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G = C3×C8.D6  order 288 = 25·32

Direct product of C3 and C8.D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C8.D6, C24.41D6, Dic12⋊2C6, C12.90D12, C62.64D4, C8.1(S3×C6), C24⋊C2⋊2C6, C24.1(C2×C6), C6.14(C6×D4), (C2×Dic6)⋊8C6, C4○D12.4C6, D12.8(C2×C6), (C3×C12).83D4, C12.13(C3×D4), (C2×C6).48D12, C4.15(C3×D12), C2.16(C6×D12), (C3×Dic12)⋊3C2, (C6×Dic6)⋊13C2, C6.102(C2×D12), (C2×C12).239D6, (C3×M4(2))⋊4S3, M4(2)⋊2(C3×S3), (C3×M4(2))⋊2C6, (C3×C24).3C22, Dic6.8(C2×C6), C22.6(C3×D12), C12.33(C22×C6), C12.220(C22×S3), (C6×C12).116C22, (C3×C12).165C23, (C3×D12).47C22, C32⋊16(C8.C22), (C32×M4(2))⋊2C2, (C3×Dic6).48C22, C4.31(S3×C2×C6), (C2×C6).7(C3×D4), (C3×C24⋊C2)⋊4C2, (C2×C4).14(S3×C6), C3⋊1(C3×C8.C22), (C2×C12).27(C2×C6), (C3×C6).184(C2×D4), (C3×C4○D12).10C2, SmallGroup(288,680)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C3×C8.D6
Chief series
C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C3×C4○D12 — C3×C8.D6
Lower central
C3 — C6 — C12 — C3×C8.D6
Upper central
C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C8.D6
 G = < a,b,c,d | a3=b8=1, c6=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >

Subgroups: 314 in 130 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C24, C24, Dic6, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×C12, S3×C6, C62, C24⋊C2, Dic12, C3×M4(2), C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C4○D12, C6×Q8, C3×C4○D4, C3×C24, C3×Dic6, C3×Dic6, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, C8.D6, C3×C8.C22, C3×C24⋊C2, C3×Dic12, C32×M4(2), C6×Dic6, C3×C4○D12, C3×C8.D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C8.C22, S3×C6, C2×D12, C6×D4, C3×D12, S3×C2×C6, C8.D6, C3×C8.C22, C6×D12, C3×C8.D6

Smallest permutation representation of C3×C8.D6
►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 13 4 22 7 19 10 16)(2 20 5 17 8 14 11 23)(3 15 6 24 9 21 12 18)(25 37 34 40 31 43 28 46)(26 44 35 47 32 38 29 41)(27 39 36 42 33 45 30 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 7 27)(2 26 8 32)(3 31 9 25)(4 36 10 30)(5 29 11 35)(6 34 12 28)(13 42 19 48)(14 47 20 41)(15 40 21 46)(16 45 22 39)(17 38 23 44)(18 43 24 37)
 
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), (1,13,4,22,7,19,10,16)(2,20,5,17,8,14,11,23)(3,15,6,24,9,21,12,18)(25,37,34,40,31,43,28,46)(26,44,35,47,32,38,29,41)(27,39,36,42,33,45,30,48), (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), (1,33,7,27)(2,26,8,32)(3,31,9,25)(4,36,10,30)(5,29,11,35)(6,34,12,28)(13,42,19,48)(14,47,20,41)(15,40,21,46)(16,45,22,39)(17,38,23,44)(18,43,24,37)>;
 
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), (1,13,4,22,7,19,10,16)(2,20,5,17,8,14,11,23)(3,15,6,24,9,21,12,18)(25,37,34,40,31,43,28,46)(26,44,35,47,32,38,29,41)(27,39,36,42,33,45,30,48), (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), (1,33,7,27)(2,26,8,32)(3,31,9,25)(4,36,10,30)(5,29,11,35)(6,34,12,28)(13,42,19,48)(14,47,20,41)(15,40,21,46)(16,45,22,39)(17,38,23,44)(18,43,24,37) );
 
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)], [(1,13,4,22,7,19,10,16),(2,20,5,17,8,14,11,23),(3,15,6,24,9,21,12,18),(25,37,34,40,31,43,28,46),(26,44,35,47,32,38,29,41),(27,39,36,42,33,45,30,48)], [(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)], [(1,33,7,27),(2,26,8,32),(3,31,9,25),(4,36,10,30),(5,29,11,35),(6,34,12,28),(13,42,19,48),(14,47,20,41),(15,40,21,46),(16,45,22,39),(17,38,23,44),(18,43,24,37)]])
 

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E6A6B6C···6G6H6I6J6K6L8A8B12A···12J12K12L12M12N···12S24A···24P
order12223333344444666···6666668812···1212121212···1224···24
size112121122222121212112···24441212442···244412···124···4

63 irreducible representations

dim111111111111222222222222224444
type+++++++++++++--
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C3×S3D12C3×D4D12C3×D4S3×C6S3×C6C3×D12C3×D12C8.C22C8.D6C3×C8.C22C3×C8.D6
kernelC3×C8.D6C3×C24⋊C2C3×Dic12C32×M4(2)C6×Dic6C3×C4○D12C8.D6C24⋊C2Dic12C3×M4(2)C2×Dic6C4○D12C3×M4(2)C3×C12C62C24C2×C12M4(2)C12C12C2×C6C2×C6C8C2×C4C4C22C32C3C3C1
# reps122111244222111212222242441224

Matrix representation of C3×C8.D6 ►in GL4(𝔽73) generated by

64000
06400
00640
00064
,
0100
46000
028046
2669720
,
70000
0300
5619490
5456024
,
1029510
4410022
46336329
33274463
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[0,46,0,26,1,0,28,69,0,0,0,72,0,0,46,0],[70,0,56,54,0,3,19,56,0,0,49,0,0,0,0,24],[10,44,46,33,29,10,33,27,51,0,63,44,0,22,29,63] >;
 

C3×C8.D6 in GAP, Magma, Sage, TeX 

C_3\times C_8.D_6
 
% in TeX
 
G:=Group("C3xC8.D6");
 
// GroupNames label
 
G:=SmallGroup(288,680);
 
// by ID
 
G=gap.SmallGroup(288,680);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,142,2524,102,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^8=1,c^6=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
 
// generators/relations
 

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