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G = C3×Q32order 96 = 25·3

Direct product of C3 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C3×Q32, C16.C6, Q16.C6, C48.2C2, C6.17D8, C12.38D4, C24.21C22, C8.4(C2×C6), C2.5(C3×D8), C4.3(C3×D4), (C3×Q16).2C2, SmallGroup(96,63)

Series: Derived Chief Lower central Upper central

C1C8 — C3×Q32
C1C2C4C8C24C3×Q16 — C3×Q32
C1C2C4C8 — C3×Q32
C1C6C12C24 — C3×Q32

Generators and relations for C3×Q32
 G = < a,b,c | a3=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

4C4
4C4
2Q8
2Q8
4C12
4C12
2C3×Q8
2C3×Q8

Smallest permutation representation of C3×Q32
Regular action on 96 points
Generators in S96
(1 39 62)(2 40 63)(3 41 64)(4 42 49)(5 43 50)(6 44 51)(7 45 52)(8 46 53)(9 47 54)(10 48 55)(11 33 56)(12 34 57)(13 35 58)(14 36 59)(15 37 60)(16 38 61)(17 95 73)(18 96 74)(19 81 75)(20 82 76)(21 83 77)(22 84 78)(23 85 79)(24 86 80)(25 87 65)(26 88 66)(27 89 67)(28 90 68)(29 91 69)(30 92 70)(31 93 71)(32 94 72)
(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)
(1 80 9 72)(2 79 10 71)(3 78 11 70)(4 77 12 69)(5 76 13 68)(6 75 14 67)(7 74 15 66)(8 73 16 65)(17 38 25 46)(18 37 26 45)(19 36 27 44)(20 35 28 43)(21 34 29 42)(22 33 30 41)(23 48 31 40)(24 47 32 39)(49 83 57 91)(50 82 58 90)(51 81 59 89)(52 96 60 88)(53 95 61 87)(54 94 62 86)(55 93 63 85)(56 92 64 84)

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

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

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

C3×Q32 is a maximal subgroup of   C3⋊SD64  C3⋊Q64  Q32⋊S3  D485C2

33 conjugacy classes

class 1  2 3A3B4A4B4C6A6B8A8B12A12B12C12D12E12F16A16B16C16D24A24B24C24D48A···48H
order12334446688121212121212161616162424242448···48
size11112881122228888222222222···2

33 irreducible representations

dim111111222222
type+++++-
imageC1C2C2C3C6C6D4D8C3×D4Q32C3×D8C3×Q32
kernelC3×Q32C48C3×Q16Q32C16Q16C12C6C4C3C2C1
# reps112224122448

Matrix representation of C3×Q32 in GL2(𝔽31) generated by

250
025
,
030
15
,
2425
297
G:=sub<GL(2,GF(31))| [25,0,0,25],[0,1,30,5],[24,29,25,7] >;

C3×Q32 in GAP, Magma, Sage, TeX

C_3\times Q_{32}
% in TeX

G:=Group("C3xQ32");
// GroupNames label

G:=SmallGroup(96,63);
// by ID

G=gap.SmallGroup(96,63);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,288,169,295,867,441,165,2164,1090,88]);
// Polycyclic

G:=Group<a,b,c|a^3=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Q32 in TeX

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