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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, (C3×C12).30D4, (C2×C12).73D6, C6.33(D6⋊C4), (C2×Dic3).Dic3, C62.29(C2×C4), (C2×Dic6).5S3, (C6×Dic3).1C4, (C6×Dic6).1C2, C4.Dic3.2S3, C12.26(C3⋊D4), C2.4(D6⋊Dic3), (C6×C12).22C22, C4.10(D6⋊S3), C4.20(C3⋊D12), C31(C12.10D4), C33(C12.47D4), C22.4(S3×Dic3), C12.58D6.2C2, C6.3(C6.D4), C323(C4.10D4), (C2×C4).3S32, (C2×C6).67(C4×S3), (C2×C6).5(C2×Dic3), (C3×C4.Dic3).1C2, (C3×C6).27(C22⋊C4), SmallGroup(288,208)

Series: Derived Chief Lower central Upper central

C1C62 — C12.14D12
C1C3C32C3×C6C3×C12C6×C12C6×Dic6 — C12.14D12
C32C3×C6C62 — C12.14D12
C1C2C2×C4

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 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, M4(2) [×2], C2×Q8, C3×C6, C3×C6, C3⋊C8 [×5], C24, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×2], C4.10D4, C3×Dic3 [×2], C3×C12 [×2], C62, C4.Dic3, C4.Dic3 [×3], C3×M4(2), C2×Dic6, C6×Q8, C3×C3⋊C8, C324C8, C3×Dic6 [×2], C6×Dic3 [×2], C6×C12, C12.47D4, C12.10D4, C3×C4.Dic3, C12.58D6, C6×Dic6, C12.14D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], C4.10D4, S32, D6⋊C4, C6.D4, S3×Dic3, 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 35 7 29 13 47 19 41)(2 28 20 34 14 40 8 46)(3 33 9 27 15 45 21 39)(4 26 22 32 16 38 10 44)(5 31 11 25 17 43 23 37)(6 48 24 30 18 36 12 42)

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim111112222222244444444
type+++++++-++-+-+--
imageC1C2C2C2C4S3S3D4Dic3D6D12C3⋊D4C4×S3C4.10D4S32D6⋊S3C3⋊D12S3×Dic3C12.47D4C12.10D4C12.14D12
kernelC12.14D12C3×C4.Dic3C12.58D6C6×Dic6C6×Dic3C4.Dic3C2×Dic6C3×C12C2×Dic3C2×C12C12C12C2×C6C32C2×C4C4C4C22C3C3C1
# 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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