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G = C3×C12.47D4  order 288 = 25·32

Direct product of C3 and C12.47D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12.47D4, C12.87D12, C12.56(C3×D4), C4.12(C3×D12), (C2×Dic3).C12, C6.50(D6⋊C4), (C2×C12).221D6, (C3×C12).158D4, C62.38(C2×C4), C22.5(S3×C12), (C6×Dic3).2C4, (C6×Dic6).5C2, (C2×Dic6).6C6, C4.Dic3.3C6, (C6×C12).45C22, M4(2).2(C3×S3), (C3×M4(2)).8C6, C12.139(C3⋊D4), C32⋊5(C4.10D4), (C3×M4(2)).10S3, (C32×M4(2)).2C2, (C2×C4).2(S3×C6), (C2×C6).60(C4×S3), (C2×C6).3(C2×C12), C2.10(C3×D6⋊C4), C4.22(C3×C3⋊D4), C6.9(C3×C22⋊C4), (C2×C12).15(C2×C6), C3⋊1(C3×C4.10D4), (C3×C4.Dic3).7C2, (C3×C6).49(C22⋊C4), SmallGroup(288,258)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C3×C12.47D4
Chief series
C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C6×Dic6 — C3×C12.47D4
Lower central
C3 — C6 — C2×C6 — C3×C12.47D4
Upper central
C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C12.47D4
 G = < a,b,c,d | a3=b12=1, c4=d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c3 >

Subgroups: 186 in 86 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, M4(2), M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4.10D4, C3×Dic3, C3×C12, C62, C4.Dic3, C3×M4(2), C3×M4(2), C2×Dic6, C6×Q8, C3×C3⋊C8, C3×C24, C3×Dic6, C6×Dic3, C6×C12, C12.47D4, C3×C4.10D4, C3×C4.Dic3, C32×M4(2), C6×Dic6, C3×C12.47D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C4.10D4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C12.47D4, C3×C4.10D4, C3×D6⋊C4, C3×C12.47D4

Smallest permutation representation of C3×C12.47D4
►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 40 10 43 7 46 4 37)(2 39 11 42 8 45 5 48)(3 38 12 41 9 44 6 47)(13 26 16 35 19 32 22 29)(14 25 17 34 20 31 23 28)(15 36 18 33 21 30 24 27)

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

63 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C4D6A6B6C···6G6H6I6J8A8B8C8D12A···12J12K12L12M12N12O12P12Q24A···24P24Q24R24S24T
order122333334444666···6666888812···121212121212121224···2424242424
size11211222221212112···24444412122···2444121212124···412121212

63 irreducible representations

dim11111111112222222222224444
type++++++++--
imageC1C2C2C2C3C4C6C6C6C12S3D4D6C3×S3D12C3⋊D4C3×D4C4×S3S3×C6C3×D12C3×C3⋊D4S3×C12C4.10D4C12.47D4C3×C4.10D4C3×C12.47D4
kernelC3×C12.47D4C3×C4.Dic3C32×M4(2)C6×Dic6C12.47D4C6×Dic3C4.Dic3C3×M4(2)C2×Dic6C2×Dic3C3×M4(2)C3×C12C2×C12M4(2)C12C12C12C2×C6C2×C4C4C4C22C32C3C3C1
# reps11112422281212224224441224

Matrix representation of C3×C12.47D4 ►in GL4(𝔽73) generated by

8000
0800
0080
0008
,
30170
0300
00490
00049
,
2745026
005246
4323460
7246210
,
720420
0001
66010
07200
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[3,0,0,0,0,3,0,0,17,0,49,0,0,0,0,49],[27,0,43,72,45,0,23,46,0,52,46,21,26,46,0,0],[72,0,66,0,0,0,0,72,42,0,1,0,0,1,0,0] >;
 

C3×C12.47D4 in GAP, Magma, Sage, TeX 

C_3\times C_{12}._{47}D_4
 
% in TeX
 
G:=Group("C3xC12.47D4");
 
// GroupNames label
 
G:=SmallGroup(288,258);
 
// by ID
 
G=gap.SmallGroup(288,258);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,365,92,1683,136,1271,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=1,c^4=d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^3>;
 
// generators/relations
 

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