metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C10)⋊8D8, (C5×D4)⋊13D4, D4⋊5(C5⋊D4), C5⋊5(C22⋊D8), C10.72(C2×D8), (C22×D4)⋊1D5, C20⋊7D4⋊25C2, C22⋊3(D4⋊D5), (C2×C20).300D4, C20.205(C2×D4), C10.71C22≀C2, (C2×D4).198D10, D4⋊Dic5⋊38C2, (C2×D20)⋊14C22, C4⋊Dic5⋊21C22, C20.55D4⋊15C2, (C2×C20).472C23, (C22×C4).149D10, (C22×C10).196D4, C2.4(C24⋊2D5), C23.85(C5⋊D4), C10.102(C8⋊C22), (D4×C10).240C22, C2.22(D4.D10), (C22×C20).197C22, (D4×C2×C10)⋊1C2, (C2×D4⋊D5)⋊23C2, C2.26(C2×D4⋊D5), C4.58(C2×C5⋊D4), (C2×C5⋊2C8)⋊10C22, (C2×C10).553(C2×D4), (C2×C4).83(C5⋊D4), (C2×C4).558(C22×D5), C22.216(C2×C5⋊D4), SmallGroup(320,844)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C10)⋊8D8
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, cac-1=dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 718 in 198 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C22×C10, C22⋊D8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×C10, C20.55D4, D4⋊Dic5, C20⋊7D4, C2×D4⋊D5, D4×C2×C10, (C2×C10)⋊8D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C22≀C2, C2×D8, C8⋊C22, C5⋊D4, C22×D5, C22⋊D8, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, D4.D10, C24⋊2D5, (C2×C10)⋊8D8
►On 80 points
(21 26)(22 27)(23 28)(24 29)(25 30)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(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)
(1 25 12 72 33 65 41 58)(2 24 13 71 34 64 42 57)(3 23 14 80 35 63 43 56)(4 22 15 79 36 62 44 55)(5 21 16 78 37 61 45 54)(6 30 17 77 38 70 46 53)(7 29 18 76 39 69 47 52)(8 28 19 75 40 68 48 51)(9 27 20 74 31 67 49 60)(10 26 11 73 32 66 50 59)
(1 53)(2 52)(3 51)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 61)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 80)
G:=sub<Sym(80)| (21,26)(22,27)(23,28)(24,29)(25,30)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,25,12,72,33,65,41,58)(2,24,13,71,34,64,42,57)(3,23,14,80,35,63,43,56)(4,22,15,79,36,62,44,55)(5,21,16,78,37,61,45,54)(6,30,17,77,38,70,46,53)(7,29,18,76,39,69,47,52)(8,28,19,75,40,68,48,51)(9,27,20,74,31,67,49,60)(10,26,11,73,32,66,50,59), (1,53)(2,52)(3,51)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,61)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,80)>;
G:=Group( (21,26)(22,27)(23,28)(24,29)(25,30)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,25,12,72,33,65,41,58)(2,24,13,71,34,64,42,57)(3,23,14,80,35,63,43,56)(4,22,15,79,36,62,44,55)(5,21,16,78,37,61,45,54)(6,30,17,77,38,70,46,53)(7,29,18,76,39,69,47,52)(8,28,19,75,40,68,48,51)(9,27,20,74,31,67,49,60)(10,26,11,73,32,66,50,59), (1,53)(2,52)(3,51)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,61)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,80) );
G=PermutationGroup([[(21,26),(22,27),(23,28),(24,29),(25,30),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(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)], [(1,25,12,72,33,65,41,58),(2,24,13,71,34,64,42,57),(3,23,14,80,35,63,43,56),(4,22,15,79,36,62,44,55),(5,21,16,78,37,61,45,54),(6,30,17,77,38,70,46,53),(7,29,18,76,39,69,47,52),(8,28,19,75,40,68,48,51),(9,27,20,74,31,67,49,60),(10,26,11,73,32,66,50,59)], [(1,53),(2,52),(3,51),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,61),(12,70),(13,69),(14,68),(15,67),(16,66),(17,65),(18,64),(19,63),(20,62),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,80)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 40 | 2 | 2 | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D8 | D10 | D10 | C5⋊D4 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4⋊D5 | D4.D10 |
| kernel | (C2×C10)⋊8D8 | C20.55D4 | D4⋊Dic5 | C20⋊7D4 | C2×D4⋊D5 | D4×C2×C10 | C2×C20 | C5×D4 | C22×C10 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C2×C4 | D4 | C23 | C10 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 2 | 4 | 2 | 4 | 4 | 16 | 4 | 1 | 4 | 4 |
Matrix representation of (C2×C10)⋊8D8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 7 | 31 |
| 17 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 28 | 26 |
| 0 | 0 | 3 | 13 |
| 17 | 1 | 0 | 0 |
| 40 | 24 | 0 | 0 |
| 0 | 0 | 13 | 15 |
| 0 | 0 | 38 | 28 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,4,7,0,0,0,31],[17,40,0,0,1,0,0,0,0,0,28,3,0,0,26,13],[17,40,0,0,1,24,0,0,0,0,13,38,0,0,15,28] >;
(C2×C10)⋊8D8 in GAP, Magma, Sage, TeX
(C_2\times C_{10})\rtimes_8D_8 % in TeX
G:=Group("(C2xC10):8D8"); // GroupNames label
G:=SmallGroup(320,844);
// by ID
G=gap.SmallGroup(320,844);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations