direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C40, C40⋊3C10, D10.4C20, C20.64D10, Dic5.4C20, C5⋊3(C2×C40), (C5×C40)⋊5C2, C5⋊2C8⋊6C10, C52⋊15(C2×C8), C2.1(D5×C20), C10.8(C2×C20), (C4×D5).7C10, C4.12(D5×C10), C10.28(C4×D5), C20.13(C2×C10), (D5×C20).14C2, (D5×C10).12C4, (C5×C20).42C22, (C5×Dic5).12C4, (C5×C5⋊2C8)⋊13C2, (C5×C10).50(C2×C4), SmallGroup(400,76)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C40 |
Generators and relations for D5×C40
G = < a,b,c | a40=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C40
►On 80 points
Generators in S80
(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 33 25 17 9)(2 34 26 18 10)(3 35 27 19 11)(4 36 28 20 12)(5 37 29 21 13)(6 38 30 22 14)(7 39 31 23 15)(8 40 32 24 16)(41 49 57 65 73)(42 50 58 66 74)(43 51 59 67 75)(44 52 60 68 76)(45 53 61 69 77)(46 54 62 70 78)(47 55 63 71 79)(48 56 64 72 80)
(1 79)(2 80)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)
G:=sub<Sym(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), (1,33,25,17,9)(2,34,26,18,10)(3,35,27,19,11)(4,36,28,20,12)(5,37,29,21,13)(6,38,30,22,14)(7,39,31,23,15)(8,40,32,24,16)(41,49,57,65,73)(42,50,58,66,74)(43,51,59,67,75)(44,52,60,68,76)(45,53,61,69,77)(46,54,62,70,78)(47,55,63,71,79)(48,56,64,72,80), (1,79)(2,80)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)>;
G:=Group( (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), (1,33,25,17,9)(2,34,26,18,10)(3,35,27,19,11)(4,36,28,20,12)(5,37,29,21,13)(6,38,30,22,14)(7,39,31,23,15)(8,40,32,24,16)(41,49,57,65,73)(42,50,58,66,74)(43,51,59,67,75)(44,52,60,68,76)(45,53,61,69,77)(46,54,62,70,78)(47,55,63,71,79)(48,56,64,72,80), (1,79)(2,80)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78) );
G=PermutationGroup([[(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)], [(1,33,25,17,9),(2,34,26,18,10),(3,35,27,19,11),(4,36,28,20,12),(5,37,29,21,13),(6,38,30,22,14),(7,39,31,23,15),(8,40,32,24,16),(41,49,57,65,73),(42,50,58,66,74),(43,51,59,67,75),(44,52,60,68,76),(45,53,61,69,77),(46,54,62,70,78),(47,55,63,71,79),(48,56,64,72,80)], [(1,79),(2,80),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | ··· | 20AJ | 40A | ··· | 40P | 40Q | ··· | 40BD | 40BE | ··· | 40BT |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | D5 | D10 | C4×D5 | C5×D5 | C8×D5 | D5×C10 | D5×C20 | D5×C40 |
| kernel | D5×C40 | C5×C5⋊2C8 | C5×C40 | D5×C20 | C5×Dic5 | D5×C10 | C8×D5 | C5×D5 | C5⋊2C8 | C40 | C4×D5 | Dic5 | D10 | D5 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 32 | 2 | 2 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of D5×C40 ►in GL2(𝔽41) generated by
| 13 | 0 |
| 0 | 13 |
| 37 | 0 |
| 30 | 10 |
| 31 | 9 |
| 30 | 10 |
G:=sub<GL(2,GF(41))| [13,0,0,13],[37,30,0,10],[31,30,9,10] >;
D5×C40 in GAP, Magma, Sage, TeX
D_5\times C_{40} % in TeX
G:=Group("D5xC40"); // GroupNames label
G:=SmallGroup(400,76);
// by ID
G=gap.SmallGroup(400,76);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,127,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^40=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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