direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4⋊1D4, C20⋊6D4, C42⋊6C10, C4⋊1(C5×D4), (C4×C20)⋊13C2, (C2×D4)⋊3C10, C2.9(D4×C10), (D4×C10)⋊12C2, C10.72(C2×D4), C23.4(C2×C10), (C2×C10).82C23, (C2×C20).125C22, (C22×C10).4C22, C22.17(C22×C10), (C2×C4).23(C2×C10), SmallGroup(160,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊1D4
G = < a,b,c,d | a5=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, C10, C10, C42, C2×D4, C20, C2×C10, C2×C10, C4⋊1D4, C2×C20, C5×D4, C22×C10, C4×C20, D4×C10, C5×C4⋊1D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C4⋊1D4, C5×D4, C22×C10, D4×C10, C5×C4⋊1D4
►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 11 31 6)(2 12 32 7)(3 13 33 8)(4 14 34 9)(5 15 35 10)(16 26 76 21)(17 27 77 22)(18 28 78 23)(19 29 79 24)(20 30 80 25)(36 71 51 56)(37 72 52 57)(38 73 53 58)(39 74 54 59)(40 75 55 60)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)
(1 41 21 36)(2 42 22 37)(3 43 23 38)(4 44 24 39)(5 45 25 40)(6 61 76 56)(7 62 77 57)(8 63 78 58)(9 64 79 59)(10 65 80 60)(11 66 16 71)(12 67 17 72)(13 68 18 73)(14 69 19 74)(15 70 20 75)(26 51 31 46)(27 52 32 47)(28 53 33 48)(29 54 34 49)(30 55 35 50)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 71)(27 72)(28 73)(29 74)(30 75)(31 66)(32 67)(33 68)(34 69)(35 70)(36 76)(37 77)(38 78)(39 79)(40 80)
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), (1,11,31,6)(2,12,32,7)(3,13,33,8)(4,14,34,9)(5,15,35,10)(16,26,76,21)(17,27,77,22)(18,28,78,23)(19,29,79,24)(20,30,80,25)(36,71,51,56)(37,72,52,57)(38,73,53,58)(39,74,54,59)(40,75,55,60)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65), (1,41,21,36)(2,42,22,37)(3,43,23,38)(4,44,24,39)(5,45,25,40)(6,61,76,56)(7,62,77,57)(8,63,78,58)(9,64,79,59)(10,65,80,60)(11,66,16,71)(12,67,17,72)(13,68,18,73)(14,69,19,74)(15,70,20,75)(26,51,31,46)(27,52,32,47)(28,53,33,48)(29,54,34,49)(30,55,35,50), (1,61)(2,62)(3,63)(4,64)(5,65)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,71)(27,72)(28,73)(29,74)(30,75)(31,66)(32,67)(33,68)(34,69)(35,70)(36,76)(37,77)(38,78)(39,79)(40,80)>;
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), (1,11,31,6)(2,12,32,7)(3,13,33,8)(4,14,34,9)(5,15,35,10)(16,26,76,21)(17,27,77,22)(18,28,78,23)(19,29,79,24)(20,30,80,25)(36,71,51,56)(37,72,52,57)(38,73,53,58)(39,74,54,59)(40,75,55,60)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65), (1,41,21,36)(2,42,22,37)(3,43,23,38)(4,44,24,39)(5,45,25,40)(6,61,76,56)(7,62,77,57)(8,63,78,58)(9,64,79,59)(10,65,80,60)(11,66,16,71)(12,67,17,72)(13,68,18,73)(14,69,19,74)(15,70,20,75)(26,51,31,46)(27,52,32,47)(28,53,33,48)(29,54,34,49)(30,55,35,50), (1,61)(2,62)(3,63)(4,64)(5,65)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,71)(27,72)(28,73)(29,74)(30,75)(31,66)(32,67)(33,68)(34,69)(35,70)(36,76)(37,77)(38,78)(39,79)(40,80) );
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)], [(1,11,31,6),(2,12,32,7),(3,13,33,8),(4,14,34,9),(5,15,35,10),(16,26,76,21),(17,27,77,22),(18,28,78,23),(19,29,79,24),(20,30,80,25),(36,71,51,56),(37,72,52,57),(38,73,53,58),(39,74,54,59),(40,75,55,60),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65)], [(1,41,21,36),(2,42,22,37),(3,43,23,38),(4,44,24,39),(5,45,25,40),(6,61,76,56),(7,62,77,57),(8,63,78,58),(9,64,79,59),(10,65,80,60),(11,66,16,71),(12,67,17,72),(13,68,18,73),(14,69,19,74),(15,70,20,75),(26,51,31,46),(27,52,32,47),(28,53,33,48),(29,54,34,49),(30,55,35,50)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,71),(27,72),(28,73),(29,74),(30,75),(31,66),(32,67),(33,68),(34,69),(35,70),(36,76),(37,77),(38,78),(39,79),(40,80)]])
C5×C4⋊1D4 is a maximal subgroup of
C20.9D8 C42⋊3Dic5 C20.16D8 C42.72D10 C20⋊2D8 C20⋊D8 C42.74D10 Dic10⋊9D4 C20⋊4SD16 D20⋊5D4 C42.166D10 C42⋊26D10 C42.238D10 D20⋊11D4 Dic10⋊11D4 C42.168D10 C42⋊28D10 C5×D42
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | C5×D4 |
| kernel | C5×C4⋊1D4 | C4×C20 | D4×C10 | C4⋊1D4 | C42 | C2×D4 | C20 | C4 |
| # reps | 1 | 1 | 6 | 4 | 4 | 24 | 6 | 24 |
Matrix representation of C5×C4⋊1D4 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 4 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 9 | 37 | 0 | 0 |
| 20 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[32,0,0,0,4,9,0,0,0,0,40,1,0,0,39,1],[40,0,0,0,0,40,0,0,0,0,40,1,0,0,39,1],[9,20,0,0,37,32,0,0,0,0,1,40,0,0,0,40] >;
C5×C4⋊1D4 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes_1D_4
% in TeX
G:=Group("C5xC4:1D4"); // GroupNames label
G:=SmallGroup(160,188);
// by ID
G=gap.SmallGroup(160,188);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,247,1514,374]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations