metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40.7C8, C40.65D4, C8.29D20, C40.16Q8, C8.14Dic10, C42.9Dic5, (C4×C8).11D5, C8.1(C5⋊2C8), C5⋊3(C8.C8), (C2×C40).42C4, C20.71(C2×C8), (C4×C20).31C4, (C4×C40).13C2, C10.13(C4⋊C8), C20.70(C4⋊C4), (C2×C8).9Dic5, (C2×C8).321D10, C2.5(C20⋊3C8), C20.4C8.5C2, C4.19(C4⋊Dic5), (C2×C40).397C22, (C2×C10).37M4(2), C22.2(C4.Dic5), C4.8(C2×C5⋊2C8), (C2×C20).474(C2×C4), (C2×C4).69(C2×Dic5), SmallGroup(320,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.7C8
G = < a,b | a40=1, b8=a20, bab-1=a-1 >
Smallest permutation representation of C40.7C8
►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 48 6 43 11 78 16 73 21 68 26 63 31 58 36 53)(2 47 7 42 12 77 17 72 22 67 27 62 32 57 37 52)(3 46 8 41 13 76 18 71 23 66 28 61 33 56 38 51)(4 45 9 80 14 75 19 70 24 65 29 60 34 55 39 50)(5 44 10 79 15 74 20 69 25 64 30 59 35 54 40 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), (1,48,6,43,11,78,16,73,21,68,26,63,31,58,36,53)(2,47,7,42,12,77,17,72,22,67,27,62,32,57,37,52)(3,46,8,41,13,76,18,71,23,66,28,61,33,56,38,51)(4,45,9,80,14,75,19,70,24,65,29,60,34,55,39,50)(5,44,10,79,15,74,20,69,25,64,30,59,35,54,40,49)>;
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), (1,48,6,43,11,78,16,73,21,68,26,63,31,58,36,53)(2,47,7,42,12,77,17,72,22,67,27,62,32,57,37,52)(3,46,8,41,13,76,18,71,23,66,28,61,33,56,38,51)(4,45,9,80,14,75,19,70,24,65,29,60,34,55,39,50)(5,44,10,79,15,74,20,69,25,64,30,59,35,54,40,49) );
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)], [(1,48,6,43,11,78,16,73,21,68,26,63,31,58,36,53),(2,47,7,42,12,77,17,72,22,67,27,62,32,57,37,52),(3,46,8,41,13,76,18,71,23,66,28,61,33,56,38,51),(4,45,9,80,14,75,19,70,24,65,29,60,34,55,39,50),(5,44,10,79,15,74,20,69,25,64,30,59,35,54,40,49)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | ··· | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 4 | 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 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D5 | M4(2) | Dic5 | Dic5 | D10 | C5⋊2C8 | Dic10 | D20 | C8.C8 | C4.Dic5 | C40.7C8 |
| kernel | C40.7C8 | C20.4C8 | C4×C40 | C4×C20 | C2×C40 | C40 | C40 | C40 | C4×C8 | C2×C10 | C42 | C2×C8 | C2×C8 | C8 | C8 | C8 | C5 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C40.7C8 ►in GL2(𝔽41) generated by
| 15 | 0 |
| 0 | 11 |
| 0 | 27 |
| 1 | 0 |
G:=sub<GL(2,GF(41))| [15,0,0,11],[0,1,27,0] >;
C40.7C8 in GAP, Magma, Sage, TeX
C_{40}._7C_8 % in TeX
G:=Group("C40.7C8"); // GroupNames label
G:=SmallGroup(320,21);
// by ID
G=gap.SmallGroup(320,21);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,1123,136,102,12550]);
// Polycyclic
G:=Group<a,b|a^40=1,b^8=a^20,b*a*b^-1=a^-1>;
// generators/relations
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