direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8.C4, C8.1C20, C40.10C4, C20.68D4, M4(2).2C10, (C2×C8).5C10, C4.8(C2×C20), C22.(C5×Q8), C4.19(C5×D4), (C2×C10).2Q8, (C2×C40).15C2, C20.66(C2×C4), C10.21(C4⋊C4), (C5×M4(2)).4C2, (C2×C20).119C22, C2.5(C5×C4⋊C4), (C2×C4).22(C2×C10), SmallGroup(160,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.C4
G = < a,b,c | a5=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C8.C4
►On 80 points
Generators in S80
(1 41 31 37 9)(2 42 32 38 10)(3 43 25 39 11)(4 44 26 40 12)(5 45 27 33 13)(6 46 28 34 14)(7 47 29 35 15)(8 48 30 36 16)(17 67 76 55 64)(18 68 77 56 57)(19 69 78 49 58)(20 70 79 50 59)(21 71 80 51 60)(22 72 73 52 61)(23 65 74 53 62)(24 66 75 54 63)
(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 61 3 59 5 57 7 63)(2 60 4 58 6 64 8 62)(9 52 11 50 13 56 15 54)(10 51 12 49 14 55 16 53)(17 48 23 42 21 44 19 46)(18 47 24 41 22 43 20 45)(25 70 27 68 29 66 31 72)(26 69 28 67 30 65 32 71)(33 77 35 75 37 73 39 79)(34 76 36 74 38 80 40 78)
G:=sub<Sym(80)| (1,41,31,37,9)(2,42,32,38,10)(3,43,25,39,11)(4,44,26,40,12)(5,45,27,33,13)(6,46,28,34,14)(7,47,29,35,15)(8,48,30,36,16)(17,67,76,55,64)(18,68,77,56,57)(19,69,78,49,58)(20,70,79,50,59)(21,71,80,51,60)(22,72,73,52,61)(23,65,74,53,62)(24,66,75,54,63), (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), (1,61,3,59,5,57,7,63)(2,60,4,58,6,64,8,62)(9,52,11,50,13,56,15,54)(10,51,12,49,14,55,16,53)(17,48,23,42,21,44,19,46)(18,47,24,41,22,43,20,45)(25,70,27,68,29,66,31,72)(26,69,28,67,30,65,32,71)(33,77,35,75,37,73,39,79)(34,76,36,74,38,80,40,78)>;
G:=Group( (1,41,31,37,9)(2,42,32,38,10)(3,43,25,39,11)(4,44,26,40,12)(5,45,27,33,13)(6,46,28,34,14)(7,47,29,35,15)(8,48,30,36,16)(17,67,76,55,64)(18,68,77,56,57)(19,69,78,49,58)(20,70,79,50,59)(21,71,80,51,60)(22,72,73,52,61)(23,65,74,53,62)(24,66,75,54,63), (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), (1,61,3,59,5,57,7,63)(2,60,4,58,6,64,8,62)(9,52,11,50,13,56,15,54)(10,51,12,49,14,55,16,53)(17,48,23,42,21,44,19,46)(18,47,24,41,22,43,20,45)(25,70,27,68,29,66,31,72)(26,69,28,67,30,65,32,71)(33,77,35,75,37,73,39,79)(34,76,36,74,38,80,40,78) );
G=PermutationGroup([[(1,41,31,37,9),(2,42,32,38,10),(3,43,25,39,11),(4,44,26,40,12),(5,45,27,33,13),(6,46,28,34,14),(7,47,29,35,15),(8,48,30,36,16),(17,67,76,55,64),(18,68,77,56,57),(19,69,78,49,58),(20,70,79,50,59),(21,71,80,51,60),(22,72,73,52,61),(23,65,74,53,62),(24,66,75,54,63)], [(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)], [(1,61,3,59,5,57,7,63),(2,60,4,58,6,64,8,62),(9,52,11,50,13,56,15,54),(10,51,12,49,14,55,16,53),(17,48,23,42,21,44,19,46),(18,47,24,41,22,43,20,45),(25,70,27,68,29,66,31,72),(26,69,28,67,30,65,32,71),(33,77,35,75,37,73,39,79),(34,76,36,74,38,80,40,78)]])
C5×C8.C4 is a maximal subgroup of
C40.7Q8 C40.6Q8 D40.6C4 C40.8D4 D40.5C4 M4(2).25D10 D40⋊16C4 D40⋊13C4 C8.20D20 C8.21D20 C8.24D20
70 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P | 40Q | ··· | 40AF |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | Q8 | C8.C4 | C5×D4 | C5×Q8 | C5×C8.C4 |
| kernel | C5×C8.C4 | C2×C40 | C5×M4(2) | C40 | C8.C4 | C2×C8 | M4(2) | C8 | C20 | C2×C10 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 1 | 1 | 4 | 4 | 4 | 16 |
Matrix representation of C5×C8.C4 ►in GL2(𝔽41) generated by
| 37 | 0 |
| 0 | 37 |
| 14 | 0 |
| 0 | 3 |
| 0 | 1 |
| 32 | 0 |
G:=sub<GL(2,GF(41))| [37,0,0,37],[14,0,0,3],[0,32,1,0] >;
C5×C8.C4 in GAP, Magma, Sage, TeX
C_5\times C_8.C_4
% in TeX
G:=Group("C5xC8.C4"); // GroupNames label
G:=SmallGroup(160,58);
// by ID
G=gap.SmallGroup(160,58);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,127,2403,117,88]);
// Polycyclic
G:=Group<a,b,c|a^5=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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