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## G = C3×C24.C4order 288 = 25·32

### Direct product of C3 and C24.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C24.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4.Dic3 — C3×C24.C4
 Lower central C3 — C6 — C12 — C3×C24.C4
 Upper central C1 — C12 — C2×C12 — C2×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, C3, C4, C22, C6, C6, C8, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C3×C6, C3×C6, C3⋊C8, C24, C24, C2×C12, C2×C12, C8.C4, C3×C12, C62, C4.Dic3, C2×C24, C2×C24, C3×M4(2), C3×C3⋊C8, C3×C24, C6×C12, C24.C4, C3×C8.C4, C3×C4.Dic3, C6×C24, C3×C24.C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C8.C4, C3×Dic3, 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 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 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6M 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12R 24A ··· 24AF 24AG ··· 24AN order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 2 1 1 2 2 2 1 1 2 1 1 2 ··· 2 2 2 2 2 12 12 12 12 1 1 1 1 2 ··· 2 2 ··· 2 12 ··· 12

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + + - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 Dic3 D6 C3×S3 D12 C3×D4 Dic6 C3×Q8 C8.C4 C3×Dic3 S3×C6 C3×D12 C3×Dic6 C24.C4 C3×C8.C4 C3×C24.C4 kernel C3×C24.C4 C3×C4.Dic3 C6×C24 C24.C4 C3×C24 C4.Dic3 C2×C24 C24 C2×C24 C3×C12 C62 C24 C2×C12 C2×C8 C12 C12 C2×C6 C2×C6 C32 C8 C2×C4 C4 C22 C3 C3 C1 # reps 1 2 1 2 4 4 2 8 1 1 1 2 1 2 2 2 2 2 4 4 2 4 4 8 8 16

Matrix representation of C3×C24.C4 in GL4(𝔽5) generated by

 3 0 4 0 0 1 0 2 3 0 1 0 0 1 0 3
,
 3 0 2 0 0 1 0 3 4 0 2 0 0 4 0 4
,
 3 0 3 0 0 3 0 1 1 0 4 0 0 3 0 4
,
 0 4 0 4 3 0 4 0 0 4 0 2 4 0 1 0
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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