direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×M6(2), C4.C48, C96⋊7C2, C32⋊3C6, C8.3C24, C48.6C4, C24.9C8, C22.C48, C12.4C16, C16.2C12, C48.30C22, (C2×C4).5C24, C16.7(C2×C6), (C2×C16).8C6, C2.3(C2×C48), (C2×C6).1C16, (C2×C48).18C2, (C2×C12).14C8, C6.13(C2×C16), (C2×C8).13C12, C4.13(C2×C24), C8.22(C2×C12), (C2×C24).32C4, C24.91(C2×C4), C12.53(C2×C8), SmallGroup(192,176)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M6(2)
G = < a,b,c | a3=b32=c2=1, ab=ba, ac=ca, cbc=b17 >
Smallest permutation representation of C3×M6(2)
►On 96 points
Generators in S96
(1 83 38)(2 84 39)(3 85 40)(4 86 41)(5 87 42)(6 88 43)(7 89 44)(8 90 45)(9 91 46)(10 92 47)(11 93 48)(12 94 49)(13 95 50)(14 96 51)(15 65 52)(16 66 53)(17 67 54)(18 68 55)(19 69 56)(20 70 57)(21 71 58)(22 72 59)(23 73 60)(24 74 61)(25 75 62)(26 76 63)(27 77 64)(28 78 33)(29 79 34)(30 80 35)(31 81 36)(32 82 37)
(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)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)(33 49)(35 51)(37 53)(39 55)(41 57)(43 59)(45 61)(47 63)(66 82)(68 84)(70 86)(72 88)(74 90)(76 92)(78 94)(80 96)
G:=sub<Sym(96)| (1,83,38)(2,84,39)(3,85,40)(4,86,41)(5,87,42)(6,88,43)(7,89,44)(8,90,45)(9,91,46)(10,92,47)(11,93,48)(12,94,49)(13,95,50)(14,96,51)(15,65,52)(16,66,53)(17,67,54)(18,68,55)(19,69,56)(20,70,57)(21,71,58)(22,72,59)(23,73,60)(24,74,61)(25,75,62)(26,76,63)(27,77,64)(28,78,33)(29,79,34)(30,80,35)(31,81,36)(32,82,37), (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), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(33,49)(35,51)(37,53)(39,55)(41,57)(43,59)(45,61)(47,63)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96)>;
G:=Group( (1,83,38)(2,84,39)(3,85,40)(4,86,41)(5,87,42)(6,88,43)(7,89,44)(8,90,45)(9,91,46)(10,92,47)(11,93,48)(12,94,49)(13,95,50)(14,96,51)(15,65,52)(16,66,53)(17,67,54)(18,68,55)(19,69,56)(20,70,57)(21,71,58)(22,72,59)(23,73,60)(24,74,61)(25,75,62)(26,76,63)(27,77,64)(28,78,33)(29,79,34)(30,80,35)(31,81,36)(32,82,37), (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), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(33,49)(35,51)(37,53)(39,55)(41,57)(43,59)(45,61)(47,63)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96) );
G=PermutationGroup([[(1,83,38),(2,84,39),(3,85,40),(4,86,41),(5,87,42),(6,88,43),(7,89,44),(8,90,45),(9,91,46),(10,92,47),(11,93,48),(12,94,49),(13,95,50),(14,96,51),(15,65,52),(16,66,53),(17,67,54),(18,68,55),(19,69,56),(20,70,57),(21,71,58),(22,72,59),(23,73,60),(24,74,61),(25,75,62),(26,76,63),(27,77,64),(28,78,33),(29,79,34),(30,80,35),(31,81,36),(32,82,37)], [(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)], [(2,18),(4,20),(6,22),(8,24),(10,26),(12,28),(14,30),(16,32),(33,49),(35,51),(37,53),(39,55),(41,57),(43,59),(45,61),(47,63),(66,82),(68,84),(70,86),(72,88),(74,90),(76,92),(78,94),(80,96)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 32A | ··· | 32P | 48A | ··· | 48P | 48Q | ··· | 48X | 96A | ··· | 96AF |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C16 | C16 | C24 | C24 | C48 | C48 | M6(2) | C3×M6(2) |
| kernel | C3×M6(2) | C96 | C2×C48 | M6(2) | C48 | C2×C24 | C32 | C2×C16 | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 16 | 8 | 16 |
Matrix representation of C3×M6(2) ►in GL3(𝔽97) generated by
| 35 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 96 | 0 | 0 |
| 0 | 33 | 42 |
| 0 | 36 | 64 |
| 96 | 0 | 0 |
| 0 | 1 | 79 |
| 0 | 0 | 96 |
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[96,0,0,0,33,36,0,42,64],[96,0,0,0,1,0,0,79,96] >;
C3×M6(2) in GAP, Magma, Sage, TeX
C_3\times M_6(2)
% in TeX
G:=Group("C3xM6(2)"); // GroupNames label
G:=SmallGroup(192,176);
// by ID
G=gap.SmallGroup(192,176);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,1373,80,102,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^17>;
// generators/relations
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