metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:1Dic5, C20.49C42, (C4xC20):3C4, (C2xC20).6Q8, (C4xDic5):3C4, (C2xC4).13D20, C20.59(C4:C4), C5:3(C4.9C42), (C2xC20).104D4, (C2xC4).2Dic10, C4.24(C4xDic5), C4.12(C4:Dic5), (C22xC10).42D4, (C22xC4).57D10, C42:C2.1D5, C20.56(C22:C4), C4.8(C10.D4), C23.16(C5:D4), C4.33(D10:C4), (C22xC20).120C22, C23.21D10.7C2, C22.3(C10.D4), C22.10(C23.D5), C22.17(D10:C4), C2.8(C10.10C42), C10.26(C2.C42), (C2xC5:2C8):1C4, (C2xC4).139(C4xD5), (C2xC10).31(C4:C4), (C2xC20).229(C2xC4), (C2xC4).20(C5:D4), (C2xC4).72(C2xDic5), (C2xC4.Dic5).7C2, (C5xC42:C2).1C2, (C2xC10).153(C22:C4), SmallGroup(320,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:1Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 278 in 94 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, C23, C10, C10, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, Dic5, C20, C20, C2xC10, C2xC10, C42:C2, C42:C2, C2xM4(2), C5:2C8, C2xDic5, C2xC20, C2xC20, C22xC10, C4.9C42, C2xC5:2C8, C4.Dic5, C4xDic5, C4:Dic5, C23.D5, C4xC20, C5xC22:C4, C5xC4:C4, C22xC20, C2xC4.Dic5, C23.21D10, C5xC42:C2, C42:1Dic5
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, D5, C42, C22:C4, C4:C4, Dic5, D10, C2.C42, Dic10, C4xD5, D20, C2xDic5, C5:D4, C4.9C42, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, C10.10C42, C42:1Dic5
►On 80 points
(1 53 13 66)(2 59 14 62)(3 55 15 68)(4 51 11 64)(5 57 12 70)(6 45 40 75)(7 41 36 71)(8 47 37 77)(9 43 38 73)(10 49 39 79)(16 63 35 60)(17 69 31 56)(18 65 32 52)(19 61 33 58)(20 67 34 54)(21 74 28 44)(22 80 29 50)(23 76 30 46)(24 72 26 42)(25 78 27 48)
(1 21 33 39)(2 22 34 40)(3 23 35 36)(4 24 31 37)(5 25 32 38)(6 14 29 20)(7 15 30 16)(8 11 26 17)(9 12 27 18)(10 13 28 19)(41 68 46 63)(42 69 47 64)(43 70 48 65)(44 61 49 66)(45 62 50 67)(51 72 56 77)(52 73 57 78)(53 74 58 79)(54 75 59 80)(55 76 60 71)
(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 28)(2 27)(3 26)(4 30)(5 29)(6 32)(7 31)(8 35)(9 34)(10 33)(11 36)(12 40)(13 39)(14 38)(15 37)(16 24)(17 23)(18 22)(19 21)(20 25)(41 77 46 72)(42 76 47 71)(43 75 48 80)(44 74 49 79)(45 73 50 78)(51 63 56 68)(52 62 57 67)(53 61 58 66)(54 70 59 65)(55 69 60 64)
G:=sub<Sym(80)| (1,53,13,66)(2,59,14,62)(3,55,15,68)(4,51,11,64)(5,57,12,70)(6,45,40,75)(7,41,36,71)(8,47,37,77)(9,43,38,73)(10,49,39,79)(16,63,35,60)(17,69,31,56)(18,65,32,52)(19,61,33,58)(20,67,34,54)(21,74,28,44)(22,80,29,50)(23,76,30,46)(24,72,26,42)(25,78,27,48), (1,21,33,39)(2,22,34,40)(3,23,35,36)(4,24,31,37)(5,25,32,38)(6,14,29,20)(7,15,30,16)(8,11,26,17)(9,12,27,18)(10,13,28,19)(41,68,46,63)(42,69,47,64)(43,70,48,65)(44,61,49,66)(45,62,50,67)(51,72,56,77)(52,73,57,78)(53,74,58,79)(54,75,59,80)(55,76,60,71), (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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,32)(7,31)(8,35)(9,34)(10,33)(11,36)(12,40)(13,39)(14,38)(15,37)(16,24)(17,23)(18,22)(19,21)(20,25)(41,77,46,72)(42,76,47,71)(43,75,48,80)(44,74,49,79)(45,73,50,78)(51,63,56,68)(52,62,57,67)(53,61,58,66)(54,70,59,65)(55,69,60,64)>;
G:=Group( (1,53,13,66)(2,59,14,62)(3,55,15,68)(4,51,11,64)(5,57,12,70)(6,45,40,75)(7,41,36,71)(8,47,37,77)(9,43,38,73)(10,49,39,79)(16,63,35,60)(17,69,31,56)(18,65,32,52)(19,61,33,58)(20,67,34,54)(21,74,28,44)(22,80,29,50)(23,76,30,46)(24,72,26,42)(25,78,27,48), (1,21,33,39)(2,22,34,40)(3,23,35,36)(4,24,31,37)(5,25,32,38)(6,14,29,20)(7,15,30,16)(8,11,26,17)(9,12,27,18)(10,13,28,19)(41,68,46,63)(42,69,47,64)(43,70,48,65)(44,61,49,66)(45,62,50,67)(51,72,56,77)(52,73,57,78)(53,74,58,79)(54,75,59,80)(55,76,60,71), (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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,32)(7,31)(8,35)(9,34)(10,33)(11,36)(12,40)(13,39)(14,38)(15,37)(16,24)(17,23)(18,22)(19,21)(20,25)(41,77,46,72)(42,76,47,71)(43,75,48,80)(44,74,49,79)(45,73,50,78)(51,63,56,68)(52,62,57,67)(53,61,58,66)(54,70,59,65)(55,69,60,64) );
G=PermutationGroup([[(1,53,13,66),(2,59,14,62),(3,55,15,68),(4,51,11,64),(5,57,12,70),(6,45,40,75),(7,41,36,71),(8,47,37,77),(9,43,38,73),(10,49,39,79),(16,63,35,60),(17,69,31,56),(18,65,32,52),(19,61,33,58),(20,67,34,54),(21,74,28,44),(22,80,29,50),(23,76,30,46),(24,72,26,42),(25,78,27,48)], [(1,21,33,39),(2,22,34,40),(3,23,35,36),(4,24,31,37),(5,25,32,38),(6,14,29,20),(7,15,30,16),(8,11,26,17),(9,12,27,18),(10,13,28,19),(41,68,46,63),(42,69,47,64),(43,70,48,65),(44,61,49,66),(45,62,50,67),(51,72,56,77),(52,73,57,78),(53,74,58,79),(54,75,59,80),(55,76,60,71)], [(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)], [(1,28),(2,27),(3,26),(4,30),(5,29),(6,32),(7,31),(8,35),(9,34),(10,33),(11,36),(12,40),(13,39),(14,38),(15,37),(16,24),(17,23),(18,22),(19,21),(20,25),(41,77,46,72),(42,76,47,71),(43,75,48,80),(44,74,49,79),(45,73,50,78),(51,63,56,68),(52,62,57,67),(53,61,58,66),(54,70,59,65),(55,69,60,64)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D5 | Dic5 | D10 | Dic10 | C4xD5 | D20 | C5:D4 | C5:D4 | C4.9C42 | C42:1Dic5 |
| kernel | C42:1Dic5 | C2xC4.Dic5 | C23.21D10 | C5xC42:C2 | C2xC5:2C8 | C4xDic5 | C4xC20 | C2xC20 | C2xC20 | C22xC10 | C42:C2 | C42 | C22xC4 | C2xC4 | C2xC4 | C2xC4 | C2xC4 | C23 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 4 | 4 | 4 | 2 | 8 |
Matrix representation of C42:1Dic5 ►in GL4(F41) generated by
| 9 | 0 | 11 | 38 |
| 0 | 9 | 3 | 29 |
| 5 | 29 | 32 | 0 |
| 12 | 36 | 0 | 32 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 35 | 40 | 25 | 2 |
| 1 | 0 | 39 | 13 |
| 0 | 0 | 6 | 1 |
| 0 | 0 | 40 | 0 |
| 18 | 20 | 37 | 17 |
| 35 | 23 | 0 | 4 |
| 0 | 0 | 39 | 16 |
| 0 | 0 | 28 | 2 |
G:=sub<GL(4,GF(41))| [9,0,5,12,0,9,29,36,11,3,32,0,38,29,0,32],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[35,1,0,0,40,0,0,0,25,39,6,40,2,13,1,0],[18,35,0,0,20,23,0,0,37,0,39,28,17,4,16,2] >;
C42:1Dic5 in GAP, Magma, Sage, TeX
C_4^2\rtimes_1{\rm Dic}_5 % in TeX
G:=Group("C4^2:1Dic5"); // GroupNames label
G:=SmallGroup(320,89);
// by ID
G=gap.SmallGroup(320,89);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations