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G = D12.3D4  order 192 = 26·3

3rd non-split extension by D12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.3D4, Dic6.3D4, M4(2).2D6, C3:C8.25D4, C8:D6:6C2, D12.C4:6C2, (C2xD4).16D6, C4.D4:4S3, C4.149(S3xD4), C12.94(C2xD4), D12:6C22:1C2, C3:1(D4.4D4), (C2xC12).6C23, C12.53D4:2C2, C6.10(C4:D4), C4oD12.4C22, (C6xD4).16C22, C12.46D4:10C2, C2.13(Dic3:D4), (C2xD12).39C22, C4.Dic3.3C22, C22.14(C4oD12), (C3xM4(2)).19C22, (C2xD4:S3):1C2, (C2xC3:C8).2C22, (C3xC4.D4):2C2, (C2xC4).6(C22xS3), (C2xC6).31(C4oD4), SmallGroup(192,308)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — D12.3D4
Chief series
C1 — C3 — C6 — C12 — C2xC12 — C4oD12 — D12.C4 — D12.3D4
Lower central
C3 — C6 — C2xC12 — D12.3D4
Upper central
C1 — C2 — C2xC4 — C4.D4

Generators and relations for D12.3D4
 G = < a,b,c,d | a12=b2=1, c4=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a9c3 >

Subgroups: 368 in 108 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, D6, C2xC6, C2xC6, C2xC8, M4(2), M4(2), D8, SD16, C2xD4, C2xD4, C4oD4, C3:C8, C3:C8, C24, Dic6, C4xS3, D12, D12, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C4.D4, C4.D4, C8.C4, C8oD4, C2xD8, C8:C22, S3xC8, C8:S3, C24:C2, D24, C2xC3:C8, C4.Dic3, D4:S3, D4.S3, C3xM4(2), C2xD12, C4oD12, C6xD4, D4.4D4, C12.53D4, C12.46D4, C3xC4.D4, D12.C4, C8:D6, C2xD4:S3, D12:6C22, D12.3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C22xS3, C4:D4, C4oD12, S3xD4, D4.4D4, Dic3:D4, D12.3D4

Character table of D12.3D4

 class 12A2B2C2D2E34A4B4C6A6B6C6D8A8B8C8D8E8F8G12A12B24A24B24C24D
 size 11281224222122488446681224448888
ρ1111111111111111111111111111    trivial
ρ211111-111111111-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ3111-111111111-1-1-1-1-1-11-1-1111-1-11    linear of order 2
ρ4111-11-1111111-1-11111-11-111-111-1    linear of order 2
ρ5111-1-11111-111-1-111-1-1-1-1111-111-1    linear of order 2
ρ6111-1-1-1111-111-1-1-1-111111111-1-11    linear of order 2
ρ71111-11111-11111-1-111-11-111-1-1-1-1    linear of order 2
ρ81111-1-1111-1111111-1-11-1-1111111    linear of order 2
ρ9222200-1220-1-1-1-12200200-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-202022-2-22-20000000002-20000    orthogonal lifted from D4
ρ11222-200-1220-1-111-2-200200-1-1-111-1    orthogonal lifted from D6
ρ12222-200-1220-1-1112200-200-1-11-1-11    orthogonal lifted from D6
ρ1322-20002-2202-20000-2-2020-220000    orthogonal lifted from D4
ρ14222200-1220-1-1-1-1-2-200-200-1-11111    orthogonal lifted from D6
ρ1522-20-2022-222-20000000002-20000    orthogonal lifted from D4
ρ1622-20002-2202-20000220-20-220000    orthogonal lifted from D4
ρ172220002-2-2022002i-2i00000-2-202i-2i0    complex lifted from C4oD4
ρ182220002-2-202200-2i2i00000-2-20-2i2i0    complex lifted from C4oD4
ρ19222000-1-2-20-1-1√-3-√-32i-2i0000011-√3-ii√3    complex lifted from C4oD12
ρ20222000-1-2-20-1-1-√-3√-32i-2i0000011√3-ii-√3    complex lifted from C4oD12
ρ21222000-1-2-20-1-1-√-3√-3-2i2i0000011-√3i-i√3    complex lifted from C4oD12
ρ22222000-1-2-20-1-1√-3-√-3-2i2i0000011√3i-i-√3    complex lifted from C4oD12
ρ2344-4000-24-40-22000000000-220000    orthogonal lifted from S3xD4
ρ2444-4000-2-440-220000000002-20000    orthogonal lifted from S3xD4
ρ254-400004000-4000002√2-2√2000000000    orthogonal lifted from D4.4D4
ρ264-400004000-400000-2√22√2000000000    orthogonal lifted from D4.4D4
ρ278-80000-400040000000000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of D12.3D4
►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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

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

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

Matrix representation of D12.3D4 ►in GL6(F73)

0720000
1720000
001200
00727200
006565723
00460481
,
7210000
010000
006107041
0060380
0048251255
00574100
,
7200000
0720000
00460480
000001
007070270
00727200
,
7200000
0720000
00120332
006703541
0025486118
0057000

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,72,65,46,0,0,2,72,65,0,0,0,0,0,72,48,0,0,0,0,3,1],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,61,6,48,57,0,0,0,0,25,41,0,0,70,38,12,0,0,0,41,0,55,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,70,72,0,0,0,0,70,72,0,0,48,0,27,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,67,25,57,0,0,0,0,48,0,0,0,3,35,61,0,0,0,32,41,18,0] >;
 

D12.3D4 in GAP, Magma, Sage, TeX 

D_{12}._3D_4
 
% in TeX
 
G:=Group("D12.3D4");
 
// GroupNames label
 
G:=SmallGroup(192,308);
 
// by ID
 
G=gap.SmallGroup(192,308);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,297,136,1684,851,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^3>;
 
// generators/relations
 

Export

Character table of D12.3D4 in TeX

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