Copied to
clipboard

G = C12⋊D12order 288 = 25·32

1st semidirect product of C12 and D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C121D12, Dic31D12, C62.81C23, (C3×C12)⋊6D4, (C6×D12)⋊6C2, (C2×D12)⋊5S3, C6.24(S3×D4), C126(C3⋊D4), (C3×Dic3)⋊6D4, (C4×Dic3)⋊7S3, C2.26(S3×D12), C6.25(C2×D12), C41(C3⋊D12), C31(C123D4), C32(C4⋊D12), (C2×C12).140D6, C322(C41D4), (Dic3×C12)⋊12C2, (C22×S3).16D6, (C6×C12).107C22, (C2×Dic3).101D6, (C6×Dic3).143C22, (C2×C4).83S32, C6.17(C2×C3⋊D4), (C2×C3⋊D12)⋊4C2, C22.119(C2×S32), (C3×C6).107(C2×D4), (C2×C12⋊S3)⋊12C2, (S3×C2×C6).31C22, C2.20(C2×C3⋊D12), (C2×C6).100(C22×S3), (C22×C3⋊S3).25C22, SmallGroup(288,559)

Series: Derived Chief Lower central Upper central

C1C62 — C12⋊D12
C1C3C32C3×C6C62S3×C2×C6C2×C3⋊D12 — C12⋊D12
C32C62 — C12⋊D12
C1C22C2×C4

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

Subgroups: 1250 in 243 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×12], S3 [×10], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×2], D4 [×12], C23 [×4], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×26], C2×C6 [×2], C2×C6 [×7], C42, C2×D4 [×6], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], D12 [×18], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C41D4, C3×Dic3 [×4], C3×C12 [×2], S3×C6 [×6], C2×C3⋊S3 [×6], C62, C4×Dic3, C4×C12, C2×D12, C2×D12 [×7], C2×C3⋊D4 [×4], C6×D4, C3⋊D12 [×8], C3×D12 [×2], C6×Dic3 [×2], C12⋊S3 [×2], C6×C12, S3×C2×C6 [×2], C22×C3⋊S3 [×2], C4⋊D12, C123D4, Dic3×C12, C2×C3⋊D12 [×4], C6×D12, C2×C12⋊S3, C12⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×6], C3⋊D4 [×2], C22×S3 [×2], C41D4, S32, C2×D12 [×3], S3×D4 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C4⋊D12, C123D4, S3×D12 [×2], C2×C3⋊D12, C12⋊D12

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

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A12B12C12D12E···12J12K···12R
order122222223334444446···666666661212121212···1212···12
size1111121236362242266662···24441212121222224···46···6

48 irreducible representations

dim11111222222222244444
type+++++++++++++++++++
imageC1C2C2C2C2S3S3D4D4D6D6D6D12D12C3⋊D4S32S3×D4C3⋊D12C2×S32S3×D12
kernelC12⋊D12Dic3×C12C2×C3⋊D12C6×D12C2×C12⋊S3C4×Dic3C2×D12C3×Dic3C3×C12C2×Dic3C2×C12C22×S3Dic3C12C12C2×C4C6C4C22C2
# reps11411114222284412214

Matrix representation of C12⋊D12 in GL8(𝔽13)

120000000
012000000
00130000
008120000
000012000
000001200
0000001212
00000010
,
01000000
120000000
001200000
000120000
000011200
00001000
00000010
0000001212
,
120000000
01000000
00100000
008120000
000012000
000012100
00000010
0000001212

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

C12⋊D12 in GAP, Magma, Sage, TeX

C_{12}\rtimes D_{12}
% in TeX

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

G:=SmallGroup(288,559);
// by ID

G=gap.SmallGroup(288,559);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,100,1356,9414]);
// Polycyclic

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

׿
×
𝔽