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G = C12:2D12  order 288 = 25·32

2nd semidirect product of C12 and D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12:2D12, C62.86C23, (C3xC12):8D4, (C6xD12):8C2, C6.43(S3xD4), (C2xD12):11S3, C12:3(C3:D4), C4:Dic3:15S3, C6.80(C2xD12), C4:3(C3:D12), C3:1(D6:3D4), C3:5(C12:D4), D6:Dic3:13C2, (C2xC12).142D6, C32:10(C4:D4), (C2xDic3).35D6, (C22xS3).21D6, C6.12(D4:2S3), C2.18(D6:D6), (C6xC12).109C22, C6.36(Q8:3S3), C2.19(D12:S3), (C6xDic3).19C22, (C2xC4).120S32, (C2xC3:S3):10D4, C6.18(C2xC3:D4), (C2xC3:D12):6C2, (C3xC4:Dic3):19C2, C22.124(C2xS32), (C3xC6).111(C2xD4), (S3xC2xC6).36C22, (C3xC6).53(C4oD4), C2.21(C2xC3:D12), (C2xC6).105(C22xS3), (C22xC3:S3).74C22, (C2xC3:Dic3).139C22, (C2xC4xC3:S3):1C2, SmallGroup(288,564)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C12:2D12
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — C2xC3:D12 — C12:2D12
Lower central
C32 — C62 — C12:2D12
Upper central
C1 — C22 — C2xC4

Generators and relations for C12:2D12
 G = < a,b,c | a12=b12=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >

Subgroups: 946 in 215 conjugacy classes, 54 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3xC6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C4:D4, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C2xC3:S3, C62, C4:Dic3, D6:C4, C6.D4, C3xC4:C4, S3xC2xC4, C2xD12, C2xD12, C2xC3:D4, C6xD4, C3:D12, C3xD12, C6xDic3, C4xC3:S3, C2xC3:Dic3, C6xC12, S3xC2xC6, C22xC3:S3, C12:D4, D6:3D4, D6:Dic3, C3xC4:Dic3, C2xC3:D12, C6xD12, C2xC4xC3:S3, C12:2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C3:D4, C22xS3, C4:D4, S32, C2xD12, S3xD4, D4:2S3, Q8:3S3, C2xC3:D4, C3:D12, C2xS32, C12:D4, D6:3D4, D12:S3, D6:D6, C2xC3:D12, C12:2D12

Smallest permutation representation of C12:2D12
►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 43 34 13 5 39 26 21 9 47 30 17)(2 42 35 24 6 38 27 20 10 46 31 16)(3 41 36 23 7 37 28 19 11 45 32 15)(4 40 25 22 8 48 29 18 12 44 33 14)

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222223334444446···6666666612···1212121212
size11111212181822422121218182···2444121212124···412121212

42 irreducible representations

dim111111222222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C4oD4D12C3:D4S32S3xD4D4:2S3Q8:3S3C3:D12C2xS32D12:S3D6:D6
kernelC12:2D12D6:Dic3C3xC4:Dic3C2xC3:D12C6xD12C2xC4xC3:S3C4:Dic3C2xD12C3xC12C2xC3:S3C2xDic3C2xC12C22xS3C3xC6C12C12C2xC4C6C6C6C4C22C2C2
# reps121211112222224412112122

Matrix representation of C12:2D12 ►in GL8(F13)

83000000
05000000
001200000
000120000
00001100
000012000
000000120
000000012
,
115000000
22000000
001210000
001200000
00001000
0000121200
0000001211
00000011
,
10000000
01000000
001200000
001210000
00001000
0000121200
000000120
00000011

G:=sub<GL(8,GF(13))| [8,0,0,0,0,0,0,0,3,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[11,2,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1] >;
 

C12:2D12 in GAP, Magma, Sage, TeX 

C_{12}\rtimes_2D_{12}
 
% in TeX
 
G:=Group("C12:2D12");
 
// GroupNames label
 
G:=SmallGroup(288,564);
 
// by ID
 
G=gap.SmallGroup(288,564);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^-1>;
 
// generators/relations
 

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