direct product, metacyclic, supersoluble, monomial
Aliases: C5×C40⋊C2, C40⋊6D5, C40⋊2C10, C52⋊5SD16, D20.1C10, C20.60D10, C10.19D20, Dic10⋊1C10, C8⋊2(C5×D5), (C5×C40)⋊4C2, C5⋊1(C5×SD16), C10.1(C5×D4), C2.3(C5×D20), C4.8(D5×C10), C20.8(C2×C10), (C5×D20).4C2, (C5×C10).17D4, (C5×Dic10)⋊10C2, (C5×C20).37C22, SmallGroup(400,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C40⋊C2
G = < a,b,c | a5=b40=c2=1, ab=ba, ac=ca, cbc=b19 >
20C2
2C5
2C5
10C4
10C22
2C10
2C10
4D5
20C10
5Q8
5D4
2C20
2Dic5
2C20
2D10
10C20
10C2×C10
5SD16
2C40
2C40
Smallest permutation representation of C5×C40⋊C2
►On 80 points
Generators in S80
(1 9 17 25 33)(2 10 18 26 34)(3 11 19 27 35)(4 12 20 28 36)(5 13 21 29 37)(6 14 22 30 38)(7 15 23 31 39)(8 16 24 32 40)(41 73 65 57 49)(42 74 66 58 50)(43 75 67 59 51)(44 76 68 60 52)(45 77 69 61 53)(46 78 70 62 54)(47 79 71 63 55)(48 80 72 64 56)
(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 41)(2 60)(3 79)(4 58)(5 77)(6 56)(7 75)(8 54)(9 73)(10 52)(11 71)(12 50)(13 69)(14 48)(15 67)(16 46)(17 65)(18 44)(19 63)(20 42)(21 61)(22 80)(23 59)(24 78)(25 57)(26 76)(27 55)(28 74)(29 53)(30 72)(31 51)(32 70)(33 49)(34 68)(35 47)(36 66)(37 45)(38 64)(39 43)(40 62)
G:=sub<Sym(80)| (1,9,17,25,33)(2,10,18,26,34)(3,11,19,27,35)(4,12,20,28,36)(5,13,21,29,37)(6,14,22,30,38)(7,15,23,31,39)(8,16,24,32,40)(41,73,65,57,49)(42,74,66,58,50)(43,75,67,59,51)(44,76,68,60,52)(45,77,69,61,53)(46,78,70,62,54)(47,79,71,63,55)(48,80,72,64,56), (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), (1,41)(2,60)(3,79)(4,58)(5,77)(6,56)(7,75)(8,54)(9,73)(10,52)(11,71)(12,50)(13,69)(14,48)(15,67)(16,46)(17,65)(18,44)(19,63)(20,42)(21,61)(22,80)(23,59)(24,78)(25,57)(26,76)(27,55)(28,74)(29,53)(30,72)(31,51)(32,70)(33,49)(34,68)(35,47)(36,66)(37,45)(38,64)(39,43)(40,62)>;
G:=Group( (1,9,17,25,33)(2,10,18,26,34)(3,11,19,27,35)(4,12,20,28,36)(5,13,21,29,37)(6,14,22,30,38)(7,15,23,31,39)(8,16,24,32,40)(41,73,65,57,49)(42,74,66,58,50)(43,75,67,59,51)(44,76,68,60,52)(45,77,69,61,53)(46,78,70,62,54)(47,79,71,63,55)(48,80,72,64,56), (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), (1,41)(2,60)(3,79)(4,58)(5,77)(6,56)(7,75)(8,54)(9,73)(10,52)(11,71)(12,50)(13,69)(14,48)(15,67)(16,46)(17,65)(18,44)(19,63)(20,42)(21,61)(22,80)(23,59)(24,78)(25,57)(26,76)(27,55)(28,74)(29,53)(30,72)(31,51)(32,70)(33,49)(34,68)(35,47)(36,66)(37,45)(38,64)(39,43)(40,62) );
G=PermutationGroup([[(1,9,17,25,33),(2,10,18,26,34),(3,11,19,27,35),(4,12,20,28,36),(5,13,21,29,37),(6,14,22,30,38),(7,15,23,31,39),(8,16,24,32,40),(41,73,65,57,49),(42,74,66,58,50),(43,75,67,59,51),(44,76,68,60,52),(45,77,69,61,53),(46,78,70,62,54),(47,79,71,63,55),(48,80,72,64,56)], [(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)], [(1,41),(2,60),(3,79),(4,58),(5,77),(6,56),(7,75),(8,54),(9,73),(10,52),(11,71),(12,50),(13,69),(14,48),(15,67),(16,46),(17,65),(18,44),(19,63),(20,42),(21,61),(22,80),(23,59),(24,78),(25,57),(26,76),(27,55),(28,74),(29,53),(30,72),(31,51),(32,70),(33,49),(34,68),(35,47),(36,66),(37,45),(38,64),(39,43),(40,62)]])
115 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | 10P | 10Q | 10R | 20A | ··· | 20X | 20Y | 20Z | 20AA | 20AB | 40A | ··· | 40AV |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 20 | 2 | 20 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D5 | SD16 | D10 | D20 | C5×D4 | C5×D5 | C40⋊C2 | C5×SD16 | D5×C10 | C5×D20 | C5×C40⋊C2 |
| kernel | C5×C40⋊C2 | C5×C40 | C5×Dic10 | C5×D20 | C40⋊C2 | C40 | Dic10 | D20 | C5×C10 | C40 | C52 | C20 | C10 | C10 | C8 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C5×C40⋊C2 ►in GL2(𝔽41) generated by
| 16 | 0 |
| 0 | 16 |
| 29 | 0 |
| 0 | 24 |
| 0 | 32 |
| 9 | 0 |
G:=sub<GL(2,GF(41))| [16,0,0,16],[29,0,0,24],[0,9,32,0] >;
C5×C40⋊C2 in GAP, Magma, Sage, TeX
C_5\times C_{40}\rtimes C_2 % in TeX
G:=Group("C5xC40:C2"); // GroupNames label
G:=SmallGroup(400,78);
// by ID
G=gap.SmallGroup(400,78);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,265,127,1443,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^5=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations
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