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G = C4×D12order 96 = 25·3

Direct product of C4 and D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×D12, C125D4, C425S3, C31(C4×D4), C42(C4×S3), C124(C2×C4), (C4×C12)⋊7C2, D61(C2×C4), C6.2(C2×D4), C42(D6⋊C4), D6⋊C417C2, (C2×C4).75D6, C2.1(C2×D12), C42(C4⋊Dic3), C4⋊Dic316C2, C6.4(C4○D4), C6.4(C22×C4), (C2×D12).10C2, C2.3(C4○D12), (C2×C6).14C23, (C2×C12).86C22, C22.11(C22×S3), (C22×S3).15C22, (C2×Dic3).25C22, (S3×C2×C4)⋊7C2, C2.6(S3×C2×C4), SmallGroup(96,80)

Series: Derived Chief Lower central Upper central

C1C6 — C4×D12
C1C3C6C2×C6C22×S3C2×D12 — C4×D12
C3C6 — C4×D12
C1C2×C4C42

Generators and relations for C4×D12
 G = < a,b,c | a4=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 218 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], Dic3 [×2], C12 [×4], C12, D6 [×4], D6 [×4], C2×C6, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C4×D4, C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C4×D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], D12 [×2], C22×S3, C4×D4, S3×C2×C4, C2×D12, C4○D12, C4×D12

Smallest permutation representation of C4×D12
On 48 points
Generators in S48
(1 25 17 47)(2 26 18 48)(3 27 19 37)(4 28 20 38)(5 29 21 39)(6 30 22 40)(7 31 23 41)(8 32 24 42)(9 33 13 43)(10 34 14 44)(11 35 15 45)(12 36 16 46)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 27)(28 36)(29 35)(30 34)(31 33)(37 47)(38 46)(39 45)(40 44)(41 43)

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

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

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

C4×D12 is a maximal subgroup of
C4.17D24  D122C8  C86D12  C89D12  C42.16D6  D24⋊C4  D12⋊C8  D63M4(2)  C122M4(2)  C12⋊SD16  D123Q8  C4⋊D24  D12.19D4  D124Q8  D12.3Q8  C42.48D6  C42.56D6  D12.23D4  D12.4Q8  C122D8  C125SD16  D125Q8  D126Q8  C42.276D6  C42.277D6  C429D6  C42.91D6  C4210D6  C4212D6  C42.93D6  C42.95D6  C42.99D6  C42.100D6  C4×S3×D4  C4213D6  C4214D6  C42.228D6  D1223D4  D1224D4  D45D12  D46D12  C42.113D6  C42.116D6  C42.117D6  C42.119D6  C42.126D6  Q86D12  Q87D12  D1210Q8  C42.131D6  C42.132D6  C42.133D6  C42.135D6  C42.136D6  D1210D4  Dic610D4  C4222D6  C42.143D6  D127Q8  C42.150D6  C42.152D6  C42.153D6  C4225D6  C4226D6  C42.161D6  C42.163D6  D1211D4  Dic611D4  D1212D4  D128Q8  D129Q8  C42.177D6  C42.179D6  Dic34D12  Dic35D12  Dic54D12  D6017C4
C4×D12 is a maximal quotient of
C2.(C4×D12)  (C2×C4)⋊9D12  D6⋊C4⋊C4  D6⋊C43C4  C86D12  D2411C4  C89D12  C42.16D6  D24⋊C4  Dic12⋊C4  D244C4  C124(C4⋊C4)  (C2×C4)⋊6D12  (C2×C42)⋊3S3  Dic34D12  Dic35D12  Dic54D12  D6017C4

36 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C12A···12L
order12222222344444444444466612···12
size1111666621111222266662222···2

36 irreducible representations

dim11111112222222
type++++++++++
imageC1C2C2C2C2C2C4S3D4D6C4○D4C4×S3D12C4○D12
kernelC4×D12C4⋊Dic3D6⋊C4C4×C12S3×C2×C4C2×D12D12C42C12C2×C4C6C4C4C2
# reps11212181232444

Matrix representation of C4×D12 in GL3(𝔽13) generated by

800
080
008
,
1200
0310
036
,
1200
011
0012
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[12,0,0,0,3,3,0,10,6],[12,0,0,0,1,0,0,1,12] >;

C4×D12 in GAP, Magma, Sage, TeX

C_4\times D_{12}
% in TeX

G:=Group("C4xD12");
// GroupNames label

G:=SmallGroup(96,80);
// by ID

G=gap.SmallGroup(96,80);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,50,2309]);
// Polycyclic

G:=Group<a,b,c|a^4=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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