direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C16⋊C4, C16⋊2C20, C80⋊12C4, C42.1C20, C20.65C42, M5(2).2C10, C20.56M4(2), (C2×C8).2C20, (C4×C20).21C4, (C2×C40).29C4, C4.11(C4×C20), C8.19(C2×C20), C8⋊C4.4C10, C40.128(C2×C4), C4.6(C5×M4(2)), C10.17(C8⋊C4), (C5×M5(2)).6C2, (C2×C40).308C22, (C2×C10).34M4(2), C22.4(C5×M4(2)), C2.3(C5×C8⋊C4), (C5×C8⋊C4).9C2, (C2×C8).45(C2×C10), (C2×C4).66(C2×C20), (C2×C20).500(C2×C4), SmallGroup(320,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C16⋊C4
G = < a,b,c | a5=b16=c4=1, ab=ba, ac=ca, cbc-1=b13 >
Smallest permutation representation of C5×C16⋊C4
►On 80 points
Generators in S80
(1 64 18 45 77)(2 49 19 46 78)(3 50 20 47 79)(4 51 21 48 80)(5 52 22 33 65)(6 53 23 34 66)(7 54 24 35 67)(8 55 25 36 68)(9 56 26 37 69)(10 57 27 38 70)(11 58 28 39 71)(12 59 29 40 72)(13 60 30 41 73)(14 61 31 42 74)(15 62 32 43 75)(16 63 17 44 76)
(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 6 10 14)(3 11)(4 16 12 8)(7 15)(17 29 25 21)(19 23 27 31)(20 28)(24 32)(34 38 42 46)(35 43)(36 48 44 40)(39 47)(49 53 57 61)(50 58)(51 63 59 55)(54 62)(66 70 74 78)(67 75)(68 80 76 72)(71 79)
G:=sub<Sym(80)| (1,64,18,45,77)(2,49,19,46,78)(3,50,20,47,79)(4,51,21,48,80)(5,52,22,33,65)(6,53,23,34,66)(7,54,24,35,67)(8,55,25,36,68)(9,56,26,37,69)(10,57,27,38,70)(11,58,28,39,71)(12,59,29,40,72)(13,60,30,41,73)(14,61,31,42,74)(15,62,32,43,75)(16,63,17,44,76), (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), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,29,25,21)(19,23,27,31)(20,28)(24,32)(34,38,42,46)(35,43)(36,48,44,40)(39,47)(49,53,57,61)(50,58)(51,63,59,55)(54,62)(66,70,74,78)(67,75)(68,80,76,72)(71,79)>;
G:=Group( (1,64,18,45,77)(2,49,19,46,78)(3,50,20,47,79)(4,51,21,48,80)(5,52,22,33,65)(6,53,23,34,66)(7,54,24,35,67)(8,55,25,36,68)(9,56,26,37,69)(10,57,27,38,70)(11,58,28,39,71)(12,59,29,40,72)(13,60,30,41,73)(14,61,31,42,74)(15,62,32,43,75)(16,63,17,44,76), (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), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,29,25,21)(19,23,27,31)(20,28)(24,32)(34,38,42,46)(35,43)(36,48,44,40)(39,47)(49,53,57,61)(50,58)(51,63,59,55)(54,62)(66,70,74,78)(67,75)(68,80,76,72)(71,79) );
G=PermutationGroup([[(1,64,18,45,77),(2,49,19,46,78),(3,50,20,47,79),(4,51,21,48,80),(5,52,22,33,65),(6,53,23,34,66),(7,54,24,35,67),(8,55,25,36,68),(9,56,26,37,69),(10,57,27,38,70),(11,58,28,39,71),(12,59,29,40,72),(13,60,30,41,73),(14,61,31,42,74),(15,62,32,43,75),(16,63,17,44,76)], [(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)], [(2,6,10,14),(3,11),(4,16,12,8),(7,15),(17,29,25,21),(19,23,27,31),(20,28),(24,32),(34,38,42,46),(35,43),(36,48,44,40),(39,47),(49,53,57,61),(50,58),(51,63,59,55),(54,62),(66,70,74,78),(67,75),(68,80,76,72),(71,79)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 16A | ··· | 16H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 40A | ··· | 40P | 40Q | ··· | 40X | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | C20 | M4(2) | M4(2) | C5×M4(2) | C5×M4(2) | C16⋊C4 | C5×C16⋊C4 |
| kernel | C5×C16⋊C4 | C5×C8⋊C4 | C5×M5(2) | C80 | C4×C20 | C2×C40 | C16⋊C4 | C8⋊C4 | M5(2) | C16 | C42 | C2×C8 | C20 | C2×C10 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 4 | 8 | 32 | 8 | 8 | 2 | 2 | 8 | 8 | 2 | 8 |
Matrix representation of C5×C16⋊C4 ►in GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 115 | 239 | 105 | 135 |
| 0 | 0 | 0 | 240 |
| 0 | 64 | 0 | 0 |
| 2 | 240 | 115 | 126 |
| 1 | 240 | 2 | 128 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 64 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[115,0,0,2,239,0,64,240,105,0,0,115,135,240,0,126],[1,0,0,0,240,240,0,0,2,0,177,0,128,0,0,64] >;
C5×C16⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_{16}\rtimes C_4 % in TeX
G:=Group("C5xC16:C4"); // GroupNames label
G:=SmallGroup(320,152);
// by ID
G=gap.SmallGroup(320,152);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,1149,288,2530,136,7004,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations
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