metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40.1C8, C8.1(C5⋊C8), (C2×C8).12F5, C10.8(C4⋊C8), C5⋊2(C8.C8), (C2×C40).12C4, C20.42(C2×C8), C5⋊2C8.39D4, C4.27(C4⋊F5), C20.27(C4⋊C4), C5⋊2C8.13Q8, C2.5(C20⋊C8), C20.C8.3C2, (C4×Dic5).39C4, (C8×Dic5).20C2, (C2×C10).3M4(2), C22.2(C4.F5), C4.8(C2×C5⋊C8), (C2×C4).121(C2×F5), (C2×C20).138(C2×C4), (C2×C5⋊2C8).337C22, SmallGroup(320,227)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.1C8
G = < a,b | a40=1, b8=a20, bab-1=a23 >
Smallest permutation representation of C40.1C8
►On 80 points
Generators in S80
(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 47 6 42 11 77 16 72 21 67 26 62 31 57 36 52)(2 54 15 65 12 44 25 55 22 74 35 45 32 64 5 75)(3 61 24 48 13 51 34 78 23 41 4 68 33 71 14 58)(7 49 20 60 17 79 30 50 27 69 40 80 37 59 10 70)(8 56 29 43 18 46 39 73 28 76 9 63 38 66 19 53)
G:=sub<Sym(80)| (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), (1,47,6,42,11,77,16,72,21,67,26,62,31,57,36,52)(2,54,15,65,12,44,25,55,22,74,35,45,32,64,5,75)(3,61,24,48,13,51,34,78,23,41,4,68,33,71,14,58)(7,49,20,60,17,79,30,50,27,69,40,80,37,59,10,70)(8,56,29,43,18,46,39,73,28,76,9,63,38,66,19,53)>;
G:=Group( (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), (1,47,6,42,11,77,16,72,21,67,26,62,31,57,36,52)(2,54,15,65,12,44,25,55,22,74,35,45,32,64,5,75)(3,61,24,48,13,51,34,78,23,41,4,68,33,71,14,58)(7,49,20,60,17,79,30,50,27,69,40,80,37,59,10,70)(8,56,29,43,18,46,39,73,28,76,9,63,38,66,19,53) );
G=PermutationGroup([[(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)], [(1,47,6,42,11,77,16,72,21,67,26,62,31,57,36,52),(2,54,15,65,12,44,25,55,22,74,35,45,32,64,5,75),(3,61,24,48,13,51,34,78,23,41,4,68,33,71,14,58),(7,49,20,60,17,79,30,50,27,69,40,80,37,59,10,70),(8,56,29,43,18,46,39,73,28,76,9,63,38,66,19,53)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 10 | 10 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 4 | 4 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) | C8.C8 | F5 | C5⋊C8 | C2×F5 | C4⋊F5 | C4.F5 | C40.1C8 |
| kernel | C40.1C8 | C8×Dic5 | C20.C8 | C4×Dic5 | C2×C40 | C40 | C5⋊2C8 | C5⋊2C8 | C2×C10 | C5 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C40.1C8 ►in GL4(𝔽241) generated by
| 167 | 74 | 0 | 0 |
| 167 | 8 | 0 | 0 |
| 0 | 0 | 0 | 211 |
| 0 | 0 | 30 | 84 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 167 | 0 | 0 |
| 0 | 233 | 0 | 0 |
G:=sub<GL(4,GF(241))| [167,167,0,0,74,8,0,0,0,0,0,30,0,0,211,84],[0,0,8,0,0,0,167,233,1,0,0,0,0,1,0,0] >;
C40.1C8 in GAP, Magma, Sage, TeX
C_{40}._1C_8 % in TeX
G:=Group("C40.1C8"); // GroupNames label
G:=SmallGroup(320,227);
// by ID
G=gap.SmallGroup(320,227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,1123,136,102,6278,3156]);
// Polycyclic
G:=Group<a,b|a^40=1,b^8=a^20,b*a*b^-1=a^23>;
// generators/relations
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