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G = C3×C8○D12  order 288 = 25·32

Direct product of C3 and C8○D12

direct product, metabelian, supersoluble, monomial

Aliases: C3×C8○D12, C24.96D6, D12.2C12, Dic6.2C12, (S3×C8)⋊6C6, C8⋊S3⋊7C6, (C6×C24)⋊20C2, (C2×C24)⋊15S3, (C2×C24)⋊12C6, C8.18(S3×C6), (S3×C24)⋊15C2, C4.10(S3×C12), C12.88(C4×S3), C24.36(C2×C6), C4○D12.6C6, (C3×D12).4C4, D6.1(C2×C12), C3⋊D4.2C12, C32⋊9(C8○D4), C12.20(C2×C12), (C2×C12).440D6, C4.Dic3⋊11C6, C22.2(S3×C12), C62.78(C2×C4), (C3×Dic6).4C4, (C3×C24).73C22, C6.14(C22×C12), C12.37(C22×C6), Dic3.3(C2×C12), (S3×C12).59C22, (C3×C12).169C23, (C6×C12).313C22, C12.225(C22×S3), (C2×C8)⋊7(C3×S3), C3⋊1(C3×C8○D4), C4.37(S3×C2×C6), C3⋊C8.11(C2×C6), C6.113(S3×C2×C4), C2.15(S3×C2×C12), (C2×C6).33(C4×S3), (C2×C4).77(S3×C6), (C3×C3⋊D4).4C4, (C3×C8⋊S3)⋊15C2, (C4×S3).15(C2×C6), (S3×C6).13(C2×C4), (C2×C6).19(C2×C12), (C3×C3⋊C8).42C22, (C2×C12).114(C2×C6), (C3×C12).100(C2×C4), (C3×C4○D12).12C2, (C3×C4.Dic3)⋊23C2, (C3×C6).85(C22×C4), (C3×Dic3).19(C2×C4), SmallGroup(288,672)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C3×C8○D12
Chief series
C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×C4○D12 — C3×C8○D12
Lower central
C3 — C6 — C3×C8○D12
Upper central
C1 — C24 — C2×C24

Generators and relations for C3×C8○D12
 G = < a,b,c,d | a3=b8=d2=1, c6=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c5 >

Subgroups: 250 in 135 conjugacy classes, 74 normal (46 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8○D4, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C2×C24, C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C8○D12, C3×C8○D4, S3×C24, C3×C8⋊S3, C3×C4.Dic3, C6×C24, C3×C4○D12, C3×C8○D12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C8○D4, S3×C6, S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, C8○D12, C3×C8○D4, S3×C2×C12, C3×C8○D12

Smallest permutation representation of C3×C8○D12
►On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

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

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

108 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C···6M6N6O6P6Q8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E···12R12S12T12U12V24A···24H24I···24AJ24AK···24AR
order122223333344444666···6666688888888881212121212···121212121224···2424···2424···24
size112661122211266112···26666111122666611112···266661···12···26···6

108 irreducible representations

dim11111111111111111122222222222222
type+++++++++
imageC1C2C2C2C2C2C3C4C4C4C6C6C6C6C6C12C12C12S3D6D6C3×S3C4×S3C4×S3C8○D4S3×C6S3×C6S3×C12S3×C12C8○D12C3×C8○D4C3×C8○D12
kernelC3×C8○D12S3×C24C3×C8⋊S3C3×C4.Dic3C6×C24C3×C4○D12C8○D12C3×Dic6C3×D12C3×C3⋊D4S3×C8C8⋊S3C4.Dic3C2×C24C4○D12Dic6D12C3⋊D4C2×C24C24C2×C12C2×C8C12C2×C6C32C8C2×C4C4C22C3C3C1
# reps122111222444222448121222442448816

Matrix representation of C3×C8○D12 ►in GL2(𝔽73) generated by

80
08
,
220
022
,
30
049
,
049
30
G:=sub<GL(2,GF(73))| [8,0,0,8],[22,0,0,22],[3,0,0,49],[0,3,49,0] >;
 

C3×C8○D12 in GAP, Magma, Sage, TeX 

C_3\times C_8\circ D_{12}
 
% in TeX
 
G:=Group("C3xC8oD12");
 
// GroupNames label
 
G:=SmallGroup(288,672);
 
// by ID
 
G=gap.SmallGroup(288,672);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,142,102,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^6=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^5>;
 
// generators/relations
 

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