direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C6xSD32, C24.69D4, C12.45D8, C48:11C22, C24.64C23, (C2xC16):7C6, C16:3(C2xC6), C4.8(C6xD4), C4.7(C3xD8), (C2xC48):14C2, Q16:1(C2xC6), (C2xQ16):6C6, D8.1(C2xC6), (C2xD8).4C6, (C2xC6).56D8, C2.13(C6xD8), C8.10(C3xD4), C6.85(C2xD8), (C6xQ16):20C2, (C6xD8).11C2, C8.4(C22xC6), (C2xC12).427D4, C12.315(C2xD4), C22.15(C3xD8), (C3xQ16):15C22, (C3xD8).11C22, (C2xC24).405C22, (C2xC8).85(C2xC6), (C2xC4).83(C3xD4), SmallGroup(192,939)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6xSD32
G = < a,b,c | a6=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Subgroups: 210 in 90 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C12, C12, C2xC6, C2xC6, C16, C2xC8, D8, D8, Q16, Q16, C2xD4, C2xQ8, C24, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C2xC16, SD32, C2xD8, C2xQ16, C48, C2xC24, C3xD8, C3xD8, C3xQ16, C3xQ16, C6xD4, C6xQ8, C2xSD32, C2xC48, C3xSD32, C6xD8, C6xQ16, C6xSD32
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, D8, C2xD4, C3xD4, C22xC6, SD32, C2xD8, C3xD8, C6xD4, C2xSD32, C3xSD32, C6xD8, C6xSD32
►On 96 points
(1 56 21 46 92 66)(2 57 22 47 93 67)(3 58 23 48 94 68)(4 59 24 33 95 69)(5 60 25 34 96 70)(6 61 26 35 81 71)(7 62 27 36 82 72)(8 63 28 37 83 73)(9 64 29 38 84 74)(10 49 30 39 85 75)(11 50 31 40 86 76)(12 51 32 41 87 77)(13 52 17 42 88 78)(14 53 18 43 89 79)(15 54 19 44 90 80)(16 55 20 45 91 65)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 35)(34 42)(36 40)(37 47)(39 45)(41 43)(44 48)(49 55)(50 62)(51 53)(52 60)(54 58)(57 63)(59 61)(65 75)(67 73)(68 80)(69 71)(70 78)(72 76)(77 79)(81 95)(82 86)(83 93)(85 91)(87 89)(88 96)(90 94)
G:=sub<Sym(96)| (1,56,21,46,92,66)(2,57,22,47,93,67)(3,58,23,48,94,68)(4,59,24,33,95,69)(5,60,25,34,96,70)(6,61,26,35,81,71)(7,62,27,36,82,72)(8,63,28,37,83,73)(9,64,29,38,84,74)(10,49,30,39,85,75)(11,50,31,40,86,76)(12,51,32,41,87,77)(13,52,17,42,88,78)(14,53,18,43,89,79)(15,54,19,44,90,80)(16,55,20,45,91,65), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)(81,95)(82,86)(83,93)(85,91)(87,89)(88,96)(90,94)>;
G:=Group( (1,56,21,46,92,66)(2,57,22,47,93,67)(3,58,23,48,94,68)(4,59,24,33,95,69)(5,60,25,34,96,70)(6,61,26,35,81,71)(7,62,27,36,82,72)(8,63,28,37,83,73)(9,64,29,38,84,74)(10,49,30,39,85,75)(11,50,31,40,86,76)(12,51,32,41,87,77)(13,52,17,42,88,78)(14,53,18,43,89,79)(15,54,19,44,90,80)(16,55,20,45,91,65), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)(81,95)(82,86)(83,93)(85,91)(87,89)(88,96)(90,94) );
G=PermutationGroup([[(1,56,21,46,92,66),(2,57,22,47,93,67),(3,58,23,48,94,68),(4,59,24,33,95,69),(5,60,25,34,96,70),(6,61,26,35,81,71),(7,62,27,36,82,72),(8,63,28,37,83,73),(9,64,29,38,84,74),(10,49,30,39,85,75),(11,50,31,40,86,76),(12,51,32,41,87,77),(13,52,17,42,88,78),(14,53,18,43,89,79),(15,54,19,44,90,80),(16,55,20,45,91,65)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(33,35),(34,42),(36,40),(37,47),(39,45),(41,43),(44,48),(49,55),(50,62),(51,53),(52,60),(54,58),(57,63),(59,61),(65,75),(67,73),(68,80),(69,71),(70,78),(72,76),(77,79),(81,95),(82,86),(83,93),(85,91),(87,89),(88,96),(90,94)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | D8 | D8 | C3xD4 | C3xD4 | SD32 | C3xD8 | C3xD8 | C3xSD32 |
| kernel | C6xSD32 | C2xC48 | C3xSD32 | C6xD8 | C6xQ16 | C2xSD32 | C2xC16 | SD32 | C2xD8 | C2xQ16 | C24 | C2xC12 | C12 | C2xC6 | C8 | C2xC4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C6xSD32 ►in GL3(F97) generated by
| 96 | 0 | 0 |
| 0 | 62 | 0 |
| 0 | 0 | 62 |
| 96 | 0 | 0 |
| 0 | 63 | 77 |
| 0 | 10 | 43 |
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 1 | 96 |
G:=sub<GL(3,GF(97))| [96,0,0,0,62,0,0,0,62],[96,0,0,0,63,10,0,77,43],[96,0,0,0,1,1,0,0,96] >;
C6xSD32 in GAP, Magma, Sage, TeX
C_6\times {\rm SD}_{32} % in TeX
G:=Group("C6xSD32"); // GroupNames label
G:=SmallGroup(192,939);
// by ID
G=gap.SmallGroup(192,939);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,2524,1271,242,6053,3036,124]);
// 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^7>;
// generators/relations