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G = C12.D12order 288 = 25·32

13rd non-split extension by C12 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.13D12, (C2×D12).5S3, (C6×D12).1C2, (C2×C12).71D6, (C3×C12).28D4, C4.Dic32S3, C6.32(D6⋊C4), C62.27(C2×C4), (C22×S3).Dic3, C4.9(D6⋊S3), C12.58D62C2, C31(C12.D4), C12.24(C3⋊D4), C2.3(D6⋊Dic3), (C6×C12).20C22, C322(C4.D4), C4.19(C3⋊D12), C33(C12.46D4), C22.3(S3×Dic3), C6.2(C6.D4), (C2×C4).1S32, (S3×C2×C6).1C4, (C2×C6).66(C4×S3), (C2×C6).4(C2×Dic3), (C3×C4.Dic3)⋊13C2, (C3×C6).25(C22⋊C4), SmallGroup(288,206)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C12.D12
 G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b11 >

Subgroups: 370 in 102 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×6], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×5], M4(2) [×2], C2×D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×5], C24, D12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4.D4, C3×C12 [×2], S3×C6 [×4], C62, C4.Dic3, C4.Dic3 [×3], C3×M4(2), C2×D12, C6×D4, C3×C3⋊C8, C324C8, C3×D12 [×2], C6×C12, S3×C2×C6 [×2], C12.46D4, C12.D4, C3×C4.Dic3, C12.58D6, C6×D12, C12.D12
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.D4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C12.46D4, C12.D4, D6⋊Dic3, C12.D12

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

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

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

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

39 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B6A6B6C6D6E6F6G6H6I6J6K6L8A8B8C8D12A12B12C···12I24A24B24C24D
order12222333446666666666668888121212···1224242424
size112121222422222244441212121212123636224···412121212

39 irreducible representations

dim111112222222244444444
type++++++++-+++-+-+
imageC1C2C2C2C4S3S3D4D6Dic3D12C3⋊D4C4×S3C4.D4S32D6⋊S3C3⋊D12S3×Dic3C12.46D4C12.D4C12.D12
kernelC12.D12C3×C4.Dic3C12.58D6C6×D12S3×C2×C6C4.Dic3C2×D12C3×C12C2×C12C22×S3C12C12C2×C6C32C2×C4C4C4C22C3C3C1
# reps111141122226211111224

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

24000
07000
015490
15003
,
06400
24000
1008
027700
,
10057
02760
029460
200072
G:=sub<GL(4,GF(73))| [24,0,0,15,0,70,15,0,0,0,49,0,0,0,0,3],[0,24,1,0,64,0,0,27,0,0,0,70,0,0,8,0],[1,0,0,20,0,27,29,0,0,6,46,0,57,0,0,72] >;

C12.D12 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,206);
// by ID

G=gap.SmallGroup(288,206);
# by ID

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

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,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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