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G = C3×C24.C4order 288 = 25·32

Direct product of C3 and C24.C4

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C24.C4, C24.1C12, C12.93D12, C62.12Q8, C24.13Dic3, (C3×C24).7C4, (C6×C24).10C2, (C2×C24).22S3, (C2×C24).11C6, C12.34(C3×D4), C4.18(C3×D12), C4.8(C6×Dic3), C8.1(C3×Dic3), C12.43(C2×C12), (C2×C12).434D6, (C3×C12).136D4, (C2×C6).11Dic6, C4.Dic3.1C6, C6.20(C4⋊Dic3), C327(C8.C4), C12.65(C2×Dic3), C22.2(C3×Dic6), (C6×C12).312C22, C6.7(C3×C4⋊C4), (C2×C8).5(C3×S3), (C2×C6).6(C3×Q8), C31(C3×C8.C4), (C2×C4).73(S3×C6), C2.5(C3×C4⋊Dic3), (C3×C6).33(C4⋊C4), (C2×C12).113(C2×C6), (C3×C12).129(C2×C4), (C3×C4.Dic3).5C2, SmallGroup(288,253)

Series: Derived Chief Lower central Upper central

C1C12 — C3×C24.C4
C1C3C6C12C2×C12C6×C12C3×C4.Dic3 — C3×C24.C4
C3C6C12 — C3×C24.C4
C1C12C2×C12C2×C24

Generators and relations for C3×C24.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 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C8 [×2], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C2×C8, M4(2) [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×4], C24 [×4], C2×C12 [×2], C2×C12, C8.C4, C3×C12 [×2], C62, C4.Dic3 [×2], C2×C24 [×2], C2×C24, C3×M4(2) [×2], C3×C3⋊C8 [×2], C3×C24 [×2], C6×C12, C24.C4, C3×C8.C4, C3×C4.Dic3 [×2], C6×C24, C3×C24.C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C8.C4, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C3×Dic6, C3×D12, C6×Dic3, C24.C4, C3×C8.C4, C3×C4⋊Dic3, C3×C24.C4

Smallest permutation representation of C3×C24.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 37 28 40 31 43 34 46)(26 38 29 41 32 44 35 47)(27 39 30 42 33 45 36 48)
(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 27 10 36 7 33 4 30)(2 32 11 29 8 26 5 35)(3 25 12 34 9 31 6 28)(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)(25,37,28,40,31,43,34,46)(26,38,29,41,32,44,35,47)(27,39,30,42,33,45,36,48), (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), (1,27,10,36,7,33,4,30)(2,32,11,29,8,26,5,35)(3,25,12,34,9,31,6,28)(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)(25,37,28,40,31,43,34,46)(26,38,29,41,32,44,35,47)(27,39,30,42,33,45,36,48), (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), (1,27,10,36,7,33,4,30)(2,32,11,29,8,26,5,35)(3,25,12,34,9,31,6,28)(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),(25,37,28,40,31,43,34,46),(26,38,29,41,32,44,35,47),(27,39,30,42,33,45,36,48)], [(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)], [(1,27,10,36,7,33,4,30),(2,32,11,29,8,26,5,35),(3,25,12,34,9,31,6,28),(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+++++--++-
imageC1C2C2C3C4C6C6C12S3D4Q8Dic3D6C3×S3D12C3×D4Dic6C3×Q8C8.C4C3×Dic3S3×C6C3×D12C3×Dic6C24.C4C3×C8.C4C3×C24.C4
kernelC3×C24.C4C3×C4.Dic3C6×C24C24.C4C3×C24C4.Dic3C2×C24C24C2×C24C3×C12C62C24C2×C12C2×C8C12C12C2×C6C2×C6C32C8C2×C4C4C22C3C3C1
# reps121244281112122222442448816

Matrix representation of C3×C24.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] >;

C3×C24.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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