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G = C12.14D12order 288 = 25·32

14th non-split extension by C12 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.14D12, (C3xC12).30D4, (C2xC12).73D6, C6.33(D6:C4), (C2xDic3).Dic3, C62.29(C2xC4), (C2xDic6).5S3, (C6xDic3).1C4, (C6xDic6).1C2, C4.Dic3.2S3, C12.26(C3:D4), C2.4(D6:Dic3), (C6xC12).22C22, C4.10(D6:S3), C4.20(C3:D12), C3:1(C12.10D4), C3:3(C12.47D4), C22.4(S3xDic3), C12.58D6.2C2, C6.3(C6.D4), C32:3(C4.10D4), (C2xC4).3S32, (C2xC6).67(C4xS3), (C2xC6).5(C2xDic3), (C3xC4.Dic3).1C2, (C3xC6).27(C22:C4), SmallGroup(288,208)

Series: Derived Chief Lower central Upper central

C1C62 — C12.14D12
C1C3C32C3xC6C3xC12C6xC12C6xDic6 — C12.14D12
C32C3xC6C62 — C12.14D12
C1C2C2xC4

Generators and relations for C12.14D12
 G = < a,b,c | a12=1, b12=a6, c2=a3, bab-1=a-1, cac-1=a5, cbc-1=a3b11 >

Subgroups: 242 in 86 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2xC4, C2xC4, Q8, C32, Dic3, C12, C12, C2xC6, C2xC6, M4(2), C2xQ8, C3xC6, C3xC6, C3:C8, C24, Dic6, C2xDic3, C2xC12, C2xC12, C3xQ8, C4.10D4, C3xDic3, C3xC12, C62, C4.Dic3, C4.Dic3, C3xM4(2), C2xDic6, C6xQ8, C3xC3:C8, C32:4C8, C3xDic6, C6xDic3, C6xC12, C12.47D4, C12.10D4, C3xC4.Dic3, C12.58D6, C6xDic6, C12.14D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C4xS3, D12, C2xDic3, C3:D4, C4.10D4, S32, D6:C4, C6.D4, S3xDic3, D6:S3, C3:D12, C12.47D4, C12.10D4, D6:Dic3, C12.14D12

Smallest permutation representation of C12.14D12
On 48 points
Generators in S48
(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)(25 47 45 43 41 39 37 35 33 31 29 27)(26 28 30 32 34 36 38 40 42 44 46 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 45 7 39 13 33 19 27)(2 38 20 44 14 26 8 32)(3 43 9 37 15 31 21 25)(4 36 22 42 16 48 10 30)(5 41 11 35 17 29 23 47)(6 34 24 40 18 46 12 28)

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

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

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

39 conjugacy classes

class 1 2A2B3A3B3C4A4B4C4D6A6B6C6D6E6F6G6H8A8B8C8D12A12B12C···12I12J12K12L12M24A24B24C24D
order1223334444666666668888121212···121212121224242424
size1122242212122222444412123636224···41212121212121212

39 irreducible representations

dim111112222222244444444
type+++++++-++-+-+--
imageC1C2C2C2C4S3S3D4Dic3D6D12C3:D4C4xS3C4.10D4S32D6:S3C3:D12S3xDic3C12.47D4C12.10D4C12.14D12
kernelC12.14D12C3xC4.Dic3C12.58D6C6xDic6C6xDic3C4.Dic3C2xDic6C3xC12C2xDic3C2xC12C12C12C2xC6C32C2xC4C4C4C22C3C3C1
# reps111141122226211111224

Matrix representation of C12.14D12 in GL4(F73) generated by

321208
04900
00700
00024
,
8122120
21657065
00064
00490
,
65616540
00490
06500
52838
G:=sub<GL(4,GF(73))| [3,0,0,0,21,49,0,0,20,0,70,0,8,0,0,24],[8,21,0,0,12,65,0,0,21,70,0,49,20,65,64,0],[65,0,0,52,61,0,65,8,65,49,0,3,40,0,0,8] >;

C12.14D12 in GAP, Magma, Sage, TeX

C_{12}._{14}D_{12}
% in TeX

G:=Group("C12.14D12");
// GroupNames label

G:=SmallGroup(288,208);
// by ID

G=gap.SmallGroup(288,208);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations

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