metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1Dic5, C20.49C42, (C4×C20)⋊3C4, (C2×C20).6Q8, (C4×Dic5)⋊3C4, (C2×C4).13D20, C20.59(C4⋊C4), C5⋊3(C4.9C42), (C2×C20).104D4, (C2×C4).2Dic10, C4.24(C4×Dic5), C4.12(C4⋊Dic5), (C22×C10).42D4, (C22×C4).57D10, C42⋊C2.1D5, C20.56(C22⋊C4), C4.8(C10.D4), C23.16(C5⋊D4), C4.33(D10⋊C4), (C22×C20).120C22, C23.21D10.7C2, C22.3(C10.D4), C22.10(C23.D5), C22.17(D10⋊C4), C2.8(C10.10C42), C10.26(C2.C42), (C2×C5⋊2C8)⋊1C4, (C2×C4).139(C4×D5), (C2×C10).31(C4⋊C4), (C2×C20).229(C2×C4), (C2×C4).20(C5⋊D4), (C2×C4).72(C2×Dic5), (C2×C4.Dic5).7C2, (C5×C42⋊C2).1C2, (C2×C10).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 [×3], C4 [×4], C4 [×4], C22 [×3], C22, C5, C8 [×2], C2×C4 [×6], C2×C4 [×4], C23, C10, C10 [×3], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, Dic5 [×2], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10, C42⋊C2, C42⋊C2, C2×M4(2), C5⋊2C8 [×2], C2×Dic5 [×2], C2×C20 [×6], C2×C20 [×2], C22×C10, C4.9C42, C2×C5⋊2C8 [×2], C4.Dic5 [×2], C4×Dic5 [×2], C4⋊Dic5, C23.D5, C4×C20 [×2], C5×C22⋊C4, C5×C4⋊C4, C22×C20, C2×C4.Dic5, C23.21D10, C5×C42⋊C2, C42⋊1Dic5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C4.9C42, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C10.10C42, C42⋊1Dic5
►On 80 points
(1 59 19 61)(2 55 20 67)(3 51 16 63)(4 57 17 69)(5 53 18 65)(6 60 14 62)(7 56 15 68)(8 52 11 64)(9 58 12 70)(10 54 13 66)(21 46 36 76)(22 42 37 72)(23 48 38 78)(24 44 39 74)(25 50 40 80)(26 47 31 77)(27 43 32 73)(28 49 33 79)(29 45 34 75)(30 41 35 71)
(1 33 10 39)(2 34 6 40)(3 35 7 36)(4 31 8 37)(5 32 9 38)(11 22 17 26)(12 23 18 27)(13 24 19 28)(14 25 20 29)(15 21 16 30)(41 68 46 63)(42 69 47 64)(43 70 48 65)(44 61 49 66)(45 62 50 67)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 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 26)(2 30)(3 29)(4 28)(5 27)(6 21)(7 25)(8 24)(9 23)(10 22)(11 33)(12 32)(13 31)(14 35)(15 34)(16 40)(17 39)(18 38)(19 37)(20 36)(41 75 46 80)(42 74 47 79)(43 73 48 78)(44 72 49 77)(45 71 50 76)(51 62 56 67)(52 61 57 66)(53 70 58 65)(54 69 59 64)(55 68 60 63)
G:=sub<Sym(80)| (1,59,19,61)(2,55,20,67)(3,51,16,63)(4,57,17,69)(5,53,18,65)(6,60,14,62)(7,56,15,68)(8,52,11,64)(9,58,12,70)(10,54,13,66)(21,46,36,76)(22,42,37,72)(23,48,38,78)(24,44,39,74)(25,50,40,80)(26,47,31,77)(27,43,32,73)(28,49,33,79)(29,45,34,75)(30,41,35,71), (1,33,10,39)(2,34,6,40)(3,35,7,36)(4,31,8,37)(5,32,9,38)(11,22,17,26)(12,23,18,27)(13,24,19,28)(14,25,20,29)(15,21,16,30)(41,68,46,63)(42,69,47,64)(43,70,48,65)(44,61,49,66)(45,62,50,67)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80), (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), (1,26)(2,30)(3,29)(4,28)(5,27)(6,21)(7,25)(8,24)(9,23)(10,22)(11,33)(12,32)(13,31)(14,35)(15,34)(16,40)(17,39)(18,38)(19,37)(20,36)(41,75,46,80)(42,74,47,79)(43,73,48,78)(44,72,49,77)(45,71,50,76)(51,62,56,67)(52,61,57,66)(53,70,58,65)(54,69,59,64)(55,68,60,63)>;
G:=Group( (1,59,19,61)(2,55,20,67)(3,51,16,63)(4,57,17,69)(5,53,18,65)(6,60,14,62)(7,56,15,68)(8,52,11,64)(9,58,12,70)(10,54,13,66)(21,46,36,76)(22,42,37,72)(23,48,38,78)(24,44,39,74)(25,50,40,80)(26,47,31,77)(27,43,32,73)(28,49,33,79)(29,45,34,75)(30,41,35,71), (1,33,10,39)(2,34,6,40)(3,35,7,36)(4,31,8,37)(5,32,9,38)(11,22,17,26)(12,23,18,27)(13,24,19,28)(14,25,20,29)(15,21,16,30)(41,68,46,63)(42,69,47,64)(43,70,48,65)(44,61,49,66)(45,62,50,67)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80), (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), (1,26)(2,30)(3,29)(4,28)(5,27)(6,21)(7,25)(8,24)(9,23)(10,22)(11,33)(12,32)(13,31)(14,35)(15,34)(16,40)(17,39)(18,38)(19,37)(20,36)(41,75,46,80)(42,74,47,79)(43,73,48,78)(44,72,49,77)(45,71,50,76)(51,62,56,67)(52,61,57,66)(53,70,58,65)(54,69,59,64)(55,68,60,63) );
G=PermutationGroup([(1,59,19,61),(2,55,20,67),(3,51,16,63),(4,57,17,69),(5,53,18,65),(6,60,14,62),(7,56,15,68),(8,52,11,64),(9,58,12,70),(10,54,13,66),(21,46,36,76),(22,42,37,72),(23,48,38,78),(24,44,39,74),(25,50,40,80),(26,47,31,77),(27,43,32,73),(28,49,33,79),(29,45,34,75),(30,41,35,71)], [(1,33,10,39),(2,34,6,40),(3,35,7,36),(4,31,8,37),(5,32,9,38),(11,22,17,26),(12,23,18,27),(13,24,19,28),(14,25,20,29),(15,21,16,30),(41,68,46,63),(42,69,47,64),(43,70,48,65),(44,61,49,66),(45,62,50,67),(51,71,56,76),(52,72,57,77),(53,73,58,78),(54,74,59,79),(55,75,60,80)], [(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)], [(1,26),(2,30),(3,29),(4,28),(5,27),(6,21),(7,25),(8,24),(9,23),(10,22),(11,33),(12,32),(13,31),(14,35),(15,34),(16,40),(17,39),(18,38),(19,37),(20,36),(41,75,46,80),(42,74,47,79),(43,73,48,78),(44,72,49,77),(45,71,50,76),(51,62,56,67),(52,61,57,66),(53,70,58,65),(54,69,59,64),(55,68,60,63)])
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 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | C4.9C42 | C42⋊1Dic5 |
| kernel | C42⋊1Dic5 | C2×C4.Dic5 | C23.21D10 | C5×C42⋊C2 | C2×C5⋊2C8 | C4×Dic5 | C4×C20 | C2×C20 | C2×C20 | C22×C10 | C42⋊C2 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | 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