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G = C6×C4.Dic3  order 288 = 25·32

Direct product of C6 and C4.Dic3

direct product, metabelian, supersoluble, monomial

Aliases: C6×C4.Dic3, (C6×C12).27C4, C6⋊2(C3×M4(2)), (C3×C6)⋊7M4(2), C3⋊3(C6×M4(2)), (C2×C12).14C12, C12.48(C2×C12), (C2×C12).445D6, (C2×C62).13C4, C4.14(C6×Dic3), (C22×C12).22C6, C12.41(C22×C6), C62.108(C2×C4), (C22×C12).39S3, (C22×C6).16C12, C6.21(C22×C12), (C2×C12).29Dic3, C12.71(C2×Dic3), C32⋊14(C2×M4(2)), C23.5(C3×Dic3), C22.5(C6×Dic3), (C3×C12).173C23, (C6×C12).350C22, C12.229(C22×S3), (C22×C6).10Dic3, C6.41(C22×Dic3), (C6×C3⋊C8)⋊25C2, (C2×C3⋊C8)⋊12C6, C3⋊C8⋊12(C2×C6), C4.41(S3×C2×C6), (C2×C6×C12).13C2, C2.3(Dic3×C2×C6), (C3×C3⋊C8)⋊43C22, (C2×C4).82(S3×C6), (C2×C6).42(C2×C12), (C22×C4).8(C3×S3), (C2×C4).6(C3×Dic3), (C2×C12).116(C2×C6), (C3×C12).136(C2×C4), (C2×C6).26(C2×Dic3), (C3×C6).113(C22×C4), SmallGroup(288,692)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C6×C4.Dic3
Chief series
C1 — C3 — C6 — C12 — C3×C12 — C3×C3⋊C8 — C6×C3⋊C8 — C6×C4.Dic3
Lower central
C3 — C6 — C6×C4.Dic3
Upper central
C1 — C2×C12 — C22×C12

Generators and relations for C6×C4.Dic3
 G = < a,b,c,d | a6=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 250 in 163 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C22×C6, C22×C6, C2×M4(2), C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C22×C12, C22×C12, C3×C3⋊C8, C6×C12, C6×C12, C2×C62, C2×C4.Dic3, C6×M4(2), C6×C3⋊C8, C3×C4.Dic3, C2×C6×C12, C6×C4.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, M4(2), C22×C4, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C2×M4(2), C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), C22×Dic3, C22×C12, C6×Dic3, S3×C2×C6, C2×C4.Dic3, C6×M4(2), C3×C4.Dic3, Dic3×C2×C6, C6×C4.Dic3

Smallest permutation representation of C6×C4.Dic3
►On 48 points
Generators in S48
(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 45 29 37 33 41)(26 46 30 38 34 42)(27 47 31 39 35 43)(28 48 32 40 36 44)

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

108 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6AE8A···8H12A···12H12I···12AJ24A···24P
order122222333334444446···66···68···812···1212···1224···24
size111122112221111221···12···26···61···12···26···6

108 irreducible representations

dim111111111111222222222222
type+++++-+-
imageC1C2C2C2C3C4C4C6C6C6C12C12S3Dic3D6Dic3M4(2)C3×S3C3×Dic3S3×C6C3×Dic3C4.Dic3C3×M4(2)C3×C4.Dic3
kernelC6×C4.Dic3C6×C3⋊C8C3×C4.Dic3C2×C6×C12C2×C4.Dic3C6×C12C2×C62C2×C3⋊C8C4.Dic3C22×C12C2×C12C22×C6C22×C12C2×C12C2×C12C22×C6C3×C6C22×C4C2×C4C2×C4C23C6C6C2
# reps12412624821241331426628816

Matrix representation of C6×C4.Dic3 ►in GL4(𝔽73) generated by

72000
07200
00650
00065
,
27000
04600
0010
0001
,
27000
02700
0080
00064
,
0100
27000
0001
0010
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,65,0,0,0,0,65],[27,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,8,0,0,0,0,64],[0,27,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
 

C6×C4.Dic3 in GAP, Magma, Sage, TeX 

C_6\times C_4.{\rm Dic}_3
 
% in TeX
 
G:=Group("C6xC4.Dic3");
 
// GroupNames label
 
G:=SmallGroup(288,692);
 
// by ID
 
G=gap.SmallGroup(288,692);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,1094,102,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
 
// generators/relations
 

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