direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×M6(2), C4○M6(2), C8○M6(2), C32⋊4C22, C16○M6(2), C23.3C16, C16.16C23, (C2×C32)⋊8C2, C8.26(C2×C8), (C2×C8).19C8, (C2×C4).6C16, C16.23(C2×C4), (C2×C16).19C4, C4.10(C2×C16), (C22×C8).49C4, C8.68(C22×C4), (C22×C4).17C8, C2.6(C22×C16), C4.37(C22×C8), (C22×C16).18C2, C22.11(C2×C16), (C2×C16).106C22, (C2×C4).101(C2×C8), (C2×C8).254(C2×C4), SmallGroup(128,989)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×M6(2)
G = < a,b,c | a2=b32=c2=1, ab=ba, ac=ca, cbc=b17 >
Smallest permutation representation of C2×M6(2)
►On 64 points
Generators in S64
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)
(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)
(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)
G:=sub<Sym(64)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (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)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (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) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45)], [(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)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P | 16Q | ··· | 16X | 32A | ··· | 32AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | M6(2) |
| kernel | C2×M6(2) | C2×C32 | M6(2) | C22×C16 | C2×C16 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 12 | 4 | 24 | 8 | 16 |
Matrix representation of C2×M6(2) ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 22 | 0 | 0 |
| 0 | 0 | 96 |
| 0 | 89 | 0 |
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 96 |
G:=sub<GL(3,GF(97))| [96,0,0,0,1,0,0,0,1],[22,0,0,0,0,89,0,96,0],[96,0,0,0,1,0,0,0,96] >;
C2×M6(2) in GAP, Magma, Sage, TeX
C_2\times M_6(2)
% in TeX
G:=Group("C2xM6(2)"); // GroupNames label
G:=SmallGroup(128,989);
// by ID
G=gap.SmallGroup(128,989);
# by ID
G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,56,925,80,102,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^17>;
// generators/relations
Export