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, D10⋊D4⋊10C2, C23⋊D10⋊21C2, (C2×D4).221D10, C4.D20⋊28C2, C42⋊2D5⋊10C2, C4⋊Dic5⋊10C22, Dic5⋊D4⋊27C2, D10.13D4⋊8C2, (C2×C10).104C24, (C2×C20).162C23, (C22×C20)⋊11C22, Dic5.5D4⋊9C2, C5⋊2(C22.32C24), (C2×Dic10)⋊7C22, (C4×Dic5)⋊53C22, (C2×D20).29C22, C22.6(C4○D20), C2.22(D4⋊6D10), C2.17(D4⋊8D10), C23.D5⋊10C22, D10⋊C4⋊31C22, Dic5.14D4⋊9C2, (D4×C10).308C22, C23.23D10⋊2C2, C10.D4⋊33C22, (C2×Dic5).45C23, (C22×D5).38C23, (C23×D5).42C22, C22.129(C23×D5), C23.101(C22×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
Subgroups: 1054 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C5, C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], D5 [×3], C10 [×3], C10 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic5 [×5], C20 [×5], D10 [×13], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4, C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], Dic10, C4×D5, D20 [×2], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×5], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5 [×3], C22×D5 [×4], C22×C10 [×2], C22.32C24, C4×Dic5, C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×10], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20 [×2], C22×Dic5, C2×C5⋊D4 [×4], C22×C20 [×2], 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 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D5 [×7], C22.32C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊6D10, D4⋊8D10, C42⋊17D10
Generators and relations
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 >
►On 80 points
(1 55 32 60)(2 51 33 56)(3 57 34 52)(4 53 35 58)(5 59 31 54)(6 75 30 80)(7 71 26 76)(8 77 27 72)(9 73 28 78)(10 79 29 74)(11 49 16 44)(12 45 17 50)(13 41 18 46)(14 47 19 42)(15 43 20 48)(21 61 37 66)(22 67 38 62)(23 63 39 68)(24 69 40 64)(25 65 36 70)
(1 30 21 11)(2 26 22 12)(3 27 23 13)(4 28 24 14)(5 29 25 15)(6 37 16 32)(7 38 17 33)(8 39 18 34)(9 40 19 35)(10 36 20 31)(41 57 72 63)(42 58 73 64)(43 59 74 65)(44 60 75 66)(45 51 76 67)(46 52 77 68)(47 53 78 69)(48 54 79 70)(49 55 80 61)(50 56 71 62)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 9)(7 8)(11 14)(12 13)(16 19)(17 18)(21 40)(22 39)(23 38)(24 37)(25 36)(26 27)(28 30)(41 71)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)
G:=sub<Sym(80)| (1,55,32,60)(2,51,33,56)(3,57,34,52)(4,53,35,58)(5,59,31,54)(6,75,30,80)(7,71,26,76)(8,77,27,72)(9,73,28,78)(10,79,29,74)(11,49,16,44)(12,45,17,50)(13,41,18,46)(14,47,19,42)(15,43,20,48)(21,61,37,66)(22,67,38,62)(23,63,39,68)(24,69,40,64)(25,65,36,70), (1,30,21,11)(2,26,22,12)(3,27,23,13)(4,28,24,14)(5,29,25,15)(6,37,16,32)(7,38,17,33)(8,39,18,34)(9,40,19,35)(10,36,20,31)(41,57,72,63)(42,58,73,64)(43,59,74,65)(44,60,75,66)(45,51,76,67)(46,52,77,68)(47,53,78,69)(48,54,79,70)(49,55,80,61)(50,56,71,62), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,40)(22,39)(23,38)(24,37)(25,36)(26,27)(28,30)(41,71)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,63)(52,62)(53,61)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)>;
G:=Group( (1,55,32,60)(2,51,33,56)(3,57,34,52)(4,53,35,58)(5,59,31,54)(6,75,30,80)(7,71,26,76)(8,77,27,72)(9,73,28,78)(10,79,29,74)(11,49,16,44)(12,45,17,50)(13,41,18,46)(14,47,19,42)(15,43,20,48)(21,61,37,66)(22,67,38,62)(23,63,39,68)(24,69,40,64)(25,65,36,70), (1,30,21,11)(2,26,22,12)(3,27,23,13)(4,28,24,14)(5,29,25,15)(6,37,16,32)(7,38,17,33)(8,39,18,34)(9,40,19,35)(10,36,20,31)(41,57,72,63)(42,58,73,64)(43,59,74,65)(44,60,75,66)(45,51,76,67)(46,52,77,68)(47,53,78,69)(48,54,79,70)(49,55,80,61)(50,56,71,62), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,40)(22,39)(23,38)(24,37)(25,36)(26,27)(28,30)(41,71)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,63)(52,62)(53,61)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64) );
G=PermutationGroup([(1,55,32,60),(2,51,33,56),(3,57,34,52),(4,53,35,58),(5,59,31,54),(6,75,30,80),(7,71,26,76),(8,77,27,72),(9,73,28,78),(10,79,29,74),(11,49,16,44),(12,45,17,50),(13,41,18,46),(14,47,19,42),(15,43,20,48),(21,61,37,66),(22,67,38,62),(23,63,39,68),(24,69,40,64),(25,65,36,70)], [(1,30,21,11),(2,26,22,12),(3,27,23,13),(4,28,24,14),(5,29,25,15),(6,37,16,32),(7,38,17,33),(8,39,18,34),(9,40,19,35),(10,36,20,31),(41,57,72,63),(42,58,73,64),(43,59,74,65),(44,60,75,66),(45,51,76,67),(46,52,77,68),(47,53,78,69),(48,54,79,70),(49,55,80,61),(50,56,71,62)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,9),(7,8),(11,14),(12,13),(16,19),(17,18),(21,40),(22,39),(23,38),(24,37),(25,36),(26,27),(28,30),(41,71),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,63),(52,62),(53,61),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64)])
Matrix representation ►G ⊆ 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] >;
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 |
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