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G = C12:3D4order 96 = 25·3

3rd semidirect product of C12 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12:3D4, Dic3:1D4, C23.15D6, (C2xD4):6S3, (C6xD4):4C2, (C2xD12):9C2, C3:2(C4:1D4), C4:1(C3:D4), (C2xC4).52D6, C6.52(C2xD4), C2.28(S3xD4), (C4xDic3):6C2, (C2xC6).55C23, (C2xC12).35C22, C22.62(C22xS3), (C22xC6).22C22, (C22xS3).12C22, (C2xDic3).39C22, (C2xC3:D4):7C2, C2.16(C2xC3:D4), SmallGroup(96,147)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C12:3D4
C1C3C6C2xC6C22xS3C2xD12 — C12:3D4
C3C2xC6 — C12:3D4
C1C22C2xD4

Generators and relations for C12:3D4
 G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 290 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C42, C2xD4, C2xD4, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C4:1D4, C4xDic3, C2xD12, C2xC3:D4, C6xD4, C12:3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C4:1D4, S3xD4, C2xC3:D4, C12:3D4

Character table of C12:3D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E6F6G12A12B
 size 11114412122226666222444444
ρ1111111111111111111111111    trivial
ρ211111-1-111-1-11-11-11111-1-11-1-1    linear of order 2
ρ311111-11-11-1-1-11-111111-1-11-1-1    linear of order 2
ρ4111111-1-1111-1-1-1-1111111111    linear of order 2
ρ51111-11-111-1-1-11-11111-111-1-1-1    linear of order 2
ρ61111-1-111111-1-1-1-1111-1-1-1-111    linear of order 2
ρ71111-1-1-1-11111111111-1-1-1-111    linear of order 2
ρ81111-111-11-1-11-11-1111-111-1-1-1    linear of order 2
ρ92-22-20000200020-2-22-2000000    orthogonal lifted from D4
ρ102-2-220000200-20202-2-2000000    orthogonal lifted from D4
ρ112-22-200002000-202-22-2000000    orthogonal lifted from D4
ρ1222-2-2000022-20000-2-220000-22    orthogonal lifted from D4
ρ1322222-200-1-2-20000-1-1-1-111-111    orthogonal lifted from D6
ρ1422-2-200002-220000-2-2200002-2    orthogonal lifted from D4
ρ1522222200-1220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ162-2-22000020020-202-2-2000000    orthogonal lifted from D4
ρ172222-2-200-1220000-1-1-11111-1-1    orthogonal lifted from D6
ρ182222-2200-1-2-20000-1-1-11-1-1111    orthogonal lifted from D6
ρ1922-2-20000-12-2000011-1--3--3-3-31-1    complex lifted from C3:D4
ρ2022-2-20000-12-2000011-1-3-3--3--31-1    complex lifted from C3:D4
ρ2122-2-20000-1-22000011-1-3--3-3--3-11    complex lifted from C3:D4
ρ2222-2-20000-1-22000011-1--3-3--3-3-11    complex lifted from C3:D4
ρ234-4-440000-2000000-222000000    orthogonal lifted from S3xD4
ρ244-44-40000-20000002-22000000    orthogonal lifted from S3xD4

Smallest permutation representation of C12:3D4
On 48 points
Generators in S48
(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 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 47 34 24)(2 40 35 17)(3 45 36 22)(4 38 25 15)(5 43 26 20)(6 48 27 13)(7 41 28 18)(8 46 29 23)(9 39 30 16)(10 44 31 21)(11 37 32 14)(12 42 33 19)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)(25 31)(26 30)(27 29)(32 36)(33 35)

G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,47,34,24)(2,40,35,17)(3,45,36,22)(4,38,25,15)(5,43,26,20)(6,48,27,13)(7,41,28,18)(8,46,29,23)(9,39,30,16)(10,44,31,21)(11,37,32,14)(12,42,33,19), (2,12)(3,11)(4,10)(5,9)(6,8)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,31)(26,30)(27,29)(32,36)(33,35)>;

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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,47,34,24)(2,40,35,17)(3,45,36,22)(4,38,25,15)(5,43,26,20)(6,48,27,13)(7,41,28,18)(8,46,29,23)(9,39,30,16)(10,44,31,21)(11,37,32,14)(12,42,33,19), (2,12)(3,11)(4,10)(5,9)(6,8)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,31)(26,30)(27,29)(32,36)(33,35) );

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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,47,34,24),(2,40,35,17),(3,45,36,22),(4,38,25,15),(5,43,26,20),(6,48,27,13),(7,41,28,18),(8,46,29,23),(9,39,30,16),(10,44,31,21),(11,37,32,14),(12,42,33,19)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,48),(24,47),(25,31),(26,30),(27,29),(32,36),(33,35)]])

C12:3D4 is a maximal subgroup of
C23.2D12  D12:1D4  Dic3.SD16  Dic6:2D4  C4:C4.D6  D12:3D4  Dic3:D8  C24:5D4  C24:11D4  Dic3:5SD16  C24:15D4  C24:9D4  D12:18D4  2+ 1+4:6S3  C42.228D6  Dic6:24D4  C42.114D6  C42.116D6  C24:8D6  C24.45D6  C24.47D6  C12:(C4oD4)  Dic6:20D4  C6.382+ 1+4  D12:19D4  C6.442+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.662+ 1+4  C6.672+ 1+4  C6.682+ 1+4  C42.233D6  C42:20D6  C42.143D6  S3xC4:1D4  C42:28D6  Dic6:11D4  D4xC3:D4  C24.52D6  C6.1462+ 1+4  (C2xC12):17D4  C6.1482+ 1+4  C36:D4  C12:D12  C62.84C23  C62.121C23  C62.258C23  C12:D20  C60:10D4  Dic15:5D4  C60:3D4
C12:3D4 is a maximal quotient of
C24.18D6  C24.24D6  C23:3D12  (C4xDic3):8C4  (C2xD12):10C4  (C2xC12).290D4  C42.64D6  C42.214D6  C42.65D6  C12:D8  C42.74D6  C12:4SD16  C12:6SD16  C42.80D6  C12:3Q16  C24:5D4  C24:11D4  C24.22D4  C24.31D4  C24.43D4  C24:15D4  C24:9D4  C24.26D4  C24.37D4  C24.28D4  C24.30D6  C24.32D6  C36:D4  C12:D12  C62.84C23  C62.121C23  C62.258C23  C12:D20  C60:10D4  Dic15:5D4  C60:3D4

Matrix representation of C12:3D4 in GL4(F13) generated by

01200
1000
00121
00120
,
01200
1000
00411
0029
,
1000
01200
0001
0010
G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,12,12,0,0,1,0],[0,1,0,0,12,0,0,0,0,0,4,2,0,0,11,9],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;

C12:3D4 in GAP, Magma, Sage, TeX

C_{12}\rtimes_3D_4
% in TeX

G:=Group("C12:3D4");
// GroupNames label

G:=SmallGroup(96,147);
// by ID

G=gap.SmallGroup(96,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Character table of C12:3D4 in TeX

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