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G = C3⋊C32order 96 = 25·3

The semidirect product of C3 and C32 acting via C32/C16=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3⋊C32, C6.C16, C12.2C8, C24.2C4, C48.3C2, C16.2S3, C8.3Dic3, C2.(C3⋊C16), C4.2(C3⋊C8), SmallGroup(96,1)

Series: Derived Chief Lower central Upper central

C1C3 — C3⋊C32
C1C3C6C12C24C48 — C3⋊C32
C3 — C3⋊C32
C1C16

Generators and relations for C3⋊C32
 G = < a,b | a3=b32=1, bab-1=a-1 >

3C32

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

C3⋊C32 is a maximal subgroup of
S3×C32  C96⋊C2  C3⋊M6(2)  C3⋊D32  D16.S3  C3⋊SD64  C3⋊Q64  C9⋊C32  C48.S3  C153C32  C15⋊C32
C3⋊C32 is a maximal quotient of
C3⋊C64  C9⋊C32  C48.S3  C153C32  C15⋊C32

48 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D12A12B16A···16H24A24B24C24D32A···32P48A···48H
order1234468888121216···162424242432···3248···48
size1121121111221···122223···32···2

48 irreducible representations

dim11111122222
type+++-
imageC1C2C4C8C16C32S3Dic3C3⋊C8C3⋊C16C3⋊C32
kernelC3⋊C32C48C24C12C6C3C16C8C4C2C1
# reps112481611248

Matrix representation of C3⋊C32 in GL2(𝔽17) generated by

214
814
,
011
10
G:=sub<GL(2,GF(17))| [2,8,14,14],[0,1,11,0] >;

C3⋊C32 in GAP, Magma, Sage, TeX

C_3\rtimes C_{32}
% in TeX

G:=Group("C3:C32");
// GroupNames label

G:=SmallGroup(96,1);
// by ID

G=gap.SmallGroup(96,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,12,31,50,69,2309]);
// Polycyclic

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

Export

Subgroup lattice of C3⋊C32 in TeX

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