metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40.6C4, C4.18D20, C20.34D4, C8.1Dic5, C22.2Dic10, (C2×C8).5D5, (C2×C40).7C2, (C2×C10).3Q8, C5⋊3(C8.C4), C20.56(C2×C4), (C2×C4).70D10, C10.14(C4⋊C4), C4.8(C2×Dic5), C2.5(C4⋊Dic5), C4.Dic5.1C2, (C2×C20).97C22, SmallGroup(160,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.6C4
G = < a,b | a40=1, b4=a20, bab-1=a19 >
Smallest permutation representation of C40.6C4
►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 45 31 55 21 65 11 75)(2 64 32 74 22 44 12 54)(3 43 33 53 23 63 13 73)(4 62 34 72 24 42 14 52)(5 41 35 51 25 61 15 71)(6 60 36 70 26 80 16 50)(7 79 37 49 27 59 17 69)(8 58 38 68 28 78 18 48)(9 77 39 47 29 57 19 67)(10 56 40 66 30 76 20 46)
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), (1,45,31,55,21,65,11,75)(2,64,32,74,22,44,12,54)(3,43,33,53,23,63,13,73)(4,62,34,72,24,42,14,52)(5,41,35,51,25,61,15,71)(6,60,36,70,26,80,16,50)(7,79,37,49,27,59,17,69)(8,58,38,68,28,78,18,48)(9,77,39,47,29,57,19,67)(10,56,40,66,30,76,20,46)>;
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), (1,45,31,55,21,65,11,75)(2,64,32,74,22,44,12,54)(3,43,33,53,23,63,13,73)(4,62,34,72,24,42,14,52)(5,41,35,51,25,61,15,71)(6,60,36,70,26,80,16,50)(7,79,37,49,27,59,17,69)(8,58,38,68,28,78,18,48)(9,77,39,47,29,57,19,67)(10,56,40,66,30,76,20,46) );
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)], [(1,45,31,55,21,65,11,75),(2,64,32,74,22,44,12,54),(3,43,33,53,23,63,13,73),(4,62,34,72,24,42,14,52),(5,41,35,51,25,61,15,71),(6,60,36,70,26,80,16,50),(7,79,37,49,27,59,17,69),(8,58,38,68,28,78,18,48),(9,77,39,47,29,57,19,67),(10,56,40,66,30,76,20,46)]])
C40.6C4 is a maximal subgroup of
C8.Dic10 C40.7Q8 C80.6C4 D40.3C4 C40.Q8 D40.4C4 C20.4D8 D8.Dic5 Q16.Dic5 C20.58D8 D40⋊17C4 D40⋊10C4 D5×C8.C4 M4(2).25D10 M4(2).Dic5 D4.3D20 D4.4D20 D4.5D20 C40.23D4 C40.44D4 C40.29D4 D8⋊5Dic5 D8⋊4Dic5 C12.59D20 C4.18D60
C40.6C4 is a maximal quotient of
C40⋊6C8 C40⋊5C8 C20.40C42 C12.59D20 C4.18D60
46 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D5 | Dic5 | D10 | C8.C4 | D20 | Dic10 | C40.6C4 |
| kernel | C40.6C4 | C4.Dic5 | C2×C40 | C40 | C20 | C2×C10 | C2×C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 16 |
Matrix representation of C40.6C4 ►in GL2(𝔽41) generated by
| 30 | 0 |
| 0 | 15 |
| 0 | 32 |
| 1 | 0 |
G:=sub<GL(2,GF(41))| [30,0,0,15],[0,1,32,0] >;
C40.6C4 in GAP, Magma, Sage, TeX
C_{40}._6C_4 % in TeX
G:=Group("C40.6C4"); // GroupNames label
G:=SmallGroup(160,26);
// by ID
G=gap.SmallGroup(160,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,86,579,69,4613]);
// Polycyclic
G:=Group<a,b|a^40=1,b^4=a^20,b*a*b^-1=a^19>;
// generators/relations
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