metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊5D10, D20.13D4, Dic10.13D4, (C2×Q8)⋊2D10, C4.51(D4×D5), C4.4D4⋊4D5, C20.28(C2×D4), (C4×C20)⋊13C22, (C2×D4).51D10, C20.D4⋊5C2, C5⋊3(D4.9D4), D20⋊4C4⋊11C2, (Q8×C10)⋊2C22, C10.51C22≀C2, D4⋊6D10.4C2, C20.C23⋊2C2, (C22×C10).22D4, C4.Dic5⋊6C22, (C2×C20).379C23, C4○D20.19C22, (D4×C10).67C22, C23.10(C5⋊D4), C2.19(C23⋊D10), (C5×C4.4D4)⋊4C2, (C2×C10).510(C2×D4), C22.31(C2×C5⋊D4), (C2×C4).116(C22×D5), SmallGroup(320,688)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊5D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 686 in 152 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×3], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C5⋊2C8 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], D4.9D4, C4.Dic5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C4×C20, C5×C22⋊C4 [×2], C4○D20 [×2], D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, D20⋊4C4 [×2], C20.D4, C20.C23 [×2], C5×C4.4D4, D4⋊6D10, C42⋊5D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.9D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C42⋊5D10
►On 80 points
(1 31 23 69)(2 79 24 59)(3 33 25 61)(4 71 26 51)(5 35 27 63)(6 73 28 53)(7 37 29 65)(8 75 30 55)(9 39 21 67)(10 77 22 57)(11 72 44 52)(12 36 45 64)(13 74 46 54)(14 38 47 66)(15 76 48 56)(16 40 49 68)(17 78 50 58)(18 32 41 70)(19 80 42 60)(20 34 43 62)
(1 28 50 12)(2 13 41 29)(3 30 42 14)(4 15 43 21)(5 22 44 16)(6 17 45 23)(7 24 46 18)(8 19 47 25)(9 26 48 20)(10 11 49 27)(31 53 58 36)(32 37 59 54)(33 55 60 38)(34 39 51 56)(35 57 52 40)(61 75 80 66)(62 67 71 76)(63 77 72 68)(64 69 73 78)(65 79 74 70)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 30)(20 29)(31 52)(32 51)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 48)(42 47)(43 46)(44 45)(49 50)(62 70)(63 69)(64 68)(65 67)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,31,23,69)(2,79,24,59)(3,33,25,61)(4,71,26,51)(5,35,27,63)(6,73,28,53)(7,37,29,65)(8,75,30,55)(9,39,21,67)(10,77,22,57)(11,72,44,52)(12,36,45,64)(13,74,46,54)(14,38,47,66)(15,76,48,56)(16,40,49,68)(17,78,50,58)(18,32,41,70)(19,80,42,60)(20,34,43,62), (1,28,50,12)(2,13,41,29)(3,30,42,14)(4,15,43,21)(5,22,44,16)(6,17,45,23)(7,24,46,18)(8,19,47,25)(9,26,48,20)(10,11,49,27)(31,53,58,36)(32,37,59,54)(33,55,60,38)(34,39,51,56)(35,57,52,40)(61,75,80,66)(62,67,71,76)(63,77,72,68)(64,69,73,78)(65,79,74,70), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,30)(20,29)(31,52)(32,51)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,48)(42,47)(43,46)(44,45)(49,50)(62,70)(63,69)(64,68)(65,67)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,31,23,69)(2,79,24,59)(3,33,25,61)(4,71,26,51)(5,35,27,63)(6,73,28,53)(7,37,29,65)(8,75,30,55)(9,39,21,67)(10,77,22,57)(11,72,44,52)(12,36,45,64)(13,74,46,54)(14,38,47,66)(15,76,48,56)(16,40,49,68)(17,78,50,58)(18,32,41,70)(19,80,42,60)(20,34,43,62), (1,28,50,12)(2,13,41,29)(3,30,42,14)(4,15,43,21)(5,22,44,16)(6,17,45,23)(7,24,46,18)(8,19,47,25)(9,26,48,20)(10,11,49,27)(31,53,58,36)(32,37,59,54)(33,55,60,38)(34,39,51,56)(35,57,52,40)(61,75,80,66)(62,67,71,76)(63,77,72,68)(64,69,73,78)(65,79,74,70), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,30)(20,29)(31,52)(32,51)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,48)(42,47)(43,46)(44,45)(49,50)(62,70)(63,69)(64,68)(65,67)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([(1,31,23,69),(2,79,24,59),(3,33,25,61),(4,71,26,51),(5,35,27,63),(6,73,28,53),(7,37,29,65),(8,75,30,55),(9,39,21,67),(10,77,22,57),(11,72,44,52),(12,36,45,64),(13,74,46,54),(14,38,47,66),(15,76,48,56),(16,40,49,68),(17,78,50,58),(18,32,41,70),(19,80,42,60),(20,34,43,62)], [(1,28,50,12),(2,13,41,29),(3,30,42,14),(4,15,43,21),(5,22,44,16),(6,17,45,23),(7,24,46,18),(8,19,47,25),(9,26,48,20),(10,11,49,27),(31,53,58,36),(32,37,59,54),(33,55,60,38),(34,39,51,56),(35,57,52,40),(61,75,80,66),(62,67,71,76),(63,77,72,68),(64,69,73,78),(65,79,74,70)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,30),(20,29),(31,52),(32,51),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,48),(42,47),(43,46),(44,45),(49,50),(62,70),(63,69),(64,68),(65,67),(71,79),(72,78),(73,77),(74,76)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 20 | 20 | 2 | 2 | 40 | 40 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | D4.9D4 | D4×D5 | C42⋊5D10 |
| kernel | C42⋊5D10 | D20⋊4C4 | C20.D4 | C20.C23 | C5×C4.4D4 | D4⋊6D10 | Dic10 | D20 | C22×C10 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C23 | C5 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 8 |
Matrix representation of C42⋊5D10 ►in GL6(𝔽41)
| 17 | 40 | 0 | 0 | 0 | 0 |
| 1 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 25 | 16 |
| 0 | 0 | 16 | 16 | 16 | 25 |
| 0 | 0 | 16 | 25 | 16 | 16 |
| 0 | 0 | 25 | 16 | 16 | 16 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [17,1,0,0,0,0,40,24,0,0,0,0,0,0,16,16,16,25,0,0,16,16,25,16,0,0,25,16,16,16,0,0,16,25,16,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1,0,0,0] >;
C42⋊5D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5D_{10} % in TeX
G:=Group("C4^2:5D10"); // GroupNames label
G:=SmallGroup(320,688);
// by ID
G=gap.SmallGroup(320,688);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,1123,570,297,136,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations