metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C20.4C8, C40.9C4, C5⋊4M5(2), C8.22D10, C8.2Dic5, C40.22C22, C4.(C5⋊2C8), (C2×C8).7D5, C5⋊2C16⋊5C2, (C2×C10).5C8, (C2×C40).10C2, C20.60(C2×C4), (C2×C20).19C4, C10.18(C2×C8), C22.(C5⋊2C8), (C2×C4).5Dic5, C4.11(C2×Dic5), C2.4(C2×C5⋊2C8), SmallGroup(160,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.4C8
G = < a,b | a40=1, b4=a10, bab-1=a29 >
Smallest permutation representation of C20.4C8
►On 80 points
(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 58 26 63 11 68 36 73 21 78 6 43 31 48 16 53)(2 47 27 52 12 57 37 62 22 67 7 72 32 77 17 42)(3 76 28 41 13 46 38 51 23 56 8 61 33 66 18 71)(4 65 29 70 14 75 39 80 24 45 9 50 34 55 19 60)(5 54 30 59 15 64 40 69 25 74 10 79 35 44 20 49)
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), (1,58,26,63,11,68,36,73,21,78,6,43,31,48,16,53)(2,47,27,52,12,57,37,62,22,67,7,72,32,77,17,42)(3,76,28,41,13,46,38,51,23,56,8,61,33,66,18,71)(4,65,29,70,14,75,39,80,24,45,9,50,34,55,19,60)(5,54,30,59,15,64,40,69,25,74,10,79,35,44,20,49)>;
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), (1,58,26,63,11,68,36,73,21,78,6,43,31,48,16,53)(2,47,27,52,12,57,37,62,22,67,7,72,32,77,17,42)(3,76,28,41,13,46,38,51,23,56,8,61,33,66,18,71)(4,65,29,70,14,75,39,80,24,45,9,50,34,55,19,60)(5,54,30,59,15,64,40,69,25,74,10,79,35,44,20,49) );
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)], [(1,58,26,63,11,68,36,73,21,78,6,43,31,48,16,53),(2,47,27,52,12,57,37,62,22,67,7,72,32,77,17,42),(3,76,28,41,13,46,38,51,23,56,8,61,33,66,18,71),(4,65,29,70,14,75,39,80,24,45,9,50,34,55,19,60),(5,54,30,59,15,64,40,69,25,74,10,79,35,44,20,49)]])
C20.4C8 is a maximal subgroup of
C40.7C8 C20.45C42 C8.Dic10 D40⋊14C4 C40.6Q8 D40.6C4 C40.8D4 D20.3C8 C40.9Q8 C80⋊C4 C8.25D20 C40.D4 C40.92D4 D8.Dic5 Q16.Dic5 D8⋊2Dic5 D20.6C8 D5×M5(2) C40.70C23 D8.D10 Q16.D10 D8⋊D10 C40.31C23 C40.52D6 C60.7C8
C20.4C8 is a maximal quotient of
C40.10C8 C20⋊3C16 C40.91D4 C40.52D6 C60.7C8
52 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D5 | Dic5 | D10 | Dic5 | M5(2) | C5⋊2C8 | C5⋊2C8 | C20.4C8 |
| kernel | C20.4C8 | C5⋊2C16 | C2×C40 | C40 | C2×C20 | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 16 |
Matrix representation of C20.4C8 ►in GL2(𝔽41) generated by
| 11 | 0 |
| 0 | 29 |
| 0 | 3 |
| 1 | 0 |
G:=sub<GL(2,GF(41))| [11,0,0,29],[0,1,3,0] >;
C20.4C8 in GAP, Magma, Sage, TeX
C_{20}._4C_8 % in TeX
G:=Group("C20.4C8"); // GroupNames label
G:=SmallGroup(160,19);
// by ID
G=gap.SmallGroup(160,19);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,50,69,4613]);
// Polycyclic
G:=Group<a,b|a^40=1,b^4=a^10,b*a*b^-1=a^29>;
// generators/relations
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