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G = C3xC8:D6  order 288 = 25·32

Direct product of C3 and C8:D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xC8:D6, D24:2C6, C24:11D6, C12.89D12, C62.63D4, C8:1(S3xC6), C24:1(C2xC6), C24:C2:1C6, C4oD12:2C6, (C3xD24):3C2, (C6xD12):9C2, D12:4(C2xC6), (C2xD12):7C6, C6.13(C6xD4), (C3xC24):3C22, Dic6:4(C2xC6), (C2xC6).47D12, C2.15(C6xD12), C4.14(C3xD12), (C3xC12).82D4, C12.12(C3xD4), (C2xC12).238D6, C6.101(C2xD12), M4(2):1(C3xS3), (C3xM4(2)):3S3, (C3xM4(2)):1C6, C22.5(C3xD12), (C3xD12):39C22, C32:16(C8:C22), C12.32(C22xC6), (C6xC12).115C22, C12.219(C22xS3), (C3xC12).164C23, (C3xDic6):37C22, (C32xM4(2)):1C2, C4.30(S3xC2xC6), C3:1(C3xC8:C22), (C2xC6).6(C3xD4), (C3xC24:C2):3C2, (C3xC4oD12):6C2, (C2xC4).13(S3xC6), (C2xC12).26(C2xC6), (C3xC6).183(C2xD4), SmallGroup(288,679)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C3xC8:D6
Chief series
C1 — C3 — C6 — C12 — C3xC12 — C3xD12 — C6xD12 — C3xC8:D6
Lower central
C3 — C6 — C12 — C3xC8:D6
Upper central
C1 — C6 — C2xC12 — C3xM4(2)

Generators and relations for C3xC8:D6
 G = < a,b,c,d | a3=b8=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 442 in 146 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3xC6, C3xC6, C24, C24, Dic6, C4xS3, D12, D12, D12, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xC6, C8:C22, C3xDic3, C3xC12, S3xC6, C62, C24:C2, D24, C3xM4(2), C3xM4(2), C3xD8, C3xSD16, C2xD12, C4oD12, C6xD4, C3xC4oD4, C3xC24, C3xDic6, S3xC12, C3xD12, C3xD12, C3xD12, C3xC3:D4, C6xC12, S3xC2xC6, C8:D6, C3xC8:C22, C3xC24:C2, C3xD24, C32xM4(2), C6xD12, C3xC4oD12, C3xC8:D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, D12, C3xD4, C22xS3, C22xC6, C8:C22, S3xC6, C2xD12, C6xD4, C3xD12, S3xC2xC6, C8:D6, C3xC8:C22, C6xD12, C3xC8:D6

Smallest permutation representation of C3xC8:D6
►On 48 points
Generators in S48
(1 36 29)(2 37 30)(3 38 31)(4 39 32)(5 40 25)(6 33 26)(7 34 27)(8 35 28)(9 46 19)(10 47 20)(11 48 21)(12 41 22)(13 42 23)(14 43 24)(15 44 17)(16 45 18)

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

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

63 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C6A6B6C···6G6H6I6J6K···6P8A8B12A···12J12K12L12M12N12O24A···24P
order12222233333444666···66666···68812···12121212121224···24
size112121212112222212112···244412···12442···244412124···4

63 irreducible representations

dim111111111111222222222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C3xS3D12C3xD4D12C3xD4S3xC6S3xC6C3xD12C3xD12C8:C22C8:D6C3xC8:C22C3xC8:D6
kernelC3xC8:D6C3xC24:C2C3xD24C32xM4(2)C6xD12C3xC4oD12C8:D6C24:C2D24C3xM4(2)C2xD12C4oD12C3xM4(2)C3xC12C62C24C2xC12M4(2)C12C12C2xC6C2xC6C8C2xC4C4C22C32C3C3C1
# reps122111244222111212222242441224

Matrix representation of C3xC8:D6 ►in GL4(F73) generated by

8000
0800
0080
0008
,
0100
27000
00027
00720
,
64000
0900
0080
00065
,
0080
00065
64000
0900
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,27,0,0,1,0,0,0,0,0,0,72,0,0,27,0],[64,0,0,0,0,9,0,0,0,0,8,0,0,0,0,65],[0,0,64,0,0,0,0,9,8,0,0,0,0,65,0,0] >;
 

C3xC8:D6 in GAP, Magma, Sage, TeX 

C_3\times C_8\rtimes D_6
 
% in TeX
 
G:=Group("C3xC8:D6");
 
// GroupNames label
 
G:=SmallGroup(288,679);
 
// by ID
 
G=gap.SmallGroup(288,679);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,142,2524,102,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^8=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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