direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4⋊D4, C20⋊9D4, C4⋊2(C5×D4), C4⋊C4⋊2C10, (C2×C10)⋊4D4, (C2×D4)⋊2C10, C2.5(D4×C10), C22⋊1(C5×D4), (D4×C10)⋊11C2, C22⋊C4⋊3C10, (C22×C4)⋊4C10, C10.68(C2×D4), (C22×C20)⋊11C2, C23.8(C2×C10), C10.41(C4○D4), (C2×C10).76C23, (C2×C20).123C22, (C22×C10).27C22, C22.11(C22×C10), (C5×C4⋊C4)⋊11C2, C2.4(C5×C4○D4), (C2×C4).3(C2×C10), (C5×C22⋊C4)⋊11C2, SmallGroup(160,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊D4
G = < a,b,c,d | a5=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C10, C5×C4⋊D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C5×D4, C22×C10, D4×C10, C5×C4○D4, C5×C4⋊D4
►On 80 points
(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 66 26 56)(2 67 27 57)(3 68 28 58)(4 69 29 59)(5 70 30 60)(6 41 16 51)(7 42 17 52)(8 43 18 53)(9 44 19 54)(10 45 20 55)(11 46 76 36)(12 47 77 37)(13 48 78 38)(14 49 79 39)(15 50 80 40)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)
(1 36 21 41)(2 37 22 42)(3 38 23 43)(4 39 24 44)(5 40 25 45)(6 66 76 71)(7 67 77 72)(8 68 78 73)(9 69 79 74)(10 70 80 75)(11 61 16 56)(12 62 17 57)(13 63 18 58)(14 64 19 59)(15 65 20 60)(26 46 31 51)(27 47 32 52)(28 48 33 53)(29 49 34 54)(30 50 35 55)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
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), (1,66,26,56)(2,67,27,57)(3,68,28,58)(4,69,29,59)(5,70,30,60)(6,41,16,51)(7,42,17,52)(8,43,18,53)(9,44,19,54)(10,45,20,55)(11,46,76,36)(12,47,77,37)(13,48,78,38)(14,49,79,39)(15,50,80,40)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65), (1,36,21,41)(2,37,22,42)(3,38,23,43)(4,39,24,44)(5,40,25,45)(6,66,76,71)(7,67,77,72)(8,68,78,73)(9,69,79,74)(10,70,80,75)(11,61,16,56)(12,62,17,57)(13,63,18,58)(14,64,19,59)(15,65,20,60)(26,46,31,51)(27,47,32,52)(28,48,33,53)(29,49,34,54)(30,50,35,55), (6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)>;
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), (1,66,26,56)(2,67,27,57)(3,68,28,58)(4,69,29,59)(5,70,30,60)(6,41,16,51)(7,42,17,52)(8,43,18,53)(9,44,19,54)(10,45,20,55)(11,46,76,36)(12,47,77,37)(13,48,78,38)(14,49,79,39)(15,50,80,40)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65), (1,36,21,41)(2,37,22,42)(3,38,23,43)(4,39,24,44)(5,40,25,45)(6,66,76,71)(7,67,77,72)(8,68,78,73)(9,69,79,74)(10,70,80,75)(11,61,16,56)(12,62,17,57)(13,63,18,58)(14,64,19,59)(15,65,20,60)(26,46,31,51)(27,47,32,52)(28,48,33,53)(29,49,34,54)(30,50,35,55), (6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75) );
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)], [(1,66,26,56),(2,67,27,57),(3,68,28,58),(4,69,29,59),(5,70,30,60),(6,41,16,51),(7,42,17,52),(8,43,18,53),(9,44,19,54),(10,45,20,55),(11,46,76,36),(12,47,77,37),(13,48,78,38),(14,49,79,39),(15,50,80,40),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65)], [(1,36,21,41),(2,37,22,42),(3,38,23,43),(4,39,24,44),(5,40,25,45),(6,66,76,71),(7,67,77,72),(8,68,78,73),(9,69,79,74),(10,70,80,75),(11,61,16,56),(12,62,17,57),(13,63,18,58),(14,64,19,59),(15,65,20,60),(26,46,31,51),(27,47,32,52),(28,48,33,53),(29,49,34,54),(30,50,35,55)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,76),(17,77),(18,78),(19,79),(20,80),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75)]])
C5×C4⋊D4 is a maximal subgroup of
C4⋊C4⋊Dic5 (C2×C10).D8 C4⋊D4.D5 (C2×D4).D10 D20⋊16D4 D20⋊17D4 (C2×C10)⋊D8 C4⋊D4⋊D5 Dic10⋊17D4 C5⋊2C8⋊23D4 C4.(D4×D5) C20⋊(C4○D4) C10.682- 1+4 Dic10⋊19D4 Dic10⋊20D4 C4⋊C4.178D10 C10.342+ 1+4 C10.352+ 1+4 C10.362+ 1+4 C10.372+ 1+4 C4⋊C4⋊21D10 C10.382+ 1+4 C10.392+ 1+4 D20⋊19D4 C10.402+ 1+4 C10.732- 1+4 D20⋊20D4 C10.422+ 1+4 C10.432+ 1+4 C10.442+ 1+4 C10.452+ 1+4 C10.462+ 1+4 C10.1152+ 1+4 C10.472+ 1+4 C10.482+ 1+4 C10.742- 1+4 C5×D42
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C4⋊D4 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | D4×C10 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C20 | C2×C10 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 4 | 8 | 4 | 4 | 12 | 2 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of C5×C4⋊D4 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 1 | 0 | 0 |
| 39 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[40,39,0,0,1,1,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,40,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C4⋊D4 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes D_4
% in TeX
G:=Group("C5xC4:D4"); // GroupNames label
G:=SmallGroup(160,182);
// by ID
G=gap.SmallGroup(160,182);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,247,1514]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations