direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C16⋊C22, D16⋊2C10, C80⋊6C22, C40.53D4, C20.65D8, SD32⋊1C10, M5(2)⋊1C10, C40.76C23, C16⋊(C2×C10), C8.3(C5×D4), C4○D8⋊2C10, D8⋊2(C2×C10), (C5×D16)⋊6C2, C4.14(C5×D8), (C10×D8)⋊24C2, (C2×D8)⋊10C10, Q16⋊2(C2×C10), (C5×SD32)⋊5C2, C10.88(C2×D8), (C2×C10).27D8, C4.11(D4×C10), C2.16(C10×D8), C22.5(C5×D8), (C2×C20).346D4, C20.318(C2×D4), (C5×D8)⋊18C22, (C5×M5(2))⋊3C2, C8.7(C22×C10), (C5×Q16)⋊16C22, (C2×C40).278C22, (C5×C4○D8)⋊9C2, (C2×C4).47(C5×D4), (C2×C8).30(C2×C10), SmallGroup(320,1010)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 226 in 90 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C10, C10 [×4], C16 [×2], C2×C8, D8, D8 [×2], D8, SD16, Q16, C2×D4, C4○D4, C20 [×2], C20, C2×C10, C2×C10 [×5], M5(2), D16 [×2], SD32 [×2], C2×D8, C4○D8, C40 [×2], C2×C20, C2×C20, C5×D4 [×5], C5×Q8, C22×C10, C16⋊C22, C80 [×2], C2×C40, C5×D8, C5×D8 [×2], C5×D8, C5×SD16, C5×Q16, D4×C10, C5×C4○D4, C5×M5(2), C5×D16 [×2], C5×SD32 [×2], C10×D8, C5×C4○D8, C5×C16⋊C22
Quotients:
C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], C2×D8, C5×D4 [×2], C22×C10, C16⋊C22, C5×D8 [×2], D4×C10, C10×D8, C5×C16⋊C22
Generators and relations
G = < a,b,c,d | a5=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >
►On 80 points
(1 18 35 65 63)(2 19 36 66 64)(3 20 37 67 49)(4 21 38 68 50)(5 22 39 69 51)(6 23 40 70 52)(7 24 41 71 53)(8 25 42 72 54)(9 26 43 73 55)(10 27 44 74 56)(11 28 45 75 57)(12 29 46 76 58)(13 30 47 77 59)(14 31 48 78 60)(15 32 33 79 61)(16 17 34 80 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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 27)(19 25)(20 32)(21 23)(22 30)(24 28)(29 31)(33 37)(34 44)(36 42)(38 40)(39 47)(41 45)(46 48)(49 61)(50 52)(51 59)(53 57)(54 64)(56 62)(58 60)(66 72)(67 79)(68 70)(69 77)(71 75)(74 80)(76 78)
(1 9)(3 11)(5 13)(7 15)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)
G:=sub<Sym(80)| (1,18,35,65,63)(2,19,36,66,64)(3,20,37,67,49)(4,21,38,68,50)(5,22,39,69,51)(6,23,40,70,52)(7,24,41,71,53)(8,25,42,72,54)(9,26,43,73,55)(10,27,44,74,56)(11,28,45,75,57)(12,29,46,76,58)(13,30,47,77,59)(14,31,48,78,60)(15,32,33,79,61)(16,17,34,80,62), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,27)(19,25)(20,32)(21,23)(22,30)(24,28)(29,31)(33,37)(34,44)(36,42)(38,40)(39,47)(41,45)(46,48)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(66,72)(67,79)(68,70)(69,77)(71,75)(74,80)(76,78), (1,9)(3,11)(5,13)(7,15)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)>;
G:=Group( (1,18,35,65,63)(2,19,36,66,64)(3,20,37,67,49)(4,21,38,68,50)(5,22,39,69,51)(6,23,40,70,52)(7,24,41,71,53)(8,25,42,72,54)(9,26,43,73,55)(10,27,44,74,56)(11,28,45,75,57)(12,29,46,76,58)(13,30,47,77,59)(14,31,48,78,60)(15,32,33,79,61)(16,17,34,80,62), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,27)(19,25)(20,32)(21,23)(22,30)(24,28)(29,31)(33,37)(34,44)(36,42)(38,40)(39,47)(41,45)(46,48)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(66,72)(67,79)(68,70)(69,77)(71,75)(74,80)(76,78), (1,9)(3,11)(5,13)(7,15)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79) );
G=PermutationGroup([(1,18,35,65,63),(2,19,36,66,64),(3,20,37,67,49),(4,21,38,68,50),(5,22,39,69,51),(6,23,40,70,52),(7,24,41,71,53),(8,25,42,72,54),(9,26,43,73,55),(10,27,44,74,56),(11,28,45,75,57),(12,29,46,76,58),(13,30,47,77,59),(14,31,48,78,60),(15,32,33,79,61),(16,17,34,80,62)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,27),(19,25),(20,32),(21,23),(22,30),(24,28),(29,31),(33,37),(34,44),(36,42),(38,40),(39,47),(41,45),(46,48),(49,61),(50,52),(51,59),(53,57),(54,64),(56,62),(58,60),(66,72),(67,79),(68,70),(69,77),(71,75),(74,80),(76,78)], [(1,9),(3,11),(5,13),(7,15),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47),(49,57),(51,59),(53,61),(55,63),(65,73),(67,75),(69,77),(71,79)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 68 | 0 | 214 | 227 |
| 22 | 0 | 105 | 174 |
| 22 | 11 | 83 | 162 |
| 219 | 219 | 180 | 90 |
| 1 | 1 | 16 | 215 |
| 0 | 240 | 24 | 1 |
| 0 | 0 | 22 | 11 |
| 0 | 0 | 219 | 219 |
| 240 | 0 | 100 | 50 |
| 0 | 240 | 4 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[68,22,22,219,0,0,11,219,214,105,83,180,227,174,162,90],[1,0,0,0,1,240,0,0,16,24,22,219,215,1,11,219],[240,0,0,0,0,240,0,0,100,4,1,0,50,2,0,1] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10T | 16A | 16B | 16C | 16D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 16 | 16 | 16 | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | C5×D4 | C5×D4 | C5×D8 | C5×D8 | C16⋊C22 | C5×C16⋊C22 |
| kernel | C5×C16⋊C22 | C5×M5(2) | C5×D16 | C5×SD32 | C10×D8 | C5×C4○D8 | C16⋊C22 | M5(2) | D16 | SD32 | C2×D8 | C4○D8 | C40 | C2×C20 | C20 | C2×C10 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_5\times C_{16}\rtimes C_2^2 % in TeX
G:=Group("C5xC16:C2^2"); // GroupNames label
G:=SmallGroup(320,1010);
// by ID
G=gap.SmallGroup(320,1010);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,3446,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^7,d*b*d=b^9,c*d=d*c>;
// generators/relations