direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C16⋊C22, D16⋊2C10, C80⋊6C22, C20.65D8, C40.53D4, SD32⋊1C10, M5(2)⋊1C10, C40.76C23, C16⋊(C2×C10), C8.3(C5×D4), C4○D8⋊2C10, (C5×D16)⋊6C2, D8⋊2(C2×C10), C4.14(C5×D8), (C2×D8)⋊10C10, (C10×D8)⋊24C2, 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
Generators and relations for C5×C16⋊C22
G = < a,b,c,d | a5=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >
Subgroups: 226 in 90 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C16, C2×C8, D8, D8, D8, SD16, Q16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, M5(2), D16, SD32, C2×D8, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C16⋊C22, C80, C2×C40, C5×D8, C5×D8, C5×D8, C5×SD16, C5×Q16, D4×C10, C5×C4○D4, C5×M5(2), C5×D16, C5×SD32, C10×D8, C5×C4○D8, C5×C16⋊C22
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C2×D8, C5×D4, C22×C10, C16⋊C22, C5×D8, D4×C10, C10×D8, C5×C16⋊C22
►On 80 points
(1 56 70 31 46)(2 57 71 32 47)(3 58 72 17 48)(4 59 73 18 33)(5 60 74 19 34)(6 61 75 20 35)(7 62 76 21 36)(8 63 77 22 37)(9 64 78 23 38)(10 49 79 24 39)(11 50 80 25 40)(12 51 65 26 41)(13 52 66 27 42)(14 53 67 28 43)(15 54 68 29 44)(16 55 69 30 45)
(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 29)(18 20)(19 27)(21 25)(22 32)(24 30)(26 28)(33 35)(34 42)(36 40)(37 47)(39 45)(41 43)(44 48)(49 55)(50 62)(51 53)(52 60)(54 58)(57 63)(59 61)(65 67)(66 74)(68 72)(69 79)(71 77)(73 75)(76 80)
(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(66 74)(68 76)(70 78)(72 80)
G:=sub<Sym(80)| (1,56,70,31,46)(2,57,71,32,47)(3,58,72,17,48)(4,59,73,18,33)(5,60,74,19,34)(6,61,75,20,35)(7,62,76,21,36)(8,63,77,22,37)(9,64,78,23,38)(10,49,79,24,39)(11,50,80,25,40)(12,51,65,26,41)(13,52,66,27,42)(14,53,67,28,43)(15,54,68,29,44)(16,55,69,30,45), (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)(17,29)(18,20)(19,27)(21,25)(22,32)(24,30)(26,28)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,67)(66,74)(68,72)(69,79)(71,77)(73,75)(76,80), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(66,74)(68,76)(70,78)(72,80)>;
G:=Group( (1,56,70,31,46)(2,57,71,32,47)(3,58,72,17,48)(4,59,73,18,33)(5,60,74,19,34)(6,61,75,20,35)(7,62,76,21,36)(8,63,77,22,37)(9,64,78,23,38)(10,49,79,24,39)(11,50,80,25,40)(12,51,65,26,41)(13,52,66,27,42)(14,53,67,28,43)(15,54,68,29,44)(16,55,69,30,45), (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)(17,29)(18,20)(19,27)(21,25)(22,32)(24,30)(26,28)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,67)(66,74)(68,72)(69,79)(71,77)(73,75)(76,80), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(66,74)(68,76)(70,78)(72,80) );
G=PermutationGroup([[(1,56,70,31,46),(2,57,71,32,47),(3,58,72,17,48),(4,59,73,18,33),(5,60,74,19,34),(6,61,75,20,35),(7,62,76,21,36),(8,63,77,22,37),(9,64,78,23,38),(10,49,79,24,39),(11,50,80,25,40),(12,51,65,26,41),(13,52,66,27,42),(14,53,67,28,43),(15,54,68,29,44),(16,55,69,30,45)], [(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),(17,29),(18,20),(19,27),(21,25),(22,32),(24,30),(26,28),(33,35),(34,42),(36,40),(37,47),(39,45),(41,43),(44,48),(49,55),(50,62),(51,53),(52,60),(54,58),(57,63),(59,61),(65,67),(66,74),(68,72),(69,79),(71,77),(73,75),(76,80)], [(1,9),(3,11),(5,13),(7,15),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48),(50,58),(52,60),(54,62),(56,64),(66,74),(68,76),(70,78),(72,80)]])
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 |
Matrix representation of C5×C16⋊C22 ►in 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] >;
C5×C16⋊C22 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