Copied to
clipboard

G = Dic3:4D12order 288 = 25·32

1st semidirect product of Dic3 and D12 acting through Inn(Dic3)

metabelian, supersoluble, monomial

Aliases: Dic3:4D12, C62.50C23, D6:1(C4xS3), C3:2(C4xD12), C32:3(C4xD4), D6:C4:20S3, C2.1(S3xD12), C6.12(S3xD4), C3:D12:1C4, (C3xDic3):7D4, Dic3:3(C4xS3), C6.12(C2xD12), (Dic3xC12):2C2, (C4xDic3):15S3, (C2xC12).197D6, C6.53(C4oD12), (C22xS3).32D6, Dic3:Dic3:13C2, C3:1(Dic3:4D4), C6.40(D4:2S3), C6.11D12:15C2, (C6xC12).228C22, (C2xDic3).111D6, C2.4(D6.3D6), (C6xDic3).153C22, C2.16(C4xS32), (C2xC4).48S32, C6.15(S3xC2xC4), (S3xC6):2(C2xC4), (C2xS3xDic3):8C2, (C3xD6:C4):21C2, C22.30(C2xS32), (C3xC6).43(C2xD4), (C3xDic3):7(C2xC4), (C2xC6.D6):8C2, (S3xC2xC6).13C22, (C2xC3:D12).5C2, (C3xC6).64(C4oD4), (C2xC6).69(C22xS3), (C3xC6).14(C22xC4), (C22xC3:S3).13C22, (C2xC3:Dic3).37C22, (C2xC3:S3):1(C2xC4), SmallGroup(288,528)

Series: Derived Chief Lower central Upper central

C1C3xC6 — Dic3:4D12
C1C3C32C3xC6C62S3xC2xC6C2xS3xDic3 — Dic3:4D12
C32C3xC6 — Dic3:4D12
C1C22C2xC4

Generators and relations for Dic3:4D12
 G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 826 in 205 conjugacy classes, 62 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3xC6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C4xD4, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, D6:C4, C4xC12, C3xC22:C4, S3xC2xC4, C2xD12, C22xDic3, C2xC3:D4, S3xDic3, C6.D6, C3:D12, C6xDic3, C2xC3:Dic3, C6xC12, S3xC2xC6, C22xC3:S3, C4xD12, Dic3:4D4, Dic3:Dic3, Dic3xC12, C3xD6:C4, C6.11D12, C2xS3xDic3, C2xC6.D6, C2xC3:D12, Dic3:4D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22xC4, C2xD4, C4oD4, C4xS3, D12, C22xS3, C4xD4, S32, S3xC2xC4, C2xD12, C4oD12, S3xD4, D4:2S3, C2xS32, C4xD12, Dic3:4D4, C4xS32, S3xD12, D6.3D6, Dic3:4D12

Smallest permutation representation of Dic3:4D12
On 48 points
Generators in S48
(1 23 9 19 5 15)(2 16 6 20 10 24)(3 13 11 21 7 17)(4 18 8 22 12 14)(25 39 29 43 33 47)(26 48 34 44 30 40)(27 41 31 45 35 37)(28 38 36 46 32 42)
(1 27 19 45)(2 28 20 46)(3 29 21 47)(4 30 22 48)(5 31 23 37)(6 32 24 38)(7 33 13 39)(8 34 14 40)(9 35 15 41)(10 36 16 42)(11 25 17 43)(12 26 18 44)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 36)(33 35)(37 43)(38 42)(39 41)(44 48)(45 47)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G4H4I4J4K4L6A···6F6G6H6I6J6K12A12B12C12D12E···12J12K···12R12S12T
order122222223334444444444446···6666661212121212···1212···121212
size1111661818224223333666618182···2444121222224···46···61212

54 irreducible representations

dim111111111222222222224444444
type+++++++++++++++++-++
imageC1C2C2C2C2C2C2C2C4S3S3D4D6D6D6C4oD4C4xS3D12C4xS3C4oD12S32S3xD4D4:2S3C2xS32C4xS32S3xD12D6.3D6
kernelDic3:4D12Dic3:Dic3Dic3xC12C3xD6:C4C6.11D12C2xS3xDic3C2xC6.D6C2xC3:D12C3:D12C4xDic3D6:C4C3xDic3C2xDic3C2xC12C22xS3C3xC6Dic3Dic3D6C6C2xC4C6C6C22C2C2C2
# reps111111118112321244441111222

Matrix representation of Dic3:4D12 in GL6(F13)

1200000
0120000
001000
000100
000011
0000120
,
800000
080000
001000
000100
000005
000050
,
1120000
100000
0012300
008100
000001
000010
,
1200000
1210000
0012000
008100
000001
000010

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,8,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic3:4D12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_4D_{12}
% in TeX

G:=Group("Dic3:4D12");
// GroupNames label

G:=SmallGroup(288,528);
// by ID

G=gap.SmallGroup(288,528);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<