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G = (C3×C24)⋊C4  order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: (C3×C24)⋊2C4, C8⋊3(C32⋊C4), C32⋊2C8⋊5C4, (C3×C6).2C42, C32⋊4C8⋊10C4, C32⋊1(C8⋊C4), C3⋊S3.6M4(2), (C8×C3⋊S3).9C2, C2.3(C4×C32⋊C4), (C2×C32⋊C4).3C4, (C4×C32⋊C4).5C2, C4.18(C2×C32⋊C4), C3⋊S3⋊3C8.7C2, (C3×C12).11(C2×C4), (C4×C3⋊S3).79C22, C3⋊Dic3.26(C2×C4), (C2×C3⋊S3).23(C2×C4), SmallGroup(288,415)

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 [×2], C3 [×2], C4, C4 [×3], C22, S3 [×4], C6 [×2], C8, C8 [×3], C2×C4 [×3], C32, Dic3 [×2], C12 [×2], D6 [×2], C42, C2×C8 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×2], C8⋊C4, C3⋊Dic3, C3×C12, C32⋊C4 [×2], C2×C3⋊S3, S3×C8 [×2], C32⋊4C8, C3×C24, C32⋊2C8 [×2], C4×C3⋊S3, C2×C32⋊C4 [×2], C8×C3⋊S3, C3⋊S3⋊3C8, C4×C32⋊C4, (C3×C24)⋊C4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], C42, M4(2) [×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 33)(2 46 18 38)(3 35 11 43)(4 48)(5 37 21 29)(6 26 14 34)(7 39)(8 28 24 44)(9 41 17 25)(10 30)(12 32 20 40)(13 45)(15 47 23 31)(16 36)(19 27)(22 42)
 
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), (1,33)(2,46,18,38)(3,35,11,43)(4,48)(5,37,21,29)(6,26,14,34)(7,39)(8,28,24,44)(9,41,17,25)(10,30)(12,32,20,40)(13,45)(15,47,23,31)(16,36)(19,27)(22,42)>;
 
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), (1,33)(2,46,18,38)(3,35,11,43)(4,48)(5,37,21,29)(6,26,14,34)(7,39)(8,28,24,44)(9,41,17,25)(10,30)(12,32,20,40)(13,45)(15,47,23,31)(16,36)(19,27)(22,42) );
 
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)], [(1,33),(2,46,18,38),(3,35,11,43),(4,48),(5,37,21,29),(6,26,14,34),(7,39),(8,28,24,44),(9,41,17,25),(10,30),(12,32,20,40),(13,45),(15,47,23,31),(16,36),(19,27),(22,42)])
 

36 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H6A6B8A8B8C···8H12A12B12C12D24A···24H
order1222334444444466888···81212121224···24
size119944119918181818442218···1844444···4

36 irreducible representations

dim1111111124444
type++++++
imageC1C2C2C2C4C4C4C4M4(2)C32⋊C4C2×C32⋊C4C4×C32⋊C4(C3×C24)⋊C4
kernel(C3×C24)⋊C4C8×C3⋊S3C3⋊S3⋊3C8C4×C32⋊C4C32⋊4C8C3×C24C32⋊2C8C2×C32⋊C4C3⋊S3C8C4C2C1
# reps1111224442248

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

100000
010000
001000
000100
00007272
000010
,
5180000
0220000
00272700
0046000
0000046
00002727
,
18240000
26550000
000010
000001
001000
00727200

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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