direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5xC4:1D4, C42:34D10, C4:1(D4xD5), (C4xD5):7D4, C20:2(C2xD4), (C2xD4):25D10, Dic5:1(C2xD4), C20:4D4:16C2, C20:D4:25C2, (C4xC20):25C22, (D5xC42):12C2, D10.109(C2xD4), (D4xC10):17C22, (C2xD20):30C22, C10.92(C22xD4), (C2xC20).507C23, (C2xC10).258C24, (C4xDic5):65C22, C23.64(C22xD5), (C22xC10).72C23, (C23xD5).71C22, C22.279(C23xD5), (C2xDic5).278C23, (C22xD5).296C23, (C2xD4xD5):18C2, C5:2(C2xC4:1D4), C2.65(C2xD4xD5), (C5xC4:1D4):5C2, (C2xC5:D4):25C22, (C2xC4xD5).320C22, (C2xC4).596(C22xD5), SmallGroup(320,1386)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5xC4:1D4
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=c-1, ede=d-1 >
Subgroups: 2286 in 498 conjugacy classes, 131 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, D4, C23, C23, D5, D5, C10, C10, C42, C42, C22xC4, C2xD4, C2xD4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC42, C4:1D4, C4:1D4, C22xD4, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C5xD4, C22xD5, C22xD5, C22xD5, C22xC10, C2xC4:1D4, C4xDic5, C4xC20, C2xC4xD5, C2xD20, D4xD5, C2xC5:D4, D4xC10, C23xD5, D5xC42, C20:4D4, C20:D4, C5xC4:1D4, C2xD4xD5, D5xC4:1D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C4:1D4, C22xD4, C22xD5, C2xC4:1D4, D4xD5, C23xD5, C2xD4xD5, D5xC4:1D4
►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 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)
(1 54 19 49)(2 55 20 50)(3 51 16 46)(4 52 17 47)(5 53 18 48)(6 56 11 41)(7 57 12 42)(8 58 13 43)(9 59 14 44)(10 60 15 45)(21 76 36 61)(22 77 37 62)(23 78 38 63)(24 79 39 64)(25 80 40 65)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)
(1 64 14 74)(2 65 15 75)(3 61 11 71)(4 62 12 72)(5 63 13 73)(6 66 16 76)(7 67 17 77)(8 68 18 78)(9 69 19 79)(10 70 20 80)(21 41 31 51)(22 42 32 52)(23 43 33 53)(24 44 34 54)(25 45 35 55)(26 46 36 56)(27 47 37 57)(28 48 38 58)(29 49 39 59)(30 50 40 60)
(1 54)(2 55)(3 51)(4 52)(5 53)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)
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), (1,18)(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), (1,54,19,49)(2,55,20,50)(3,51,16,46)(4,52,17,47)(5,53,18,48)(6,56,11,41)(7,57,12,42)(8,58,13,43)(9,59,14,44)(10,60,15,45)(21,76,36,61)(22,77,37,62)(23,78,38,63)(24,79,39,64)(25,80,40,65)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70), (1,64,14,74)(2,65,15,75)(3,61,11,71)(4,62,12,72)(5,63,13,73)(6,66,16,76)(7,67,17,77)(8,68,18,78)(9,69,19,79)(10,70,20,80)(21,41,31,51)(22,42,32,52)(23,43,33,53)(24,44,34,54)(25,45,35,55)(26,46,36,56)(27,47,37,57)(28,48,38,58)(29,49,39,59)(30,50,40,60), (1,54)(2,55)(3,51)(4,52)(5,53)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)>;
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), (1,18)(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), (1,54,19,49)(2,55,20,50)(3,51,16,46)(4,52,17,47)(5,53,18,48)(6,56,11,41)(7,57,12,42)(8,58,13,43)(9,59,14,44)(10,60,15,45)(21,76,36,61)(22,77,37,62)(23,78,38,63)(24,79,39,64)(25,80,40,65)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70), (1,64,14,74)(2,65,15,75)(3,61,11,71)(4,62,12,72)(5,63,13,73)(6,66,16,76)(7,67,17,77)(8,68,18,78)(9,69,19,79)(10,70,20,80)(21,41,31,51)(22,42,32,52)(23,43,33,53)(24,44,34,54)(25,45,35,55)(26,46,36,56)(27,47,37,57)(28,48,38,58)(29,49,39,59)(30,50,40,60), (1,54)(2,55)(3,51)(4,52)(5,53)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70) );
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)], [(1,18),(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)], [(1,54,19,49),(2,55,20,50),(3,51,16,46),(4,52,17,47),(5,53,18,48),(6,56,11,41),(7,57,12,42),(8,58,13,43),(9,59,14,44),(10,60,15,45),(21,76,36,61),(22,77,37,62),(23,78,38,63),(24,79,39,64),(25,80,40,65),(26,71,31,66),(27,72,32,67),(28,73,33,68),(29,74,34,69),(30,75,35,70)], [(1,64,14,74),(2,65,15,75),(3,61,11,71),(4,62,12,72),(5,63,13,73),(6,66,16,76),(7,67,17,77),(8,68,18,78),(9,69,19,79),(10,70,20,80),(21,41,31,51),(22,42,32,52),(23,43,33,53),(24,44,34,54),(25,45,35,55),(26,46,36,56),(27,47,37,57),(28,48,38,58),(29,49,39,59),(30,50,40,60)], [(1,54),(2,55),(3,51),(4,52),(5,53),(6,56),(7,57),(8,58),(9,59),(10,60),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D4xD5 |
| kernel | D5xC4:1D4 | D5xC42 | C20:4D4 | C20:D4 | C5xC4:1D4 | C2xD4xD5 | C4xD5 | C4:1D4 | C42 | C2xD4 | C4 |
| # reps | 1 | 1 | 1 | 6 | 1 | 6 | 12 | 2 | 2 | 12 | 12 |
Matrix representation of D5xC4:1D4 ►in GL6(F41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,1] >;
D5xC4:1D4 in GAP, Magma, Sage, TeX
D_5\times C_4\rtimes_1D_4
% in TeX
G:=Group("D5xC4:1D4"); // GroupNames label
G:=SmallGroup(320,1386);
// by ID
G=gap.SmallGroup(320,1386);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,570,185,12550]);
// 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^-1,e*d*e=d^-1>;
// generators/relations