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G = C3×C3⋊C32order 288 = 25·32

Direct product of C3 and C3⋊C32

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×C3⋊C32, C3⋊C96, C6.C48, C48.6C6, C323C32, C24.4C12, C12.2C24, C48.10S3, C24.15Dic3, C6.4(C3⋊C16), C16.2(C3×S3), (C3×C12).9C8, (C3×C24).8C4, (C3×C6).3C16, (C3×C48).4C2, C12.10(C3⋊C8), C8.3(C3×Dic3), C2.(C3×C3⋊C16), C4.2(C3×C3⋊C8), SmallGroup(288,64)

Series: Derived Chief Lower central Upper central

C1C3 — C3×C3⋊C32
C1C3C6C12C24C48C3×C48 — C3×C3⋊C32
C3 — C3×C3⋊C32
C1C48

Generators and relations for C3×C3⋊C32
 G = < a,b,c | a3=b3=c32=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
2C12
2C24
3C32
2C48
3C96

Smallest permutation representation of C3×C3⋊C32
On 96 points
Generators in S96
(1 81 44)(2 82 45)(3 83 46)(4 84 47)(5 85 48)(6 86 49)(7 87 50)(8 88 51)(9 89 52)(10 90 53)(11 91 54)(12 92 55)(13 93 56)(14 94 57)(15 95 58)(16 96 59)(17 65 60)(18 66 61)(19 67 62)(20 68 63)(21 69 64)(22 70 33)(23 71 34)(24 72 35)(25 73 36)(26 74 37)(27 75 38)(28 76 39)(29 77 40)(30 78 41)(31 79 42)(32 80 43)
(1 81 44)(2 45 82)(3 83 46)(4 47 84)(5 85 48)(6 49 86)(7 87 50)(8 51 88)(9 89 52)(10 53 90)(11 91 54)(12 55 92)(13 93 56)(14 57 94)(15 95 58)(16 59 96)(17 65 60)(18 61 66)(19 67 62)(20 63 68)(21 69 64)(22 33 70)(23 71 34)(24 35 72)(25 73 36)(26 37 74)(27 75 38)(28 39 76)(29 77 40)(30 41 78)(31 79 42)(32 43 80)
(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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,81,44)(2,82,45)(3,83,46)(4,84,47)(5,85,48)(6,86,49)(7,87,50)(8,88,51)(9,89,52)(10,90,53)(11,91,54)(12,92,55)(13,93,56)(14,94,57)(15,95,58)(16,96,59)(17,65,60)(18,66,61)(19,67,62)(20,68,63)(21,69,64)(22,70,33)(23,71,34)(24,72,35)(25,73,36)(26,74,37)(27,75,38)(28,76,39)(29,77,40)(30,78,41)(31,79,42)(32,80,43), (1,81,44)(2,45,82)(3,83,46)(4,47,84)(5,85,48)(6,49,86)(7,87,50)(8,51,88)(9,89,52)(10,53,90)(11,91,54)(12,55,92)(13,93,56)(14,57,94)(15,95,58)(16,59,96)(17,65,60)(18,61,66)(19,67,62)(20,63,68)(21,69,64)(22,33,70)(23,71,34)(24,35,72)(25,73,36)(26,37,74)(27,75,38)(28,39,76)(29,77,40)(30,41,78)(31,79,42)(32,43,80), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)>;

G:=Group( (1,81,44)(2,82,45)(3,83,46)(4,84,47)(5,85,48)(6,86,49)(7,87,50)(8,88,51)(9,89,52)(10,90,53)(11,91,54)(12,92,55)(13,93,56)(14,94,57)(15,95,58)(16,96,59)(17,65,60)(18,66,61)(19,67,62)(20,68,63)(21,69,64)(22,70,33)(23,71,34)(24,72,35)(25,73,36)(26,74,37)(27,75,38)(28,76,39)(29,77,40)(30,78,41)(31,79,42)(32,80,43), (1,81,44)(2,45,82)(3,83,46)(4,47,84)(5,85,48)(6,49,86)(7,87,50)(8,51,88)(9,89,52)(10,53,90)(11,91,54)(12,55,92)(13,93,56)(14,57,94)(15,95,58)(16,59,96)(17,65,60)(18,61,66)(19,67,62)(20,63,68)(21,69,64)(22,33,70)(23,71,34)(24,35,72)(25,73,36)(26,37,74)(27,75,38)(28,39,76)(29,77,40)(30,41,78)(31,79,42)(32,43,80), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96) );

G=PermutationGroup([(1,81,44),(2,82,45),(3,83,46),(4,84,47),(5,85,48),(6,86,49),(7,87,50),(8,88,51),(9,89,52),(10,90,53),(11,91,54),(12,92,55),(13,93,56),(14,94,57),(15,95,58),(16,96,59),(17,65,60),(18,66,61),(19,67,62),(20,68,63),(21,69,64),(22,70,33),(23,71,34),(24,72,35),(25,73,36),(26,74,37),(27,75,38),(28,76,39),(29,77,40),(30,78,41),(31,79,42),(32,80,43)], [(1,81,44),(2,45,82),(3,83,46),(4,47,84),(5,85,48),(6,49,86),(7,87,50),(8,51,88),(9,89,52),(10,53,90),(11,91,54),(12,55,92),(13,93,56),(14,57,94),(15,95,58),(16,59,96),(17,65,60),(18,61,66),(19,67,62),(20,63,68),(21,69,64),(22,33,70),(23,71,34),(24,35,72),(25,73,36),(26,37,74),(27,75,38),(28,39,76),(29,77,40),(30,41,78),(31,79,42),(32,43,80)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)])

144 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12J16A···16H24A···24H24I···24T32A···32P48A···48P48Q···48AN96A···96AF
order1233333446666688881212121212···1216···1624···2424···2432···3248···4848···4896···96
size11112221111222111111112···21···11···12···23···31···12···23···3

144 irreducible representations

dim1111111111112222222222
type+++-
imageC1C2C3C4C6C8C12C16C24C32C48C96S3Dic3C3×S3C3⋊C8C3×Dic3C3⋊C16C3×C3⋊C8C3⋊C32C3×C3⋊C16C3×C3⋊C32
kernelC3×C3⋊C32C3×C48C3⋊C32C3×C24C48C3×C12C24C3×C6C12C32C6C3C48C24C16C12C8C6C4C3C2C1
# reps11222448816163211222448816

Matrix representation of C3×C3⋊C32 in GL3(𝔽97) generated by

3500
0610
0061
,
100
0610
0035
,
5500
001
0640
G:=sub<GL(3,GF(97))| [35,0,0,0,61,0,0,0,61],[1,0,0,0,61,0,0,0,35],[55,0,0,0,0,64,0,1,0] >;

C3×C3⋊C32 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes C_{32}
% in TeX

G:=Group("C3xC3:C32");
// GroupNames label

G:=SmallGroup(288,64);
// by ID

G=gap.SmallGroup(288,64);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-2,-3,42,58,80,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^32=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×C3⋊C32 in TeX

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