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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, 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, C32⋊4C8, C3×C24, C32⋊2C8, C4×C3⋊S3, C2×C32⋊C4, C8×C3⋊S3, C3⋊S3⋊3C8, 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 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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