direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C42, C20⋊8(C2×C4), (C4×C20)⋊8C2, C5⋊2(C2×C42), Dic5⋊8(C2×C4), (C2×C4).95D10, (C4×Dic5)⋊17C2, D10.18(C2×C4), (C2×C10).12C23, C10.15(C22×C4), C22.9(C22×D5), (C2×C20).109C22, (C2×Dic5).59C22, (C22×D5).41C22, C2.1(C2×C4×D5), (C2×C4×D5).19C2, SmallGroup(160,92)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C42 |
Generators and relations for D5×C42
G = < a,b,c,d | a4=b4=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 264 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, C10, C42, C42, C22×C4, Dic5, C20, D10, C2×C10, C2×C42, C4×D5, C2×Dic5, C2×C20, C22×D5, C4×Dic5, C4×C20, C2×C4×D5, D5×C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, D10, C2×C42, C4×D5, C22×D5, C2×C4×D5, D5×C42
►On 80 points
(1 49 9 44)(2 50 10 45)(3 46 6 41)(4 47 7 42)(5 48 8 43)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 39 14 29)(2 40 15 30)(3 36 11 26)(4 37 12 27)(5 38 13 28)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)
G:=sub<Sym(80)| (1,49,9,44)(2,50,10,45)(3,46,6,41)(4,47,7,42)(5,48,8,43)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (1,39,14,29)(2,40,15,30)(3,36,11,26)(4,37,12,27)(5,38,13,28)(6,31,16,21)(7,32,17,22)(8,33,18,23)(9,34,19,24)(10,35,20,25)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65), (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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,56)(42,60)(43,59)(44,58)(45,57)(46,51)(47,55)(48,54)(49,53)(50,52)(61,76)(62,80)(63,79)(64,78)(65,77)(66,71)(67,75)(68,74)(69,73)(70,72)>;
G:=Group( (1,49,9,44)(2,50,10,45)(3,46,6,41)(4,47,7,42)(5,48,8,43)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (1,39,14,29)(2,40,15,30)(3,36,11,26)(4,37,12,27)(5,38,13,28)(6,31,16,21)(7,32,17,22)(8,33,18,23)(9,34,19,24)(10,35,20,25)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65), (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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,56)(42,60)(43,59)(44,58)(45,57)(46,51)(47,55)(48,54)(49,53)(50,52)(61,76)(62,80)(63,79)(64,78)(65,77)(66,71)(67,75)(68,74)(69,73)(70,72) );
G=PermutationGroup([[(1,49,9,44),(2,50,10,45),(3,46,6,41),(4,47,7,42),(5,48,8,43),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75)], [(1,39,14,29),(2,40,15,30),(3,36,11,26),(4,37,12,27),(5,38,13,28),(6,31,16,21),(7,32,17,22),(8,33,18,23),(9,34,19,24),(10,35,20,25),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,71,56,61),(47,72,57,62),(48,73,58,63),(49,74,59,64),(50,75,60,65)], [(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)], [(1,18),(2,17),(3,16),(4,20),(5,19),(6,11),(7,15),(8,14),(9,13),(10,12),(21,36),(22,40),(23,39),(24,38),(25,37),(26,31),(27,35),(28,34),(29,33),(30,32),(41,56),(42,60),(43,59),(44,58),(45,57),(46,51),(47,55),(48,54),(49,53),(50,52),(61,76),(62,80),(63,79),(64,78),(65,77),(66,71),(67,75),(68,74),(69,73),(70,72)]])
D5×C42 is a maximal subgroup of
C42⋊6F5 C42.282D10 C42.182D10 D10.6C42 C42.200D10 C42.202D10 C20⋊5M4(2) C42.5F5 C42.6F5 C42.11F5 C42.12F5 C20⋊3M4(2) C42.14F5 C42.15F5 C42.7F5 C42⋊4F5 C42⋊8F5 C42⋊9F5 C42⋊5F5 C42.188D10 C42.93D10 C42.228D10 C42.229D10 C42.232D10 C42.131D10 C42.233D10 C42.234D10 C42.236D10 C42.237D10 C42.189D10 C42.238D10 C42.240D10 C42.241D10
D5×C42 is a maximal quotient of
Dic5.15C42 Dic5⋊2C42 D10⋊2C42 D10.5C42 D10.6C42 D10.7C42
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D10 | C4×D5 |
| kernel | D5×C42 | C4×Dic5 | C4×C20 | C2×C4×D5 | C4×D5 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 24 | 2 | 6 | 24 |
Matrix representation of D5×C42 ►in GL3(𝔽41) generated by
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 6 | 1 |
| 0 | 40 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 40 |
| 0 | 40 | 0 |
G:=sub<GL(3,GF(41))| [9,0,0,0,1,0,0,0,1],[1,0,0,0,9,0,0,0,9],[1,0,0,0,6,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;
D5×C42 in GAP, Magma, Sage, TeX
D_5\times C_4^2
% in TeX
G:=Group("D5xC4^2"); // GroupNames label
G:=SmallGroup(160,92);
// by ID
G=gap.SmallGroup(160,92);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations