ii1-1-i-iii    linear of order 4
ρ811-1-1-111i-i-1i1-ii-i-11-1i-i-11-i-iii    linear of order 4
ρ9222200-122222200-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ10222200-1-2-22-22-200-1-1-100001111    orthogonal lifted from D6
ρ1122-2-200-12i-2i-22i2-2i001-110000ii-i-i    complex lifted from C4xS3
ρ1222-2-200-1-2i2i-2-2i22i001-110000-i-iii    complex lifted from C4xS3
ρ133333110-3-3-11-11-1-100011-1-10000    orthogonal lifted from C2xS4
ρ14333311033-1-1-1-111000-1-1-1-10000    orthogonal lifted from S4
ρ153333-1-10-3-3-11-1111000-1-1110000    orthogonal lifted from C2xS4
ρ163333-1-1033-1-1-1-1-1-100011110000    orthogonal lifted from S4
ρ1733-3-3-110-3i3i1i-1-i-ii000i-i1-10000    complex lifted from C4xS4
ρ1833-3-3-1103i-3i1-i-1ii-i000-ii1-10000    complex lifted from C4xS4
ρ1933-3-31-103i-3i1-i-1i-ii000i-i-110000    complex lifted from C4xS4
ρ2033-3-31-10-3i3i1i-1-ii-i000-ii-110000    complex lifted from C4xS4
ρ214-4-44001000000001-1-10000-333-3    orthogonal lifted from C4.3S4
ρ224-4-4400-200000000-22200000000    orthogonal lifted from C4.3S4
ρ234-4-44001000000001-1-100003-3-33    orthogonal lifted from C4.3S4
ρ244-44-400-20000000022-200000000    symplectic lifted from Q8.D6, Schur index 2
ρ254-44-400100000000-1-110000--3-3--3-3    complex lifted from Q8.D6
ρ264-44-400100000000-1-110000-3--3-3--3    complex lifted from Q8.D6

Smallest permutation representation of GL2(F3):C4
On 32 points
Generators in S32
(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 32 7 30)(6 31 8 29)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)
(2 11 10)(4 9 12)(5 8 31)(6 29 7)(13 17 20)(15 19 18)(21 25 28)(23 27 26)
(1 14)(2 17)(3 16)(4 19)(5 23)(6 28)(7 21)(8 26)(9 15)(10 20)(11 13)(12 18)(22 32)(24 30)(25 29)(27 31)
(1 30 14 22)(2 31 15 23)(3 32 16 24)(4 29 13 21)(5 19 27 11)(6 20 28 12)(7 17 25 9)(8 18 26 10)

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

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

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

Matrix representation of GL2(F3):C4 in GL7(F73)

0010000
7272720000
1000000
0000010
0000001
00072000
00007200
,
0100000
1000000
7272720000
00007200
0001000
0000001
00000720
,
1000000
0010000
7272720000
0001000
00000720
00000072
0000100
,
72000000
00720000
07200000
00072000
00000720
00007200
0000001
,
46000000
04600000
00460000
000066766
000706666
000667066
0007770

G:=sub<GL(7,GF(73))| [0,72,1,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0],[1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,7,66,7,0,0,0,66,0,7,7,0,0,0,7,66,0,7,0,0,0,66,66,66,0] >;

GL2(F3):C4 in GAP, Magma, Sage, TeX

{\rm GL}_2({\mathbb F}_3)\rtimes C_4
% in TeX

G:=Group("GL(2,3):C4");
// GroupNames label

G:=SmallGroup(192,953);
// by ID

G=gap.SmallGroup(192,953);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1373,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

Export

Character table of GL2(F3):C4 in TeX

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