direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C3⋊C32, C3⋊C96, C6.C48, C48.6C6, C32⋊3C32, 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
| C3 — C3×C3⋊C32 |
Generators and relations for C3×C3⋊C32
G = < a,b,c | a3=b3=c32=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C3⋊C32
►On 96 points
Generators in S96
(1 42 77)(2 43 78)(3 44 79)(4 45 80)(5 46 81)(6 47 82)(7 48 83)(8 49 84)(9 50 85)(10 51 86)(11 52 87)(12 53 88)(13 54 89)(14 55 90)(15 56 91)(16 57 92)(17 58 93)(18 59 94)(19 60 95)(20 61 96)(21 62 65)(22 63 66)(23 64 67)(24 33 68)(25 34 69)(26 35 70)(27 36 71)(28 37 72)(29 38 73)(30 39 74)(31 40 75)(32 41 76)
(1 42 77)(2 78 43)(3 44 79)(4 80 45)(5 46 81)(6 82 47)(7 48 83)(8 84 49)(9 50 85)(10 86 51)(11 52 87)(12 88 53)(13 54 89)(14 90 55)(15 56 91)(16 92 57)(17 58 93)(18 94 59)(19 60 95)(20 96 61)(21 62 65)(22 66 63)(23 64 67)(24 68 33)(25 34 69)(26 70 35)(27 36 71)(28 72 37)(29 38 73)(30 74 39)(31 40 75)(32 76 41)
(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)| (1,42,77)(2,43,78)(3,44,79)(4,45,80)(5,46,81)(6,47,82)(7,48,83)(8,49,84)(9,50,85)(10,51,86)(11,52,87)(12,53,88)(13,54,89)(14,55,90)(15,56,91)(16,57,92)(17,58,93)(18,59,94)(19,60,95)(20,61,96)(21,62,65)(22,63,66)(23,64,67)(24,33,68)(25,34,69)(26,35,70)(27,36,71)(28,37,72)(29,38,73)(30,39,74)(31,40,75)(32,41,76), (1,42,77)(2,78,43)(3,44,79)(4,80,45)(5,46,81)(6,82,47)(7,48,83)(8,84,49)(9,50,85)(10,86,51)(11,52,87)(12,88,53)(13,54,89)(14,90,55)(15,56,91)(16,92,57)(17,58,93)(18,94,59)(19,60,95)(20,96,61)(21,62,65)(22,66,63)(23,64,67)(24,68,33)(25,34,69)(26,70,35)(27,36,71)(28,72,37)(29,38,73)(30,74,39)(31,40,75)(32,76,41), (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( (1,42,77)(2,43,78)(3,44,79)(4,45,80)(5,46,81)(6,47,82)(7,48,83)(8,49,84)(9,50,85)(10,51,86)(11,52,87)(12,53,88)(13,54,89)(14,55,90)(15,56,91)(16,57,92)(17,58,93)(18,59,94)(19,60,95)(20,61,96)(21,62,65)(22,63,66)(23,64,67)(24,33,68)(25,34,69)(26,35,70)(27,36,71)(28,37,72)(29,38,73)(30,39,74)(31,40,75)(32,41,76), (1,42,77)(2,78,43)(3,44,79)(4,80,45)(5,46,81)(6,82,47)(7,48,83)(8,84,49)(9,50,85)(10,86,51)(11,52,87)(12,88,53)(13,54,89)(14,90,55)(15,56,91)(16,92,57)(17,58,93)(18,94,59)(19,60,95)(20,96,61)(21,62,65)(22,66,63)(23,64,67)(24,68,33)(25,34,69)(26,70,35)(27,36,71)(28,72,37)(29,38,73)(30,74,39)(31,40,75)(32,76,41), (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([[(1,42,77),(2,43,78),(3,44,79),(4,45,80),(5,46,81),(6,47,82),(7,48,83),(8,49,84),(9,50,85),(10,51,86),(11,52,87),(12,53,88),(13,54,89),(14,55,90),(15,56,91),(16,57,92),(17,58,93),(18,59,94),(19,60,95),(20,61,96),(21,62,65),(22,63,66),(23,64,67),(24,33,68),(25,34,69),(26,35,70),(27,36,71),(28,37,72),(29,38,73),(30,39,74),(31,40,75),(32,41,76)], [(1,42,77),(2,78,43),(3,44,79),(4,80,45),(5,46,81),(6,82,47),(7,48,83),(8,84,49),(9,50,85),(10,86,51),(11,52,87),(12,88,53),(13,54,89),(14,90,55),(15,56,91),(16,92,57),(17,58,93),(18,94,59),(19,60,95),(20,96,61),(21,62,65),(22,66,63),(23,64,67),(24,68,33),(25,34,69),(26,70,35),(27,36,71),(28,72,37),(29,38,73),(30,74,39),(31,40,75),(32,76,41)], [(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 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 16A | ··· | 16H | 24A | ··· | 24H | 24I | ··· | 24T | 32A | ··· | 32P | 48A | ··· | 48P | 48Q | ··· | 48AN | 96A | ··· | 96AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C32 | C48 | C96 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3⋊C16 | C3×C3⋊C8 | C3⋊C32 | C3×C3⋊C16 | C3×C3⋊C32 |
| kernel | C3×C3⋊C32 | C3×C48 | C3⋊C32 | C3×C24 | C48 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 | C48 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 32 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C3⋊C32 ►in GL3(𝔽97) generated by
| 35 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 1 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 35 |
| 55 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 64 | 0 |
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
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