direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5xC42:2C2, C42:30D10, C4:C4:32D10, (C4xC20):31C22, (D5xC42):19C2, C42:2D5:13C2, (C2xC20).94C23, C4:Dic5:43C22, C22:C4.76D10, D10.67(C4oD4), (C2xC10).247C24, (C4xDic5):80C22, C23.53(C22xD5), C23.D10:44C2, C10.D4:31C22, (C22xC10).61C23, (C23xD5).67C22, C22.268(C23xD5), C23.D5.63C22, D10:C4.44C22, (C2xDic5).128C23, (C22xD5).295C23, (D5xC4:C4):40C2, C5:4(C2xC42:2C2), C2.94(D5xC4oD4), C4:C4:D5:40C2, (C5xC4:C4):31C22, (D5xC22:C4).3C2, (C5xC42:2C2):2C2, C10.205(C2xC4oD4), (C2xC4xD5).382C22, (C2xC4).84(C22xD5), (C5xC22:C4).72C22, SmallGroup(320,1375)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5xC42:2C2
G = < a,b,c,d,e | a5=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, ede=c2d-1 >
Subgroups: 894 in 246 conjugacy classes, 101 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2xC4, C2xC4, C23, C23, D5, D5, C10, C10, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC42, C2xC22:C4, C2xC4:C4, C42:2C2, C42:2C2, C4xD5, C2xDic5, C2xC20, C22xD5, C22xD5, C22xC10, C2xC42:2C2, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C5xC4:C4, C2xC4xD5, C23xD5, D5xC42, C42:2D5, C23.D10, D5xC22:C4, D5xC4:C4, C4:C4:D5, C5xC42:2C2, D5xC42:2C2
Quotients: C1, C2, C22, C23, D5, C4oD4, C24, D10, C42:2C2, C2xC4oD4, C22xD5, C2xC42:2C2, C23xD5, D5xC4oD4, D5xC42:2C2
►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 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 51)(42 55)(43 54)(44 53)(45 52)(46 56)(47 60)(48 59)(49 58)(50 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)
(1 69 9 64)(2 70 10 65)(3 66 6 61)(4 67 7 62)(5 68 8 63)(11 76 16 71)(12 77 17 72)(13 78 18 73)(14 79 19 74)(15 80 20 75)(21 51 26 56)(22 52 27 57)(23 53 28 58)(24 54 29 59)(25 55 30 60)(31 41 36 46)(32 42 37 47)(33 43 38 48)(34 44 39 49)(35 45 40 50)
(1 39 19 24)(2 40 20 25)(3 36 16 21)(4 37 17 22)(5 38 18 23)(6 31 11 26)(7 32 12 27)(8 33 13 28)(9 34 14 29)(10 35 15 30)(41 76 56 61)(42 77 57 62)(43 78 58 63)(44 79 59 64)(45 80 60 65)(46 71 51 66)(47 72 52 67)(48 73 53 68)(49 74 54 69)(50 75 55 70)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)(41,51)(42,55)(43,54)(44,53)(45,52)(46,56)(47,60)(48,59)(49,58)(50,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77), (1,69,9,64)(2,70,10,65)(3,66,6,61)(4,67,7,62)(5,68,8,63)(11,76,16,71)(12,77,17,72)(13,78,18,73)(14,79,19,74)(15,80,20,75)(21,51,26,56)(22,52,27,57)(23,53,28,58)(24,54,29,59)(25,55,30,60)(31,41,36,46)(32,42,37,47)(33,43,38,48)(34,44,39,49)(35,45,40,50), (1,39,19,24)(2,40,20,25)(3,36,16,21)(4,37,17,22)(5,38,18,23)(6,31,11,26)(7,32,12,27)(8,33,13,28)(9,34,14,29)(10,35,15,30)(41,76,56,61)(42,77,57,62)(43,78,58,63)(44,79,59,64)(45,80,60,65)(46,71,51,66)(47,72,52,67)(48,73,53,68)(49,74,54,69)(50,75,55,70), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75)>;
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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)(41,51)(42,55)(43,54)(44,53)(45,52)(46,56)(47,60)(48,59)(49,58)(50,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77), (1,69,9,64)(2,70,10,65)(3,66,6,61)(4,67,7,62)(5,68,8,63)(11,76,16,71)(12,77,17,72)(13,78,18,73)(14,79,19,74)(15,80,20,75)(21,51,26,56)(22,52,27,57)(23,53,28,58)(24,54,29,59)(25,55,30,60)(31,41,36,46)(32,42,37,47)(33,43,38,48)(34,44,39,49)(35,45,40,50), (1,39,19,24)(2,40,20,25)(3,36,16,21)(4,37,17,22)(5,38,18,23)(6,31,11,26)(7,32,12,27)(8,33,13,28)(9,34,14,29)(10,35,15,30)(41,76,56,61)(42,77,57,62)(43,78,58,63)(44,79,59,64)(45,80,60,65)(46,71,51,66)(47,72,52,67)(48,73,53,68)(49,74,54,69)(50,75,55,70), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75) );
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)], [(1,13),(2,12),(3,11),(4,15),(5,14),(6,16),(7,20),(8,19),(9,18),(10,17),(21,31),(22,35),(23,34),(24,33),(25,32),(26,36),(27,40),(28,39),(29,38),(30,37),(41,51),(42,55),(43,54),(44,53),(45,52),(46,56),(47,60),(48,59),(49,58),(50,57),(61,71),(62,75),(63,74),(64,73),(65,72),(66,76),(67,80),(68,79),(69,78),(70,77)], [(1,69,9,64),(2,70,10,65),(3,66,6,61),(4,67,7,62),(5,68,8,63),(11,76,16,71),(12,77,17,72),(13,78,18,73),(14,79,19,74),(15,80,20,75),(21,51,26,56),(22,52,27,57),(23,53,28,58),(24,54,29,59),(25,55,30,60),(31,41,36,46),(32,42,37,47),(33,43,38,48),(34,44,39,49),(35,45,40,50)], [(1,39,19,24),(2,40,20,25),(3,36,16,21),(4,37,17,22),(5,38,18,23),(6,31,11,26),(7,32,12,27),(8,33,13,28),(9,34,14,29),(10,35,15,30),(41,76,56,61),(42,77,57,62),(43,78,58,63),(44,79,59,64),(45,80,60,65),(46,71,51,66),(47,72,52,67),(48,73,53,68),(49,74,54,69),(50,75,55,70)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 5 | 5 | 5 | 5 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4oD4 | D10 | D10 | D10 | D5xC4oD4 |
| kernel | D5xC42:2C2 | D5xC42 | C42:2D5 | C23.D10 | D5xC22:C4 | D5xC4:C4 | C4:C4:D5 | C5xC42:2C2 | C42:2C2 | D10 | C42 | C22:C4 | C4:C4 | C2 |
| # reps | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 2 | 12 | 2 | 6 | 6 | 12 |
Matrix representation of D5xC42:2C2 ►in GL6(F41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 37 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 23 |
| 0 | 0 | 0 | 0 | 9 | 32 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 25 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,37,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,9,0,0,0,0,23,32],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[1,25,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
D5xC42:2C2 in GAP, Magma, Sage, TeX
D_5\times C_4^2\rtimes_2C_2
% in TeX
G:=Group("D5xC4^2:2C2"); // GroupNames label
G:=SmallGroup(320,1375);
// by ID
G=gap.SmallGroup(320,1375);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,346,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c*d^2,e*d*e=c^2*d^-1>;
// generators/relations