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G = C12⋊D12  order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C12⋊1D12, Dic3⋊1D12, C62.81C23, (C3×C12)⋊6D4, (C6×D12)⋊6C2, (C2×D12)⋊5S3, C6.24(S3×D4), C12⋊6(C3⋊D4), (C3×Dic3)⋊6D4, (C4×Dic3)⋊7S3, C2.26(S3×D12), C6.25(C2×D12), C4⋊1(C3⋊D12), C3⋊1(C12⋊3D4), C3⋊2(C4⋊D12), (C2×C12).140D6, C32⋊2(C4⋊1D4), (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

Derived series
C1 — C62 — C12⋊D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C12⋊D12
Lower central
C32 — C62 — C12⋊D12
Upper central
C1 — C22 — C2×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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C4⋊1D4, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C4×Dic3, C4×C12, C2×D12, C2×D12, C2×C3⋊D4, C6×D4, C3⋊D12, C3×D12, C6×Dic3, C12⋊S3, C6×C12, S3×C2×C6, C22×C3⋊S3, C4⋊D12, C12⋊3D4, Dic3×C12, C2×C3⋊D12, C6×D12, C2×C12⋊S3, C12⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C4⋊1D4, S32, C2×D12, S3×D4, C2×C3⋊D4, C3⋊D12, C2×S32, C4⋊D12, C12⋊3D4, S3×D12, 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 39 29 16 5 47 33 24 9 43 25 20)(2 44 30 21 6 40 34 17 10 48 26 13)(3 37 31 14 7 45 35 22 11 41 27 18)(4 42 32 19 8 38 36 15 12 46 28 23)

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

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
 

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