Copied to
clipboard

## 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

 Derived series C1 — C8 — C3×Q32
 Chief series C1 — C2 — C4 — C8 — C24 — C3×Q16 — C3×Q32
 Lower central C1 — C2 — C4 — C8 — C3×Q32
 Upper central C1 — C6 — C12 — C24 — 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 >

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 3A 3B 4A 4B 4C 6A 6B 8A 8B 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D 24A 24B 24C 24D 48A ··· 48H order 1 2 3 3 4 4 4 6 6 8 8 12 12 12 12 12 12 16 16 16 16 24 24 24 24 48 ··· 48 size 1 1 1 1 2 8 8 1 1 2 2 2 2 8 8 8 8 2 2 2 2 2 2 2 2 2 ··· 2

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 Q32 C3×D8 C3×Q32 kernel C3×Q32 C48 C3×Q16 Q32 C16 Q16 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 4 8

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

 25 0 0 25
,
 0 30 1 5
,
 24 25 29 7
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

׿
×
𝔽