direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C3×SD64, C32⋊2C6, C96⋊4C2, D16.C6, Q32⋊1C6, C12.40D8, C6.16D16, C24.65D4, C48.20C22, C8.6(C3×D4), C4.2(C3×D8), C16.3(C2×C6), (C3×Q32)⋊5C2, C2.4(C3×D16), (C3×D16).2C2, SmallGroup(192,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×SD64
G = < a,b,c | a3=b32=c2=1, ab=ba, ac=ca, cbc=b15 >
Smallest permutation representation of C3×SD64
►On 96 points
Generators in S96
(1 90 62)(2 91 63)(3 92 64)(4 93 33)(5 94 34)(6 95 35)(7 96 36)(8 65 37)(9 66 38)(10 67 39)(11 68 40)(12 69 41)(13 70 42)(14 71 43)(15 72 44)(16 73 45)(17 74 46)(18 75 47)(19 76 48)(20 77 49)(21 78 50)(22 79 51)(23 80 52)(24 81 53)(25 82 54)(26 83 55)(27 84 56)(28 85 57)(29 86 58)(30 87 59)(31 88 60)(32 89 61)
(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 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 43)(34 58)(35 41)(36 56)(37 39)(38 54)(40 52)(42 50)(44 48)(45 63)(47 61)(49 59)(51 57)(53 55)(60 64)(65 67)(66 82)(68 80)(69 95)(70 78)(71 93)(72 76)(73 91)(75 89)(77 87)(79 85)(81 83)(84 96)(86 94)(88 92)
G:=sub<Sym(96)| (1,90,62)(2,91,63)(3,92,64)(4,93,33)(5,94,34)(6,95,35)(7,96,36)(8,65,37)(9,66,38)(10,67,39)(11,68,40)(12,69,41)(13,70,42)(14,71,43)(15,72,44)(16,73,45)(17,74,46)(18,75,47)(19,76,48)(20,77,49)(21,78,50)(22,79,51)(23,80,52)(24,81,53)(25,82,54)(26,83,55)(27,84,56)(28,85,57)(29,86,58)(30,87,59)(31,88,60)(32,89,61), (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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,43)(34,58)(35,41)(36,56)(37,39)(38,54)(40,52)(42,50)(44,48)(45,63)(47,61)(49,59)(51,57)(53,55)(60,64)(65,67)(66,82)(68,80)(69,95)(70,78)(71,93)(72,76)(73,91)(75,89)(77,87)(79,85)(81,83)(84,96)(86,94)(88,92)>;
G:=Group( (1,90,62)(2,91,63)(3,92,64)(4,93,33)(5,94,34)(6,95,35)(7,96,36)(8,65,37)(9,66,38)(10,67,39)(11,68,40)(12,69,41)(13,70,42)(14,71,43)(15,72,44)(16,73,45)(17,74,46)(18,75,47)(19,76,48)(20,77,49)(21,78,50)(22,79,51)(23,80,52)(24,81,53)(25,82,54)(26,83,55)(27,84,56)(28,85,57)(29,86,58)(30,87,59)(31,88,60)(32,89,61), (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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,43)(34,58)(35,41)(36,56)(37,39)(38,54)(40,52)(42,50)(44,48)(45,63)(47,61)(49,59)(51,57)(53,55)(60,64)(65,67)(66,82)(68,80)(69,95)(70,78)(71,93)(72,76)(73,91)(75,89)(77,87)(79,85)(81,83)(84,96)(86,94)(88,92) );
G=PermutationGroup([[(1,90,62),(2,91,63),(3,92,64),(4,93,33),(5,94,34),(6,95,35),(7,96,36),(8,65,37),(9,66,38),(10,67,39),(11,68,40),(12,69,41),(13,70,42),(14,71,43),(15,72,44),(16,73,45),(17,74,46),(18,75,47),(19,76,48),(20,77,49),(21,78,50),(22,79,51),(23,80,52),(24,81,53),(25,82,54),(26,83,55),(27,84,56),(28,85,57),(29,86,58),(30,87,59),(31,88,60),(32,89,61)], [(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)], [(2,16),(3,31),(4,14),(5,29),(6,12),(7,27),(8,10),(9,25),(11,23),(13,21),(15,19),(18,32),(20,30),(22,28),(24,26),(33,43),(34,58),(35,41),(36,56),(37,39),(38,54),(40,52),(42,50),(44,48),(45,63),(47,61),(49,59),(51,57),(53,55),(60,64),(65,67),(66,82),(68,80),(69,95),(70,78),(71,93),(72,76),(73,91),(75,89),(77,87),(79,85),(81,83),(84,96),(86,94),(88,92)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 32A | ··· | 32H | 48A | ··· | 48H | 96A | ··· | 96P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 16 | 1 | 1 | 2 | 16 | 1 | 1 | 16 | 16 | 2 | 2 | 2 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D8 | C3×D4 | D16 | C3×D8 | SD64 | C3×D16 | C3×SD64 |
| kernel | C3×SD64 | C96 | C3×D16 | C3×Q32 | SD64 | C32 | D16 | Q32 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×SD64 ►in GL3(𝔽97) generated by
| 35 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 96 | 0 | 0 |
| 0 | 62 | 36 |
| 0 | 61 | 62 |
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 96 |
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[96,0,0,0,62,61,0,36,62],[96,0,0,0,1,0,0,0,96] >;
C3×SD64 in GAP, Magma, Sage, TeX
C_3\times {\rm SD}_{64} % in TeX
G:=Group("C3xSD64"); // GroupNames label
G:=SmallGroup(192,178);
// by ID
G=gap.SmallGroup(192,178);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,672,197,1011,514,192,2524,1271,242,6053,3036,124]);
// 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^15>;
// generators/relations
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