metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C48.5C4, C24.6C8, C3⋊2M6(2), C12.1C16, C16.22D6, C16.2Dic3, C48.27C22, C3⋊C32⋊5C2, C4.(C3⋊C16), C8.3(C3⋊C8), (C2×C6).3C16, (C2×C12).9C8, C6.9(C2×C16), (C2×C16).7S3, C22.(C3⋊C16), (C2×C24).26C4, C12.43(C2×C8), (C2×C48).14C2, C24.81(C2×C4), (C2×C8).14Dic3, C8.22(C2×Dic3), C4.15(C2×C3⋊C8), C2.4(C2×C3⋊C16), (C2×C4).5(C3⋊C8), SmallGroup(192,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊M6(2)
G = < a,b,c | a3=b32=c2=1, bab-1=a-1, ac=ca, cbc=b17 >
Smallest permutation representation of C3⋊M6(2)
►On 96 points
Generators in S96
(1 73 58)(2 59 74)(3 75 60)(4 61 76)(5 77 62)(6 63 78)(7 79 64)(8 33 80)(9 81 34)(10 35 82)(11 83 36)(12 37 84)(13 85 38)(14 39 86)(15 87 40)(16 41 88)(17 89 42)(18 43 90)(19 91 44)(20 45 92)(21 93 46)(22 47 94)(23 95 48)(24 49 96)(25 65 50)(26 51 66)(27 67 52)(28 53 68)(29 69 54)(30 55 70)(31 71 56)(32 57 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)
(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)| (1,73,58)(2,59,74)(3,75,60)(4,61,76)(5,77,62)(6,63,78)(7,79,64)(8,33,80)(9,81,34)(10,35,82)(11,83,36)(12,37,84)(13,85,38)(14,39,86)(15,87,40)(16,41,88)(17,89,42)(18,43,90)(19,91,44)(20,45,92)(21,93,46)(22,47,94)(23,95,48)(24,49,96)(25,65,50)(26,51,66)(27,67,52)(28,53,68)(29,69,54)(30,55,70)(31,71,56)(32,57,72), (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), (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( (1,73,58)(2,59,74)(3,75,60)(4,61,76)(5,77,62)(6,63,78)(7,79,64)(8,33,80)(9,81,34)(10,35,82)(11,83,36)(12,37,84)(13,85,38)(14,39,86)(15,87,40)(16,41,88)(17,89,42)(18,43,90)(19,91,44)(20,45,92)(21,93,46)(22,47,94)(23,95,48)(24,49,96)(25,65,50)(26,51,66)(27,67,52)(28,53,68)(29,69,54)(30,55,70)(31,71,56)(32,57,72), (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), (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([[(1,73,58),(2,59,74),(3,75,60),(4,61,76),(5,77,62),(6,63,78),(7,79,64),(8,33,80),(9,81,34),(10,35,82),(11,83,36),(12,37,84),(13,85,38),(14,39,86),(15,87,40),(16,41,88),(17,89,42),(18,43,90),(19,91,44),(20,45,92),(21,93,46),(22,47,94),(23,95,48),(24,49,96),(25,65,50),(26,51,66),(27,67,52),(28,53,68),(29,69,54),(30,55,70),(31,71,56),(32,57,72)], [(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)], [(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)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 24A | ··· | 24H | 32A | ··· | 32P | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 32 | ··· | 32 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | C3⋊C16 | C3⋊C16 | M6(2) | C3⋊M6(2) |
| kernel | C3⋊M6(2) | C3⋊C32 | C2×C48 | C48 | C2×C24 | C24 | C2×C12 | C12 | C2×C6 | C2×C16 | C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C3⋊M6(2) ►in GL4(𝔽7) generated by
| 1 | 1 | 0 | 4 |
| 3 | 1 | 5 | 0 |
| 1 | 3 | 5 | 6 |
| 2 | 1 | 4 | 5 |
| 2 | 4 | 6 | 0 |
| 1 | 3 | 3 | 4 |
| 4 | 0 | 0 | 4 |
| 6 | 0 | 3 | 2 |
| 5 | 1 | 0 | 4 |
| 3 | 5 | 5 | 0 |
| 1 | 3 | 2 | 6 |
| 2 | 1 | 4 | 2 |
G:=sub<GL(4,GF(7))| [1,3,1,2,1,1,3,1,0,5,5,4,4,0,6,5],[2,1,4,6,4,3,0,0,6,3,0,3,0,4,4,2],[5,3,1,2,1,5,3,1,0,5,2,4,4,0,6,2] >;
C3⋊M6(2) in GAP, Magma, Sage, TeX
C_3\rtimes M_6(2)
% in TeX
G:=Group("C3:M6(2)"); // GroupNames label
G:=SmallGroup(192,58);
// by ID
G=gap.SmallGroup(192,58);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,477,58,80,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^3=b^32=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^17>;
// generators/relations
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