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
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 20 58)(2 21 59)(3 22 60)(4 23 61)(5 24 62)(6 25 63)(7 26 64)(8 27 49)(9 28 50)(10 29 51)(11 30 52)(12 31 53)(13 32 54)(14 17 55)(15 18 56)(16 19 57)(33 93 74)(34 94 75)(35 95 76)(36 96 77)(37 81 78)(38 82 79)(39 83 80)(40 84 65)(41 85 66)(42 86 67)(43 87 68)(44 88 69)(45 89 70)(46 90 71)(47 91 72)(48 92 73)
(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 83 9 91)(2 82 10 90)(3 81 11 89)(4 96 12 88)(5 95 13 87)(6 94 14 86)(7 93 15 85)(8 92 16 84)(17 67 25 75)(18 66 26 74)(19 65 27 73)(20 80 28 72)(21 79 29 71)(22 78 30 70)(23 77 31 69)(24 76 32 68)(33 56 41 64)(34 55 42 63)(35 54 43 62)(36 53 44 61)(37 52 45 60)(38 51 46 59)(39 50 47 58)(40 49 48 57)
G:=sub<Sym(96)| (1,20,58)(2,21,59)(3,22,60)(4,23,61)(5,24,62)(6,25,63)(7,26,64)(8,27,49)(9,28,50)(10,29,51)(11,30,52)(12,31,53)(13,32,54)(14,17,55)(15,18,56)(16,19,57)(33,93,74)(34,94,75)(35,95,76)(36,96,77)(37,81,78)(38,82,79)(39,83,80)(40,84,65)(41,85,66)(42,86,67)(43,87,68)(44,88,69)(45,89,70)(46,90,71)(47,91,72)(48,92,73), (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), (1,83,9,91)(2,82,10,90)(3,81,11,89)(4,96,12,88)(5,95,13,87)(6,94,14,86)(7,93,15,85)(8,92,16,84)(17,67,25,75)(18,66,26,74)(19,65,27,73)(20,80,28,72)(21,79,29,71)(22,78,30,70)(23,77,31,69)(24,76,32,68)(33,56,41,64)(34,55,42,63)(35,54,43,62)(36,53,44,61)(37,52,45,60)(38,51,46,59)(39,50,47,58)(40,49,48,57)>;
G:=Group( (1,20,58)(2,21,59)(3,22,60)(4,23,61)(5,24,62)(6,25,63)(7,26,64)(8,27,49)(9,28,50)(10,29,51)(11,30,52)(12,31,53)(13,32,54)(14,17,55)(15,18,56)(16,19,57)(33,93,74)(34,94,75)(35,95,76)(36,96,77)(37,81,78)(38,82,79)(39,83,80)(40,84,65)(41,85,66)(42,86,67)(43,87,68)(44,88,69)(45,89,70)(46,90,71)(47,91,72)(48,92,73), (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), (1,83,9,91)(2,82,10,90)(3,81,11,89)(4,96,12,88)(5,95,13,87)(6,94,14,86)(7,93,15,85)(8,92,16,84)(17,67,25,75)(18,66,26,74)(19,65,27,73)(20,80,28,72)(21,79,29,71)(22,78,30,70)(23,77,31,69)(24,76,32,68)(33,56,41,64)(34,55,42,63)(35,54,43,62)(36,53,44,61)(37,52,45,60)(38,51,46,59)(39,50,47,58)(40,49,48,57) );
G=PermutationGroup([[(1,20,58),(2,21,59),(3,22,60),(4,23,61),(5,24,62),(6,25,63),(7,26,64),(8,27,49),(9,28,50),(10,29,51),(11,30,52),(12,31,53),(13,32,54),(14,17,55),(15,18,56),(16,19,57),(33,93,74),(34,94,75),(35,95,76),(36,96,77),(37,81,78),(38,82,79),(39,83,80),(40,84,65),(41,85,66),(42,86,67),(43,87,68),(44,88,69),(45,89,70),(46,90,71),(47,91,72),(48,92,73)], [(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)], [(1,83,9,91),(2,82,10,90),(3,81,11,89),(4,96,12,88),(5,95,13,87),(6,94,14,86),(7,93,15,85),(8,92,16,84),(17,67,25,75),(18,66,26,74),(19,65,27,73),(20,80,28,72),(21,79,29,71),(22,78,30,70),(23,77,31,69),(24,76,32,68),(33,56,41,64),(34,55,42,63),(35,54,43,62),(36,53,44,61),(37,52,45,60),(38,51,46,59),(39,50,47,58),(40,49,48,57)]])
C3×Q32 is a maximal subgroup of
C3⋊SD64 C3⋊Q64 Q32⋊S3 D48⋊5C2
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
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