metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊17D10, C10.212+ 1+4, C4⋊C4⋊50D10, (C4×D4)⋊22D5, (D4×C20)⋊24C2, C22⋊D20⋊7C2, C20⋊7D4⋊11C2, (C4×C20)⋊28C22, C22⋊C4⋊49D10, (C22×C4)⋊14D10, C23⋊D10⋊21C2, D10⋊D4⋊10C2, (C2×D4).221D10, C4.D20⋊28C2, C42⋊2D5⋊10C2, C4⋊Dic5⋊10C22, Dic5⋊D4⋊27C2, D10.13D4⋊8C2, (C2×C20).162C23, (C2×C10).104C24, (C22×C20)⋊11C22, Dic5.5D4⋊9C2, C5⋊2(C22.32C24), (C4×Dic5)⋊53C22, (C2×Dic10)⋊7C22, (C2×D20).29C22, C22.6(C4○D20), C23.D5⋊10C22, C2.22(D4⋊6D10), C2.17(D4⋊8D10), D10⋊C4⋊31C22, Dic5.14D4⋊9C2, (D4×C10).308C22, C10.D4⋊33C22, C23.23D10⋊2C2, (C2×Dic5).45C23, (C22×D5).38C23, (C23×D5).42C22, C23.101(C22×D5), C22.129(C23×D5), (C22×C10).174C23, (C22×Dic5).99C22, C4⋊C4⋊D5⋊8C2, (C4×C5⋊D4)⋊46C2, (C2×C4×D5)⋊49C22, (C5×C4⋊C4)⋊62C22, C10.46(C2×C4○D4), C2.53(C2×C4○D20), (C2×C5⋊D4)⋊5C22, (C2×D10⋊C4)⋊35C2, (C2×C10).17(C4○D4), (C5×C22⋊C4)⋊58C22, (C2×C4).162(C22×D5), SmallGroup(320,1232)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊17D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1054 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22.32C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C4.D20, C42⋊2D5, Dic5.14D4, C22⋊D20, D10⋊D4, Dic5.5D4, D10.13D4, C4⋊C4⋊D5, C2×D10⋊C4, C4×C5⋊D4, C23.23D10, C20⋊7D4, C23⋊D10, Dic5⋊D4, D4×C20, C42⋊17D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, C4○D20, C23×D5, C2×C4○D20, D4⋊6D10, D4⋊8D10, C42⋊17D10
►On 80 points
(1 60 28 55)(2 56 29 51)(3 52 30 57)(4 58 26 53)(5 54 27 59)(6 44 32 49)(7 50 33 45)(8 46 34 41)(9 42 35 47)(10 48 31 43)(11 80 16 75)(12 76 17 71)(13 72 18 77)(14 78 19 73)(15 74 20 79)(21 61 38 66)(22 67 39 62)(23 63 40 68)(24 69 36 64)(25 65 37 70)
(1 32 21 11)(2 33 22 12)(3 34 23 13)(4 35 24 14)(5 31 25 15)(6 38 16 28)(7 39 17 29)(8 40 18 30)(9 36 19 26)(10 37 20 27)(41 63 72 52)(42 64 73 53)(43 65 74 54)(44 66 75 55)(45 67 76 56)(46 68 77 57)(47 69 78 58)(48 70 79 59)(49 61 80 60)(50 62 71 51)
(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 9)(7 8)(11 14)(12 13)(16 19)(17 18)(21 36)(22 40)(23 39)(24 38)(25 37)(32 35)(33 34)(41 71)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 70)(60 69)
G:=sub<Sym(80)| (1,60,28,55)(2,56,29,51)(3,52,30,57)(4,58,26,53)(5,54,27,59)(6,44,32,49)(7,50,33,45)(8,46,34,41)(9,42,35,47)(10,48,31,43)(11,80,16,75)(12,76,17,71)(13,72,18,77)(14,78,19,73)(15,74,20,79)(21,61,38,66)(22,67,39,62)(23,63,40,68)(24,69,36,64)(25,65,37,70), (1,32,21,11)(2,33,22,12)(3,34,23,13)(4,35,24,14)(5,31,25,15)(6,38,16,28)(7,39,17,29)(8,40,18,30)(9,36,19,26)(10,37,20,27)(41,63,72,52)(42,64,73,53)(43,65,74,54)(44,66,75,55)(45,67,76,56)(46,68,77,57)(47,69,78,58)(48,70,79,59)(49,61,80,60)(50,62,71,51), (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)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,36)(22,40)(23,39)(24,38)(25,37)(32,35)(33,34)(41,71)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69)>;
G:=Group( (1,60,28,55)(2,56,29,51)(3,52,30,57)(4,58,26,53)(5,54,27,59)(6,44,32,49)(7,50,33,45)(8,46,34,41)(9,42,35,47)(10,48,31,43)(11,80,16,75)(12,76,17,71)(13,72,18,77)(14,78,19,73)(15,74,20,79)(21,61,38,66)(22,67,39,62)(23,63,40,68)(24,69,36,64)(25,65,37,70), (1,32,21,11)(2,33,22,12)(3,34,23,13)(4,35,24,14)(5,31,25,15)(6,38,16,28)(7,39,17,29)(8,40,18,30)(9,36,19,26)(10,37,20,27)(41,63,72,52)(42,64,73,53)(43,65,74,54)(44,66,75,55)(45,67,76,56)(46,68,77,57)(47,69,78,58)(48,70,79,59)(49,61,80,60)(50,62,71,51), (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)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,36)(22,40)(23,39)(24,38)(25,37)(32,35)(33,34)(41,71)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69) );
G=PermutationGroup([[(1,60,28,55),(2,56,29,51),(3,52,30,57),(4,58,26,53),(5,54,27,59),(6,44,32,49),(7,50,33,45),(8,46,34,41),(9,42,35,47),(10,48,31,43),(11,80,16,75),(12,76,17,71),(13,72,18,77),(14,78,19,73),(15,74,20,79),(21,61,38,66),(22,67,39,62),(23,63,40,68),(24,69,36,64),(25,65,37,70)], [(1,32,21,11),(2,33,22,12),(3,34,23,13),(4,35,24,14),(5,31,25,15),(6,38,16,28),(7,39,17,29),(8,40,18,30),(9,36,19,26),(10,37,20,27),(41,63,72,52),(42,64,73,53),(43,65,74,54),(44,66,75,55),(45,67,76,56),(46,68,77,57),(47,69,78,58),(48,70,79,59),(49,61,80,60),(50,62,71,51)], [(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),(6,9),(7,8),(11,14),(12,13),(16,19),(17,18),(21,36),(22,40),(23,39),(24,38),(25,37),(32,35),(33,34),(41,71),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,70),(60,69)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4⋊6D10 | D4⋊8D10 |
| kernel | C42⋊17D10 | C4.D20 | C42⋊2D5 | Dic5.14D4 | C22⋊D20 | D10⋊D4 | Dic5.5D4 | D10.13D4 | C4⋊C4⋊D5 | C2×D10⋊C4 | C4×C5⋊D4 | C23.23D10 | C20⋊7D4 | C23⋊D10 | Dic5⋊D4 | D4×C20 | C4×D4 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 4 | 4 |
Matrix representation of C42⋊17D10 ►in GL6(𝔽41)
| 25 | 31 | 0 | 0 | 0 | 0 |
| 5 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 35 | 21 | 21 |
| 0 | 0 | 7 | 24 | 40 | 38 |
| 0 | 0 | 35 | 40 | 18 | 6 |
| 0 | 0 | 2 | 1 | 35 | 23 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 19 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 13 |
| 0 | 0 | 0 | 0 | 28 | 2 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 35 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 27 | 39 | 1 | 6 |
| 0 | 0 | 2 | 0 | 35 | 6 |
| 1 | 31 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 14 | 12 | 40 | 0 |
| 0 | 0 | 39 | 39 | 6 | 1 |
G:=sub<GL(6,GF(41))| [25,5,0,0,0,0,31,16,0,0,0,0,0,0,17,7,35,2,0,0,35,24,40,1,0,0,21,40,18,35,0,0,21,38,6,23],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,11,19,0,0,0,0,13,30,0,0,0,0,0,0,39,28,0,0,0,0,13,2],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,27,2,0,0,35,0,39,0,0,0,0,0,1,35,0,0,0,0,6,6],[1,0,0,0,0,0,31,40,0,0,0,0,0,0,7,34,14,39,0,0,1,34,12,39,0,0,0,0,40,6,0,0,0,0,0,1] >;
C42⋊17D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{17}D_{10} % in TeX
G:=Group("C4^2:17D10"); // GroupNames label
G:=SmallGroup(320,1232);
// by ID
G=gap.SmallGroup(320,1232);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,675,570,12550]);
// 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,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations