direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8.C8, C8.1C40, C40.13C8, C40.22Q8, C40.109D4, C42.8C20, M5(2).4C10, C8.6(C5×Q8), (C2×C8).9C20, C4.8(C2×C40), C8.29(C5×D4), C20.82(C2×C8), (C4×C40).28C2, (C4×C20).40C4, (C4×C8).10C10, (C2×C40).52C4, C10.24(C4⋊C8), C20.99(C4⋊C4), (C5×M5(2)).8C2, (C2×C40).443C22, (C2×C10).35M4(2), C22.5(C5×M4(2)), C2.5(C5×C4⋊C8), C4.19(C5×C4⋊C4), (C2×C4).68(C2×C20), (C2×C8).97(C2×C10), (C2×C20).502(C2×C4), SmallGroup(320,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.C8
G = < a,b,c | a5=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C5×C8.C8
►On 80 points
Generators in S80
(1 66 24 41 49)(2 67 25 42 50)(3 68 26 43 51)(4 69 27 44 52)(5 70 28 45 53)(6 71 29 46 54)(7 72 30 47 55)(8 73 31 48 56)(9 74 32 33 57)(10 75 17 34 58)(11 76 18 35 59)(12 77 19 36 60)(13 78 20 37 61)(14 79 21 38 62)(15 80 22 39 63)(16 65 23 40 64)
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)(17 19 21 23 25 27 29 31)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 36 38 40 42 44 46 48)(49 55 61 51 57 63 53 59)(50 52 54 56 58 60 62 64)(65 67 69 71 73 75 77 79)(66 72 78 68 74 80 70 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)
G:=sub<Sym(80)| (1,66,24,41,49)(2,67,25,42,50)(3,68,26,43,51)(4,69,27,44,52)(5,70,28,45,53)(6,71,29,46,54)(7,72,30,47,55)(8,73,31,48,56)(9,74,32,33,57)(10,75,17,34,58)(11,76,18,35,59)(12,77,19,36,60)(13,78,20,37,61)(14,79,21,38,62)(15,80,22,39,63)(16,65,23,40,64), (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16)(17,19,21,23,25,27,29,31)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,36,38,40,42,44,46,48)(49,55,61,51,57,63,53,59)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,72,78,68,74,80,70,76), (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)>;
G:=Group( (1,66,24,41,49)(2,67,25,42,50)(3,68,26,43,51)(4,69,27,44,52)(5,70,28,45,53)(6,71,29,46,54)(7,72,30,47,55)(8,73,31,48,56)(9,74,32,33,57)(10,75,17,34,58)(11,76,18,35,59)(12,77,19,36,60)(13,78,20,37,61)(14,79,21,38,62)(15,80,22,39,63)(16,65,23,40,64), (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16)(17,19,21,23,25,27,29,31)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,36,38,40,42,44,46,48)(49,55,61,51,57,63,53,59)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,72,78,68,74,80,70,76), (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) );
G=PermutationGroup([[(1,66,24,41,49),(2,67,25,42,50),(3,68,26,43,51),(4,69,27,44,52),(5,70,28,45,53),(6,71,29,46,54),(7,72,30,47,55),(8,73,31,48,56),(9,74,32,33,57),(10,75,17,34,58),(11,76,18,35,59),(12,77,19,36,60),(13,78,20,37,61),(14,79,21,38,62),(15,80,22,39,63),(16,65,23,40,64)], [(1,7,13,3,9,15,5,11),(2,4,6,8,10,12,14,16),(17,19,21,23,25,27,29,31),(18,24,30,20,26,32,22,28),(33,39,45,35,41,47,37,43),(34,36,38,40,42,44,46,48),(49,55,61,51,57,63,53,59),(50,52,54,56,58,60,62,64),(65,67,69,71,73,75,77,79),(66,72,78,68,74,80,70,76)], [(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)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | ··· | 4G | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20AB | 40A | ··· | 40P | 40Q | ··· | 40AN | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 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 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 | D4 | Q8 | M4(2) | C5×D4 | C5×Q8 | C8.C8 | C5×M4(2) | C5×C8.C8 |
| kernel | C5×C8.C8 | C4×C40 | C5×M5(2) | C4×C20 | C2×C40 | C8.C8 | C40 | C4×C8 | M5(2) | C42 | C2×C8 | C8 | C40 | C40 | C2×C10 | C8 | C8 | C5 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 8 | 32 | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C5×C8.C8 ►in GL2(𝔽41) generated by
| 16 | 0 |
| 0 | 16 |
| 14 | 0 |
| 0 | 38 |
| 0 | 26 |
| 31 | 0 |
G:=sub<GL(2,GF(41))| [16,0,0,16],[14,0,0,38],[0,31,26,0] >;
C5×C8.C8 in GAP, Magma, Sage, TeX
C_5\times C_8.C_8
% in TeX
G:=Group("C5xC8.C8"); // GroupNames label
G:=SmallGroup(320,169);
// by ID
G=gap.SmallGroup(320,169);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,148,5043,248,102,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^8=1,c^8=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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