metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊6Dic5, C20.38C42, (C4×C20)⋊22C4, C10.19C4≀C2, C4⋊Dic5⋊12C4, C4.Dic5⋊7C4, C20.58(C4⋊C4), (C2×C20).58Q8, C4.8(C4×Dic5), (C2×C42).7D5, C5⋊4(C42⋊6C4), (C2×C4).162D20, (C2×C20).478D4, C4.20(C4⋊Dic5), (C2×C4).42Dic10, C2.3(D20⋊4C4), (C22×C10).176D4, (C22×C4).412D10, C23.72(C5⋊D4), C4.44(D10⋊C4), C20.106(C22⋊C4), C4.22(C10.D4), C22.8(C23.D5), (C22×C20).532C22, C23.21D10.1C2, C22.36(D10⋊C4), C2.3(C10.10C42), C10.20(C2.C42), C22.12(C10.D4), (C2×C4×C20).15C2, (C2×C4).98(C4×D5), (C2×C10).61(C4⋊C4), (C2×C20).390(C2×C4), (C2×C4).71(C2×Dic5), (C2×C4.Dic5).1C2, (C2×C4).231(C5⋊D4), (C2×C10).113(C22⋊C4), SmallGroup(320,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊6Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, dad-1=ab=ba, ac=ca, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 294 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C22×C10, C42⋊6C4, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C4×C20, C4×C20, C22×C20, C22×C20, C2×C4.Dic5, C23.21D10, C2×C4×C20, C42⋊6Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C4≀C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C42⋊6C4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, D20⋊4C4, C10.10C42, C42⋊6Dic5
►On 80 points
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 31)(9 32)(10 33)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 41)(21 55 69 71)(22 56 70 72)(23 57 61 73)(24 58 62 74)(25 59 63 75)(26 60 64 76)(27 51 65 77)(28 52 66 78)(29 53 67 79)(30 54 68 80)
(1 45 39 19)(2 46 40 20)(3 47 31 11)(4 48 32 12)(5 49 33 13)(6 50 34 14)(7 41 35 15)(8 42 36 16)(9 43 37 17)(10 44 38 18)(21 60 69 76)(22 51 70 77)(23 52 61 78)(24 53 62 79)(25 54 63 80)(26 55 64 71)(27 56 65 72)(28 57 66 73)(29 58 67 74)(30 59 68 75)
(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 60 6 55)(2 59 7 54)(3 58 8 53)(4 57 9 52)(5 56 10 51)(11 67 16 62)(12 66 17 61)(13 65 18 70)(14 64 19 69)(15 63 20 68)(21 50 26 45)(22 49 27 44)(23 48 28 43)(24 47 29 42)(25 46 30 41)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)
G:=sub<Sym(80)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,31)(9,32)(10,33)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,41)(21,55,69,71)(22,56,70,72)(23,57,61,73)(24,58,62,74)(25,59,63,75)(26,60,64,76)(27,51,65,77)(28,52,66,78)(29,53,67,79)(30,54,68,80), (1,45,39,19)(2,46,40,20)(3,47,31,11)(4,48,32,12)(5,49,33,13)(6,50,34,14)(7,41,35,15)(8,42,36,16)(9,43,37,17)(10,44,38,18)(21,60,69,76)(22,51,70,77)(23,52,61,78)(24,53,62,79)(25,54,63,80)(26,55,64,71)(27,56,65,72)(28,57,66,73)(29,58,67,74)(30,59,68,75), (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), (1,60,6,55)(2,59,7,54)(3,58,8,53)(4,57,9,52)(5,56,10,51)(11,67,16,62)(12,66,17,61)(13,65,18,70)(14,64,19,69)(15,63,20,68)(21,50,26,45)(22,49,27,44)(23,48,28,43)(24,47,29,42)(25,46,30,41)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75)>;
G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,31)(9,32)(10,33)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,41)(21,55,69,71)(22,56,70,72)(23,57,61,73)(24,58,62,74)(25,59,63,75)(26,60,64,76)(27,51,65,77)(28,52,66,78)(29,53,67,79)(30,54,68,80), (1,45,39,19)(2,46,40,20)(3,47,31,11)(4,48,32,12)(5,49,33,13)(6,50,34,14)(7,41,35,15)(8,42,36,16)(9,43,37,17)(10,44,38,18)(21,60,69,76)(22,51,70,77)(23,52,61,78)(24,53,62,79)(25,54,63,80)(26,55,64,71)(27,56,65,72)(28,57,66,73)(29,58,67,74)(30,59,68,75), (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), (1,60,6,55)(2,59,7,54)(3,58,8,53)(4,57,9,52)(5,56,10,51)(11,67,16,62)(12,66,17,61)(13,65,18,70)(14,64,19,69)(15,63,20,68)(21,50,26,45)(22,49,27,44)(23,48,28,43)(24,47,29,42)(25,46,30,41)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,31),(9,32),(10,33),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,41),(21,55,69,71),(22,56,70,72),(23,57,61,73),(24,58,62,74),(25,59,63,75),(26,60,64,76),(27,51,65,77),(28,52,66,78),(29,53,67,79),(30,54,68,80)], [(1,45,39,19),(2,46,40,20),(3,47,31,11),(4,48,32,12),(5,49,33,13),(6,50,34,14),(7,41,35,15),(8,42,36,16),(9,43,37,17),(10,44,38,18),(21,60,69,76),(22,51,70,77),(23,52,61,78),(24,53,62,79),(25,54,63,80),(26,55,64,71),(27,56,65,72),(28,57,66,73),(29,58,67,74),(30,59,68,75)], [(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)], [(1,60,6,55),(2,59,7,54),(3,58,8,53),(4,57,9,52),(5,56,10,51),(11,67,16,62),(12,66,17,61),(13,65,18,70),(14,64,19,69),(15,63,20,68),(21,50,26,45),(22,49,27,44),(23,48,28,43),(24,47,29,42),(25,46,30,41),(31,74,36,79),(32,73,37,78),(33,72,38,77),(34,71,39,76),(35,80,40,75)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D5 | Dic5 | D10 | C4≀C2 | Dic10 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | D20⋊4C4 |
| kernel | C42⋊6Dic5 | C2×C4.Dic5 | C23.21D10 | C2×C4×C20 | C4.Dic5 | C4⋊Dic5 | C4×C20 | C2×C20 | C2×C20 | C22×C10 | C2×C42 | C42 | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 4 | 4 | 4 | 32 |
Matrix representation of C42⋊6Dic5 ►in GL4(𝔽41) generated by
| 40 | 22 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 32 |
| 9 | 29 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 25 |
| 16 | 15 | 0 | 0 |
| 24 | 25 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,22,32,0,0,0,0,1,0,0,0,0,32],[9,0,0,0,29,32,0,0,0,0,32,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,23,0,0,0,0,25],[16,24,0,0,15,25,0,0,0,0,0,40,0,0,1,0] >;
C42⋊6Dic5 in GAP, Magma, Sage, TeX
C_4^2\rtimes_6{\rm Dic}_5 % in TeX
G:=Group("C4^2:6Dic5"); // GroupNames label
G:=SmallGroup(320,81);
// by ID
G=gap.SmallGroup(320,81);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations