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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(C4×S4), (C2×C4).3S4, (C4×Q8)⋊3S3, Q8.3(C4×S3), Q8⋊Dic35C2, (C2×Q8).10D6, C22.12(C2×S4), C2.1(C4.3S4), (C4×SL2(𝔽3))⋊1C2, SL2(𝔽3)⋊2(C2×C4), C2.2(Q8.D6), (C2×GL2(𝔽3)).1C2, (C2×SL2(𝔽3)).10C22, SmallGroup(192,953)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — GL2(𝔽3)⋊C4
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3)C2×GL2(𝔽3) — GL2(𝔽3)⋊C4
SL2(𝔽3) — GL2(𝔽3)⋊C4
C1C22C2×C4

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, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, D6⋊C4, GL2(𝔽3), C2×SL2(𝔽3), SD16⋊C4, Q8⋊Dic3, C4×SL2(𝔽3), C2×GL2(𝔽3), GL2(𝔽3)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4, 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 C4×S3
ρ1222-2-200-1-2i2i-2-2i22i001-110000-i-iii    complex lifted from C4×S3
ρ133333110-3-3-11-11-1-100011-1-10000    orthogonal lifted from C2×S4
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 C2×S4
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 C4×S4
ρ1833-3-3-1103i-3i1-i-1ii-i000-ii1-10000    complex lifted from C4×S4
ρ1933-3-31-103i-3i1-i-1i-ii000i-i-110000    complex lifted from C4×S4
ρ2033-3-31-10-3i3i1i-1-ii-i000-ii-110000    complex lifted from C4×S4
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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