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G = C3×C12.53D4  order 288 = 25·32

Direct product of C3 and C12.53D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C12.53D4, C62.7Q8, C3⋊C8.1C12, C4.13(S3×C12), C12.5(C2×C12), C12.62(C3×D4), (C2×C6).6Dic6, C12.104(C4×S3), (C3×C12).164D4, (C2×C12).315D6, C4.Dic3.2C6, (C6×C12).43C22, C32⋊9(C8.C4), M4(2).1(C3×S3), (C3×M4(2)).9S3, (C3×M4(2)).7C6, C12.145(C3⋊D4), C22.1(C3×Dic6), C6.24(Dic3⋊C4), (C32×M4(2)).1C2, (C2×C6).(C3×Q8), C6.8(C3×C4⋊C4), (C3×C3⋊C8).2C4, (C2×C3⋊C8).4C6, (C6×C3⋊C8).15C2, C3⋊2(C3×C8.C4), (C2×C4).36(S3×C6), C4.28(C3×C3⋊D4), (C3×C6).34(C4⋊C4), (C3×C12).41(C2×C4), (C2×C12).13(C2×C6), C2.5(C3×Dic3⋊C4), (C3×C4.Dic3).6C2, SmallGroup(288,256)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C3×C12.53D4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C6×C3⋊C8 — C3×C12.53D4
Lower central
C3 — C6 — C12 — C3×C12.53D4
Upper central
C1 — C12 — C2×C12 — C3×M4(2)

Generators and relations for C3×C12.53D4
 G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c3 >

Subgroups: 122 in 70 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C2×C12, C2×C12, C8.C4, C3×C12, C62, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), C3×C3⋊C8, C3×C3⋊C8, C3×C24, C6×C12, C12.53D4, C3×C8.C4, C6×C3⋊C8, C3×C4.Dic3, C32×M4(2), C3×C12.53D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C8.C4, S3×C6, Dic3⋊C4, C3×C4⋊C4, C3×Dic6, S3×C12, C3×C3⋊D4, C12.53D4, C3×C8.C4, C3×Dic3⋊C4, C3×C12.53D4

Smallest permutation representation of C3×C12.53D4
►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 33 4 36 7 27 10 30)(2 26 5 29 8 32 11 35)(3 31 6 34 9 25 12 28)(13 39 22 48 19 45 16 42)(14 44 23 41 20 38 17 47)(15 37 24 46 21 43 18 40)

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

72 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C···6G6H6I6J8A8B8C8D8E8F8G8H12A12B12C12D12E···12L12M12N12O24A···24P24Q···24X24Y24Z24AA24AB
order12233333444666···6666888888881212121212···1212121224···2424···2424242424
size11211222112112···2444446666121211112···24444···46···612121212

72 irreducible representations

dim1111111111222222222222222244
type++++++-+-
imageC1C2C2C2C3C4C6C6C6C12S3D4Q8D6C3×S3C4×S3C3⋊D4C3×D4Dic6C3×Q8C8.C4S3×C6S3×C12C3×C3⋊D4C3×Dic6C3×C8.C4C12.53D4C3×C12.53D4
kernelC3×C12.53D4C6×C3⋊C8C3×C4.Dic3C32×M4(2)C12.53D4C3×C3⋊C8C2×C3⋊C8C4.Dic3C3×M4(2)C3⋊C8C3×M4(2)C3×C12C62C2×C12M4(2)C12C12C12C2×C6C2×C6C32C2×C4C4C4C22C3C3C1
# reps1111242228111122222242444824

Matrix representation of C3×C12.53D4 ►in GL4(𝔽73) generated by

1000
0100
0080
0008
,
27000
02700
0080
00064
,
224500
01000
0001
0010
,
63000
271000
00072
00720
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[27,0,0,0,0,27,0,0,0,0,8,0,0,0,0,64],[22,0,0,0,45,10,0,0,0,0,0,1,0,0,1,0],[63,27,0,0,0,10,0,0,0,0,0,72,0,0,72,0] >;
 

C3×C12.53D4 in GAP, Magma, Sage, TeX 

C_3\times C_{12}._{53}D_4
 
% in TeX
 
G:=Group("C3xC12.53D4");
 
// GroupNames label
 
G:=SmallGroup(288,256);
 
// by ID
 
G=gap.SmallGroup(288,256);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,365,92,136,1271,102,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^3>;
 
// generators/relations
 

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