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G = C6×C4○D12order 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, D1212(C2×C6), C6.4(C23×C6), (C22×C12)⋊17S3, (C22×C12)⋊12C6, (C6×C12)⋊33C22, Dic611(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, C31(C6×C4○D4), C61(C3×C4○D4), C4.43(S3×C2×C6), (S3×C2×C12)⋊31C2, (C4×S3)⋊6(C2×C6), (C2×C4)⋊10(S3×C6), C3⋊D46(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, C3213(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

C1C6 — C6×C4○D12
C1C3C6C3×C6S3×C6S3×C2×C6S3×C2×C12 — C6×C4○D12
C3C6 — C6×C4○D12
C1C2×C12C22×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 [×2], C2 [×6], C3 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×4], C6 [×2], C6 [×4], C6 [×15], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], C32, Dic3 [×4], C12 [×8], C12 [×8], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×4], C2×C6 [×19], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×4], C2×C12 [×8], C2×C12 [×16], C3×D4 [×12], C3×Q8 [×4], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C2×C4○D4, C3×Dic3 [×4], C3×C12 [×4], S3×C6 [×4], S3×C6 [×4], C62, C62 [×2], C62 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12 [×2], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C3×Dic6 [×4], S3×C12 [×8], C3×D12 [×4], C6×Dic3 [×2], C3×C3⋊D4 [×8], C6×C12 [×2], C6×C12 [×4], S3×C2×C6 [×2], C2×C62, C2×C4○D12, C6×C4○D4, C6×Dic6, S3×C2×C12 [×2], C6×D12, C3×C4○D12 [×8], C6×C3⋊D4 [×2], C2×C6×C12, C6×C4○D12
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C4○D4 [×2], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], C2×C4○D4, S3×C6 [×7], C4○D12 [×2], C3×C4○D4 [×2], S3×C23, C23×C6, S3×C2×C6 [×7], C2×C4○D12, C6×C4○D4, C3×C4○D12 [×2], S3×C22×C6, C6×C4○D12

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

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

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

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

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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