direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42⋊2C2, C42⋊2C10, C4⋊C4⋊5C10, (C4×C20)⋊3C2, C22⋊C4.2C10, C23.3(C2×C10), C10.46(C4○D4), (C2×C20).68C22, (C2×C10).81C23, (C22×C10).3C22, C22.16(C22×C10), (C5×C4⋊C4)⋊14C2, C2.9(C5×C4○D4), (C2×C4).8(C2×C10), (C5×C22⋊C4).5C2, SmallGroup(160,187)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊2C2
G = < a,b,c,d | a5=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >
Subgroups: 84 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2×C10, C42⋊2C2, C2×C20, C22×C10, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C42⋊2C2
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C2×C10, C42⋊2C2, C22×C10, C5×C4○D4, C5×C42⋊2C2
►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 71 31 56)(2 72 32 57)(3 73 33 58)(4 74 34 59)(5 75 35 60)(6 41 11 46)(7 42 12 47)(8 43 13 48)(9 44 14 49)(10 45 15 50)(16 51 76 36)(17 52 77 37)(18 53 78 38)(19 54 79 39)(20 55 80 40)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 61 16 71)(7 62 17 72)(8 63 18 73)(9 64 19 74)(10 65 20 75)(11 66 76 56)(12 67 77 57)(13 68 78 58)(14 69 79 59)(15 70 80 60)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)
(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), (1,71,31,56)(2,72,32,57)(3,73,33,58)(4,74,34,59)(5,75,35,60)(6,41,11,46)(7,42,12,47)(8,43,13,48)(9,44,14,49)(10,45,15,50)(16,51,76,36)(17,52,77,37)(18,53,78,38)(19,54,79,39)(20,55,80,40)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(11,66,76,56)(12,67,77,57)(13,68,78,58)(14,69,79,59)(15,70,80,60)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (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), (1,71,31,56)(2,72,32,57)(3,73,33,58)(4,74,34,59)(5,75,35,60)(6,41,11,46)(7,42,12,47)(8,43,13,48)(9,44,14,49)(10,45,15,50)(16,51,76,36)(17,52,77,37)(18,53,78,38)(19,54,79,39)(20,55,80,40)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(11,66,76,56)(12,67,77,57)(13,68,78,58)(14,69,79,59)(15,70,80,60)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (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)], [(1,71,31,56),(2,72,32,57),(3,73,33,58),(4,74,34,59),(5,75,35,60),(6,41,11,46),(7,42,12,47),(8,43,13,48),(9,44,14,49),(10,45,15,50),(16,51,76,36),(17,52,77,37),(18,53,78,38),(19,54,79,39),(20,55,80,40),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,61,16,71),(7,62,17,72),(8,63,18,73),(9,64,19,74),(10,65,20,75),(11,66,76,56),(12,67,77,57),(13,68,78,58),(14,69,79,59),(15,70,80,60),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45)], [(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×C42⋊2C2 is a maximal subgroup of
C42.159D10 C42.160D10 C42⋊23D10 C42⋊24D10 C42.189D10 C42.161D10 C42.162D10 C42.163D10 C42.164D10 C42⋊25D10 C42.165D10
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 20A | ··· | 20X | 20Y | ··· | 20AJ |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C4○D4 | C5×C4○D4 |
| kernel | C5×C42⋊2C2 | C4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 |
| # reps | 1 | 1 | 3 | 3 | 4 | 4 | 12 | 12 | 6 | 24 |
Matrix representation of C5×C42⋊2C2 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 18 | 40 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 2 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[32,0,0,0,0,32,0,0,0,0,1,18,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,32,2,0,0,0,9],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,9,40] >;
C5×C42⋊2C2 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_2C_2
% in TeX
G:=Group("C5xC4^2:2C2"); // GroupNames label
G:=SmallGroup(160,187);
// by ID
G=gap.SmallGroup(160,187);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,727,1514,194]);
// 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*c^2,d*c*d=b^2*c^-1>;
// generators/relations