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G = C3xC24.C4order 288 = 25·32

Direct product of C3 and C24.C4

direct product, metacyclic, supersoluble, monomial

Aliases: C3xC24.C4, C24.1C12, C12.93D12, C62.12Q8, C24.13Dic3, (C3xC24).7C4, (C6xC24).10C2, (C2xC24).22S3, (C2xC24).11C6, C12.34(C3xD4), C4.18(C3xD12), C4.8(C6xDic3), C8.1(C3xDic3), C12.43(C2xC12), (C2xC12).434D6, (C3xC12).136D4, (C2xC6).11Dic6, C4.Dic3.1C6, C6.20(C4:Dic3), C32:7(C8.C4), C12.65(C2xDic3), C22.2(C3xDic6), (C6xC12).312C22, C6.7(C3xC4:C4), (C2xC8).5(C3xS3), (C2xC6).6(C3xQ8), C3:1(C3xC8.C4), (C2xC4).73(S3xC6), C2.5(C3xC4:Dic3), (C3xC6).33(C4:C4), (C2xC12).113(C2xC6), (C3xC12).129(C2xC4), (C3xC4.Dic3).5C2, SmallGroup(288,253)

Series: Derived Chief Lower central Upper central

C1C12 — C3xC24.C4
C1C3C6C12C2xC12C6xC12C3xC4.Dic3 — C3xC24.C4
C3C6C12 — C3xC24.C4
C1C12C2xC12C2xC24

Generators and relations for C3xC24.C4
 G = < a,b,c,d | a3=b8=1, c6=b4, d2=b4c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 122 in 71 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C8, C2xC4, C32, C12, C12, C2xC6, C2xC6, C2xC8, M4(2), C3xC6, C3xC6, C3:C8, C24, C24, C2xC12, C2xC12, C8.C4, C3xC12, C62, C4.Dic3, C2xC24, C2xC24, C3xM4(2), C3xC3:C8, C3xC24, C6xC12, C24.C4, C3xC8.C4, C3xC4.Dic3, C6xC24, C3xC24.C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, D4, Q8, Dic3, C12, D6, C2xC6, C4:C4, C3xS3, Dic6, D12, C2xDic3, C2xC12, C3xD4, C3xQ8, C8.C4, C3xDic3, S3xC6, C4:Dic3, C3xC4:C4, C3xDic6, C3xD12, C6xDic3, C24.C4, C3xC8.C4, C3xC4:Dic3, C3xC24.C4

Smallest permutation representation of C3xC24.C4
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 19 10 16 7 13 4 22)(2 20 11 17 8 14 5 23)(3 21 12 18 9 15 6 24)(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 33 10 30 7 27 4 36)(2 26 11 35 8 32 5 29)(3 31 12 28 9 25 6 34)(13 42 22 39 19 48 16 45)(14 47 23 44 20 41 17 38)(15 40 24 37 21 46 18 43)

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

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

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

90 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C···6M8A8B8C8D8E8F8G8H12A12B12C12D12E···12R24A···24AF24AG···24AN
order12233333444666···6888888881212121212···1224···2424···24
size11211222112112···222221212121211112···22···212···12

90 irreducible representations

dim11111111222222222222222222
type+++++--++-
imageC1C2C2C3C4C6C6C12S3D4Q8Dic3D6C3xS3D12C3xD4Dic6C3xQ8C8.C4C3xDic3S3xC6C3xD12C3xDic6C24.C4C3xC8.C4C3xC24.C4
kernelC3xC24.C4C3xC4.Dic3C6xC24C24.C4C3xC24C4.Dic3C2xC24C24C2xC24C3xC12C62C24C2xC12C2xC8C12C12C2xC6C2xC6C32C8C2xC4C4C22C3C3C1
# reps121244281112122222442448816

Matrix representation of C3xC24.C4 in GL4(F5) generated by

3040
0102
3010
0103
,
3020
0103
4020
0404
,
3030
0301
1040
0304
,
0404
3040
0402
4010
G:=sub<GL(4,GF(5))| [3,0,3,0,0,1,0,1,4,0,1,0,0,2,0,3],[3,0,4,0,0,1,0,4,2,0,2,0,0,3,0,4],[3,0,1,0,0,3,0,3,3,0,4,0,0,1,0,4],[0,3,0,4,4,0,4,0,0,4,0,1,4,0,2,0] >;

C3xC24.C4 in GAP, Magma, Sage, TeX

C_3\times C_{24}.C_4
% in TeX

G:=Group("C3xC24.C4");
// GroupNames label

G:=SmallGroup(288,253);
// by ID

G=gap.SmallGroup(288,253);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,176,136,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^4,d^2=b^4*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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