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## G = (C3×C24)⋊C4order 288 = 25·32

### 2nd semidirect product of C3×C24 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — (C3×C24)⋊C4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4×C32⋊C4 — (C3×C24)⋊C4
 Lower central C32 — C3×C6 — (C3×C24)⋊C4
 Upper central C1 — C4 — C8

Generators and relations for (C3×C24)⋊C4
G = < a,b,c | a3=b24=c4=1, ab=ba, cac-1=a-1b8, cbc-1=a-1b13 >

Subgroups: 288 in 62 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, C42, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C8⋊C4, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, S3×C8, C324C8, C3×C24, C322C8, C4×C3⋊S3, C2×C32⋊C4, C8×C3⋊S3, C3⋊S33C8, C4×C32⋊C4, (C3×C24)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C42, M4(2), C8⋊C4, C32⋊C4, C2×C32⋊C4, C4×C32⋊C4, (C3×C24)⋊C4

Smallest permutation representation of (C3×C24)⋊C4
On 48 points
Generators in S48
(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 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 42)(2 31 18 47)(3 44 11 28)(4 33)(5 46 21 38)(6 35 14 43)(7 48)(8 37 24 29)(9 26 17 34)(10 39)(12 41 20 25)(13 30)(15 32 23 40)(16 45)(19 36)(22 27)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 8A 8B 8C ··· 8H 12A 12B 12C 12D 24A ··· 24H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 6 6 8 8 8 ··· 8 12 12 12 12 24 ··· 24 size 1 1 9 9 4 4 1 1 9 9 18 18 18 18 4 4 2 2 18 ··· 18 4 4 4 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 M4(2) C32⋊C4 C2×C32⋊C4 C4×C32⋊C4 (C3×C24)⋊C4 kernel (C3×C24)⋊C4 C8×C3⋊S3 C3⋊S3⋊3C8 C4×C32⋊C4 C32⋊4C8 C3×C24 C32⋊2C8 C2×C32⋊C4 C3⋊S3 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 4 2 2 4 8

Matrix representation of (C3×C24)⋊C4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 51 8 0 0 0 0 0 22 0 0 0 0 0 0 27 27 0 0 0 0 46 0 0 0 0 0 0 0 0 46 0 0 0 0 27 27
,
 18 24 0 0 0 0 26 55 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 72 72 0 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[51,0,0,0,0,0,8,22,0,0,0,0,0,0,27,46,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,46,27],[18,26,0,0,0,0,24,55,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C3×C24)⋊C4 in GAP, Magma, Sage, TeX

(C_3\times C_{24})\rtimes C_4
% in TeX

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

G:=SmallGroup(288,415);
// by ID

G=gap.SmallGroup(288,415);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,80,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c|a^3=b^24=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^8,c*b*c^-1=a^-1*b^13>;
// generators/relations

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