direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×M5(2), C48⋊14C22, C23.4C24, C24.82C23, (C2×C16)⋊8C6, C16⋊4(C2×C6), (C2×C48)⋊18C2, (C2×C4).6C24, C24.89(C2×C4), C8.20(C2×C12), (C2×C8).14C12, (C2×C12).15C8, C12.50(C2×C8), (C2×C24).33C4, C4.10(C2×C24), (C22×C6).4C8, C22.6(C2×C24), C8.15(C22×C6), C2.6(C22×C24), C6.35(C22×C8), (C22×C8).18C6, C4.35(C22×C12), (C22×C24).34C2, (C22×C4).20C12, (C22×C12).38C4, (C2×C24).448C22, C12.193(C22×C4), (C2×C6).24(C2×C8), (C2×C4).85(C2×C12), (C2×C8).102(C2×C6), (C2×C12).340(C2×C4), SmallGroup(192,936)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×M5(2)
G = < a,b,c | a6=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Subgroups: 98 in 90 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C16, C2×C8, C2×C8, C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, C2×C16, M5(2), C22×C8, C48, C2×C24, C2×C24, C22×C12, C2×M5(2), C2×C48, C3×M5(2), C22×C24, C6×M5(2)
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C12, C2×C6, C2×C8, C22×C4, C24, C2×C12, C22×C6, M5(2), C22×C8, C2×C24, C22×C12, C2×M5(2), C3×M5(2), C22×C24, C6×M5(2)
►On 96 points
(1 20 60 47 67 86)(2 21 61 48 68 87)(3 22 62 33 69 88)(4 23 63 34 70 89)(5 24 64 35 71 90)(6 25 49 36 72 91)(7 26 50 37 73 92)(8 27 51 38 74 93)(9 28 52 39 75 94)(10 29 53 40 76 95)(11 30 54 41 77 96)(12 31 55 42 78 81)(13 32 56 43 79 82)(14 17 57 44 80 83)(15 18 58 45 65 84)(16 19 59 46 66 85)
(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 39)(2 48)(3 41)(4 34)(5 43)(6 36)(7 45)(8 38)(9 47)(10 40)(11 33)(12 42)(13 35)(14 44)(15 37)(16 46)(17 80)(18 73)(19 66)(20 75)(21 68)(22 77)(23 70)(24 79)(25 72)(26 65)(27 74)(28 67)(29 76)(30 69)(31 78)(32 71)(49 91)(50 84)(51 93)(52 86)(53 95)(54 88)(55 81)(56 90)(57 83)(58 92)(59 85)(60 94)(61 87)(62 96)(63 89)(64 82)
G:=sub<Sym(96)| (1,20,60,47,67,86)(2,21,61,48,68,87)(3,22,62,33,69,88)(4,23,63,34,70,89)(5,24,64,35,71,90)(6,25,49,36,72,91)(7,26,50,37,73,92)(8,27,51,38,74,93)(9,28,52,39,75,94)(10,29,53,40,76,95)(11,30,54,41,77,96)(12,31,55,42,78,81)(13,32,56,43,79,82)(14,17,57,44,80,83)(15,18,58,45,65,84)(16,19,59,46,66,85), (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), (1,39)(2,48)(3,41)(4,34)(5,43)(6,36)(7,45)(8,38)(9,47)(10,40)(11,33)(12,42)(13,35)(14,44)(15,37)(16,46)(17,80)(18,73)(19,66)(20,75)(21,68)(22,77)(23,70)(24,79)(25,72)(26,65)(27,74)(28,67)(29,76)(30,69)(31,78)(32,71)(49,91)(50,84)(51,93)(52,86)(53,95)(54,88)(55,81)(56,90)(57,83)(58,92)(59,85)(60,94)(61,87)(62,96)(63,89)(64,82)>;
G:=Group( (1,20,60,47,67,86)(2,21,61,48,68,87)(3,22,62,33,69,88)(4,23,63,34,70,89)(5,24,64,35,71,90)(6,25,49,36,72,91)(7,26,50,37,73,92)(8,27,51,38,74,93)(9,28,52,39,75,94)(10,29,53,40,76,95)(11,30,54,41,77,96)(12,31,55,42,78,81)(13,32,56,43,79,82)(14,17,57,44,80,83)(15,18,58,45,65,84)(16,19,59,46,66,85), (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), (1,39)(2,48)(3,41)(4,34)(5,43)(6,36)(7,45)(8,38)(9,47)(10,40)(11,33)(12,42)(13,35)(14,44)(15,37)(16,46)(17,80)(18,73)(19,66)(20,75)(21,68)(22,77)(23,70)(24,79)(25,72)(26,65)(27,74)(28,67)(29,76)(30,69)(31,78)(32,71)(49,91)(50,84)(51,93)(52,86)(53,95)(54,88)(55,81)(56,90)(57,83)(58,92)(59,85)(60,94)(61,87)(62,96)(63,89)(64,82) );
G=PermutationGroup([[(1,20,60,47,67,86),(2,21,61,48,68,87),(3,22,62,33,69,88),(4,23,63,34,70,89),(5,24,64,35,71,90),(6,25,49,36,72,91),(7,26,50,37,73,92),(8,27,51,38,74,93),(9,28,52,39,75,94),(10,29,53,40,76,95),(11,30,54,41,77,96),(12,31,55,42,78,81),(13,32,56,43,79,82),(14,17,57,44,80,83),(15,18,58,45,65,84),(16,19,59,46,66,85)], [(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)], [(1,39),(2,48),(3,41),(4,34),(5,43),(6,36),(7,45),(8,38),(9,47),(10,40),(11,33),(12,42),(13,35),(14,44),(15,37),(16,46),(17,80),(18,73),(19,66),(20,75),(21,68),(22,77),(23,70),(24,79),(25,72),(26,65),(27,74),(28,67),(29,76),(30,69),(31,78),(32,71),(49,91),(50,84),(51,93),(52,86),(53,95),(54,88),(55,81),(56,90),(57,83),(58,92),(59,85),(60,94),(61,87),(62,96),(63,89),(64,82)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 16A | ··· | 16P | 24A | ··· | 24P | 24Q | ··· | 24X | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 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 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | M5(2) | C3×M5(2) |
| kernel | C6×M5(2) | C2×C48 | C3×M5(2) | C22×C24 | C2×M5(2) | C2×C24 | C22×C12 | C2×C16 | M5(2) | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 2 | 6 | 2 | 4 | 8 | 2 | 12 | 4 | 12 | 4 | 24 | 8 | 8 | 16 |
Matrix representation of C6×M5(2) ►in GL3(𝔽97) generated by
| 62 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 50 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 96 |
G:=sub<GL(3,GF(97))| [62,0,0,0,36,0,0,0,36],[1,0,0,0,0,50,0,1,0],[1,0,0,0,1,0,0,0,96] >;
C6×M5(2) in GAP, Magma, Sage, TeX
C_6\times M_5(2)
% in TeX
G:=Group("C6xM5(2)"); // GroupNames label
G:=SmallGroup(192,936);
// by ID
G=gap.SmallGroup(192,936);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,1373,102,124]);
// Polycyclic
G:=Group<a,b,c|a^6=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations