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G = C3×C12⋊3D4  order 288 = 25·32

Direct product of C3 and C12⋊3D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12⋊3D4, C62.205C23, (C6×D4)⋊4C6, C12⋊3(C3×D4), (C2×D12)⋊9C6, (C6×D4)⋊13S3, (C3×C12)⋊11D4, C6.52(C6×D4), (C6×D12)⋊11C2, C12⋊8(C3⋊D4), Dic3⋊1(C3×D4), (C4×Dic3)⋊6C6, C6.201(S3×D4), (C3×Dic3)⋊10D4, (C2×C12).328D6, C32⋊7(C4⋊1D4), C23.15(S3×C6), (C22×C6).34D6, (Dic3×C12)⋊16C2, (C6×C12).123C22, (C2×C62).59C22, (C6×Dic3).161C22, (D4×C3×C6)⋊4C2, C4⋊1(C3×C3⋊D4), C2.28(C3×S3×D4), (C2×D4)⋊6(C3×S3), (C2×C3⋊D4)⋊7C6, C3⋊2(C3×C4⋊1D4), (C6×C3⋊D4)⋊21C2, (C2×C4).52(S3×C6), C2.16(C6×C3⋊D4), C22.62(S3×C2×C6), (C2×C12).34(C2×C6), (C3×C6).227(C2×D4), C6.153(C2×C3⋊D4), (S3×C2×C6).62C22, (C22×C6).33(C2×C6), (C2×C6).60(C22×C6), (C22×S3).12(C2×C6), (C2×C6).338(C22×S3), (C2×Dic3).39(C2×C6), SmallGroup(288,711)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C3×C12⋊3D4
Chief series
C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C3×C12⋊3D4
Lower central
C3 — C2×C6 — C3×C12⋊3D4
Upper central
C1 — C2×C6 — C6×D4

Generators and relations for C3×C12⋊3D4
 G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 650 in 243 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C2×D4, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4⋊1D4, C3×Dic3, C3×C12, S3×C6, C62, C62, C4×Dic3, C4×C12, C2×D12, C2×C3⋊D4, C6×D4, C6×D4, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, C2×C62, C12⋊3D4, C3×C4⋊1D4, Dic3×C12, C6×D12, C6×C3⋊D4, D4×C3×C6, C3×C12⋊3D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C4⋊1D4, S3×C6, S3×D4, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, C12⋊3D4, C3×C4⋊1D4, C3×S3×D4, C6×C3⋊D4, C3×C12⋊3D4

Smallest permutation representation of C3×C12⋊3D4
►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 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 17 46)(2 35 18 39)(3 28 19 44)(4 33 20 37)(5 26 21 42)(6 31 22 47)(7 36 23 40)(8 29 24 45)(9 34 13 38)(10 27 14 43)(11 32 15 48)(12 25 16 41)

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6O6P···6AE6AF6AG6AH6AI12A12B12C12D12E···12J12K···12R
order12222222333334444446···66···66···666661212121212···1212···12
size1111441212112222266661···12···24···41212121222224···46···6

72 irreducible representations

dim111111111122222222222244
type+++++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D4D6D6C3×S3C3×D4C3⋊D4C3×D4S3×C6S3×C6C3×C3⋊D4S3×D4C3×S3×D4
kernelC3×C12⋊3D4Dic3×C12C6×D12C6×C3⋊D4D4×C3×C6C12⋊3D4C4×Dic3C2×D12C2×C3⋊D4C6×D4C6×D4C3×Dic3C3×C12C2×C12C22×C6C2×D4Dic3C12C12C2×C4C23C4C6C2
# reps111412228214212284424824

Matrix representation of C3×C12⋊3D4 ►in GL6(𝔽13)

300000
030000
001000
000100
000010
000001
,
1000000
040000
0001200
001000
0000012
000010
,
0120000
1200000
0012000
0001200
0000012
000010
,
010000
100000
0001200
0012000
000010
0000012

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;
 

C3×C12⋊3D4 in GAP, Magma, Sage, TeX 

C_3\times C_{12}\rtimes_3D_4
 
% in TeX
 
G:=Group("C3xC12:3D4");
 
// GroupNames label
 
G:=SmallGroup(288,711);
 
// by ID
 
G=gap.SmallGroup(288,711);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,590,555,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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