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G = C6×C4○D12  order 288 = 25·32

Direct product of C6 and C4○D12

direct product, metabelian, supersoluble, monomial

Aliases: C6×C4○D12, C62.268C23, (C2×C12)⋊32D6, (C6×D12)⋊30C2, (C2×D12)⋊14C6, D12⋊12(C2×C6), C6.4(C23×C6), (C22×C12)⋊17S3, (C22×C12)⋊12C6, (C6×C12)⋊33C22, Dic6⋊11(C2×C6), (C2×Dic6)⋊15C6, (C6×Dic6)⋊31C2, C23.36(S3×C6), C6.72(S3×C23), (C3×C6).41C24, D6.1(C22×C6), (S3×C12)⋊23C22, (C3×D12)⋊40C22, (S3×C6).28C23, C12.43(C22×C6), (C22×C6).132D6, (C3×C12).183C23, C12.222(C22×S3), (C3×Dic6)⋊38C22, Dic3.2(C22×C6), (C2×C62).118C22, (C3×Dic3).29C23, (C6×Dic3).163C22, (S3×C2×C4)⋊15C6, (C2×C6×C12)⋊11C2, C3⋊1(C6×C4○D4), C6⋊1(C3×C4○D4), C4.43(S3×C2×C6), (S3×C2×C12)⋊31C2, (C4×S3)⋊6(C2×C6), (C2×C4)⋊10(S3×C6), C3⋊D4⋊6(C2×C6), (C2×C12)⋊13(C2×C6), C22.5(S3×C2×C6), C2.5(S3×C22×C6), (C3×C6)⋊7(C4○D4), (C6×C3⋊D4)⋊26C2, (C2×C3⋊D4)⋊12C6, C32⋊13(C2×C4○D4), (C22×C4)⋊10(C3×S3), (C3×C3⋊D4)⋊19C22, (S3×C2×C6).109C22, (C22×C6).71(C2×C6), (C2×C6).70(C22×C6), (C22×S3).29(C2×C6), (C2×C6).164(C22×S3), (C2×Dic3).44(C2×C6), SmallGroup(288,991)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C6×C4○D12
Chief series
C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C2×C12 — C6×C4○D12
Lower central
C3 — C6 — C6×C4○D12
Upper central
C1 — C2×C12 — C22×C12

Generators and relations for C6×C4○D12
 G = < a,b,c,d | a6=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 698 in 355 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C2×C4○D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, C6×C12, S3×C2×C6, C2×C62, C2×C4○D12, C6×C4○D4, C6×Dic6, S3×C2×C12, C6×D12, C3×C4○D12, C6×C3⋊D4, C2×C6×C12, C6×C4○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C24, C3×S3, C22×S3, C22×C6, C2×C4○D4, S3×C6, C4○D12, C3×C4○D4, S3×C23, C23×C6, S3×C2×C6, C2×C4○D12, C6×C4○D4, C3×C4○D12, S3×C22×C6, C6×C4○D12

Smallest permutation representation of C6×C4○D12
►On 48 points
Generators in S48
(1 22 5 14 9 18)(2 23 6 15 10 19)(3 24 7 16 11 20)(4 13 8 17 12 21)(25 41 33 37 29 45)(26 42 34 38 30 46)(27 43 35 39 31 47)(28 44 36 40 32 48)

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

108 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C3D3E4A4B4C4D4E4F4G4H4I4J6A···6F6G···6AE6AF···6AM12A···12H12I···12AJ12AK···12AR
order12222222223333344444444446···66···66···612···1212···1212···12
size11112266661122211112266661···12···26···61···12···26···6

108 irreducible representations

dim111111111111112222222222
type++++++++++
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C6S3D6D6C4○D4C3×S3S3×C6S3×C6C4○D12C3×C4○D4C3×C4○D12
kernelC6×C4○D12C6×Dic6S3×C2×C12C6×D12C3×C4○D12C6×C3⋊D4C2×C6×C12C2×C4○D12C2×Dic6S3×C2×C4C2×D12C4○D12C2×C3⋊D4C22×C12C22×C12C2×C12C22×C6C3×C6C22×C4C2×C4C23C6C6C2
# reps112182122421642161421228816

Matrix representation of C6×C4○D12 ►in GL3(𝔽13) generated by

1200
090
009
,
100
050
005
,
1200
020
007
,
1200
007
020
G:=sub<GL(3,GF(13))| [12,0,0,0,9,0,0,0,9],[1,0,0,0,5,0,0,0,5],[12,0,0,0,2,0,0,0,7],[12,0,0,0,0,2,0,7,0] >;
 

C6×C4○D12 in GAP, Magma, Sage, TeX 

C_6\times C_4\circ D_{12}
 
% in TeX
 
G:=Group("C6xC4oD12");
 
// GroupNames label
 
G:=SmallGroup(288,991);
 
// by ID
 
G=gap.SmallGroup(288,991);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
 
// generators/relations
 

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