direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5xC42, C20:8(C2xC4), (C4xC20):8C2, C5:2(C2xC42), Dic5:8(C2xC4), (C2xC4).95D10, (C4xDic5):17C2, D10.18(C2xC4), (C2xC10).12C23, C10.15(C22xC4), C22.9(C22xD5), (C2xC20).109C22, (C2xDic5).59C22, (C22xD5).41C22, C2.1(C2xC4xD5), (C2xC4xD5).19C2, SmallGroup(160,92)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5xC42 |
Generators and relations for D5xC42
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, C2xC4, C2xC4, C23, D5, C10, C42, C42, C22xC4, Dic5, C20, D10, C2xC10, C2xC42, C4xD5, C2xDic5, C2xC20, C22xD5, C4xDic5, C4xC20, C2xC4xD5, D5xC42
Quotients: C1, C2, C4, C22, C2xC4, C23, D5, C42, C22xC4, D10, C2xC42, C4xD5, C22xD5, C2xC4xD5, D5xC42
►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)]])
D5xC42 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
D5xC42 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 | C4xD5 |
| kernel | D5xC42 | C4xDic5 | C4xC20 | C2xC4xD5 | C4xD5 | C42 | C2xC4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 24 | 2 | 6 | 24 |
Matrix representation of D5xC42 ►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] >;
D5xC42 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