Copied to
clipboard

G = C6×C322C8order 432 = 24·33

Direct product of C6 and C322C8

direct product, metabelian, soluble, monomial, A-group

Aliases: C6×C322C8, C62.7C12, (C3×C6)⋊3C24, (C32×C6)⋊3C8, C3314(C2×C8), C327(C2×C24), (C3×C62).1C4, C3⋊Dic3.7C12, C2.3(C6×C32⋊C4), C6.23(C2×C32⋊C4), (C3×C6).14(C2×C12), (C2×C6).8(C32⋊C4), (C6×C3⋊Dic3).6C2, (C3×C3⋊Dic3).4C4, C22.2(C3×C32⋊C4), C3⋊Dic3.14(C2×C6), (C2×C3⋊Dic3).10C6, (C32×C6).12(C2×C4), (C3×C3⋊Dic3).39C22, SmallGroup(432,632)

Series: Derived Chief Lower central Upper central

C1C32 — C6×C322C8
C1C32C3×C6C3⋊Dic3C3×C3⋊Dic3C3×C322C8 — C6×C322C8
C32 — C6×C322C8
C1C2×C6

Generators and relations for C6×C322C8
 G = < a,b,c,d | a6=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 332 in 96 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, C6, C6 [×2], C6 [×12], C8 [×2], C2×C4, C32, C32 [×4], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×4], C2×C8, C3×C6, C3×C6 [×2], C3×C6 [×12], C24 [×2], C2×Dic3 [×2], C2×C12, C33, C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C62 [×4], C2×C24, C32×C6, C32×C6 [×2], C322C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C3×C62, C2×C322C8, C3×C322C8 [×2], C6×C3⋊Dic3, C6×C322C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C8 [×2], C2×C4, C12 [×2], C2×C6, C2×C8, C24 [×2], C2×C12, C32⋊C4, C2×C24, C322C8 [×2], C2×C32⋊C4, C3×C32⋊C4, C2×C322C8, C3×C322C8 [×2], C6×C32⋊C4, C6×C322C8

Smallest permutation representation of C6×C322C8
On 48 points
Generators in S48
(1 33 31 15 21 47)(2 34 32 16 22 48)(3 35 25 9 23 41)(4 36 26 10 24 42)(5 37 27 11 17 43)(6 38 28 12 18 44)(7 39 29 13 19 45)(8 40 30 14 20 46)
(1 31 21)(2 22 32)(3 23 25)(4 26 24)(5 27 17)(6 18 28)(7 19 29)(8 30 20)(9 35 41)(10 42 36)(11 43 37)(12 38 44)(13 39 45)(14 46 40)(15 47 33)(16 34 48)
(2 32 22)(4 24 26)(6 28 18)(8 20 30)(10 36 42)(12 44 38)(14 40 46)(16 48 34)
(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)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C···3H4A4B4C4D6A···6F6G···6X8A···8H12A···12H24A···24P
order1222333···344446···66···68···812···1224···24
size1111114···499991···14···49···99···99···9

72 irreducible representations

dim111111111111444444
type++++-+
imageC1C2C2C3C4C4C6C6C8C12C12C24C32⋊C4C322C8C2×C32⋊C4C3×C32⋊C4C3×C322C8C6×C32⋊C4
kernelC6×C322C8C3×C322C8C6×C3⋊Dic3C2×C322C8C3×C3⋊Dic3C3×C62C322C8C2×C3⋊Dic3C32×C6C3⋊Dic3C62C3×C6C2×C6C6C6C22C2C2
# reps1212224284416242484

Matrix representation of C6×C322C8 in GL5(𝔽73)

650000
09000
00900
00090
00009
,
10000
080012
0064250
00080
000064
,
10000
0105838
0015467
000640
00008
,
10000
035476938
066384
007676
06604738

G:=sub<GL(5,GF(73))| [65,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,8,0,0,0,0,0,64,0,0,0,0,25,8,0,0,12,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,58,54,64,0,0,38,67,0,8],[1,0,0,0,0,0,35,6,0,66,0,47,6,7,0,0,69,38,67,47,0,38,4,6,38] >;

C6×C322C8 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("C6xC3^2:2C8");
// GroupNames label

G:=SmallGroup(432,632);
// by ID

G=gap.SmallGroup(432,632);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,80,14117,362,18822,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

׿
×
𝔽