metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.9Q8, C40.90D4, C8.9Dic10, M5(2).2D5, C5⋊2C8.1C8, C4.13(C8×D5), C5⋊4(C8.C8), C20.29(C2×C8), C10.18(C4⋊C8), C20.72(C4⋊C4), (C2×C8).266D10, C8.50(C5⋊D4), (C4×Dic5).7C4, C20.4C8.7C2, (C8×Dic5).22C2, (C5×M5(2)).4C2, (C2×C40).218C22, C22.5(C8⋊D5), (C2×C10).12M4(2), C2.5(C20.8Q8), C4.28(C10.D4), (C2×C5⋊2C8).9C4, (C2×C4).135(C4×D5), (C2×C20).223(C2×C4), SmallGroup(320,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.9Q8
G = < a,b,c | a40=1, b4=a30, c2=a35b2, bab-1=a21, cac-1=a9, cbc-1=a35b3 >
Smallest permutation representation of C40.9Q8
►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 54 16 49 31 44 6 79 21 74 36 69 11 64 26 59)(2 75 17 70 32 65 7 60 22 55 37 50 12 45 27 80)(3 56 18 51 33 46 8 41 23 76 38 71 13 66 28 61)(4 77 19 72 34 67 9 62 24 57 39 52 14 47 29 42)(5 58 20 53 35 48 10 43 25 78 40 73 15 68 30 63)
(1 6 11 16 21 26 31 36)(2 15 12 25 22 35 32 5)(3 24 13 34 23 4 33 14)(7 20 17 30 27 40 37 10)(8 29 18 39 28 9 38 19)(41 72 71 62 61 52 51 42)(43 50 73 80 63 70 53 60)(44 59 74 49 64 79 54 69)(45 68 75 58 65 48 55 78)(46 77 76 67 66 57 56 47)
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), (1,54,16,49,31,44,6,79,21,74,36,69,11,64,26,59)(2,75,17,70,32,65,7,60,22,55,37,50,12,45,27,80)(3,56,18,51,33,46,8,41,23,76,38,71,13,66,28,61)(4,77,19,72,34,67,9,62,24,57,39,52,14,47,29,42)(5,58,20,53,35,48,10,43,25,78,40,73,15,68,30,63), (1,6,11,16,21,26,31,36)(2,15,12,25,22,35,32,5)(3,24,13,34,23,4,33,14)(7,20,17,30,27,40,37,10)(8,29,18,39,28,9,38,19)(41,72,71,62,61,52,51,42)(43,50,73,80,63,70,53,60)(44,59,74,49,64,79,54,69)(45,68,75,58,65,48,55,78)(46,77,76,67,66,57,56,47)>;
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), (1,54,16,49,31,44,6,79,21,74,36,69,11,64,26,59)(2,75,17,70,32,65,7,60,22,55,37,50,12,45,27,80)(3,56,18,51,33,46,8,41,23,76,38,71,13,66,28,61)(4,77,19,72,34,67,9,62,24,57,39,52,14,47,29,42)(5,58,20,53,35,48,10,43,25,78,40,73,15,68,30,63), (1,6,11,16,21,26,31,36)(2,15,12,25,22,35,32,5)(3,24,13,34,23,4,33,14)(7,20,17,30,27,40,37,10)(8,29,18,39,28,9,38,19)(41,72,71,62,61,52,51,42)(43,50,73,80,63,70,53,60)(44,59,74,49,64,79,54,69)(45,68,75,58,65,48,55,78)(46,77,76,67,66,57,56,47) );
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)], [(1,54,16,49,31,44,6,79,21,74,36,69,11,64,26,59),(2,75,17,70,32,65,7,60,22,55,37,50,12,45,27,80),(3,56,18,51,33,46,8,41,23,76,38,71,13,66,28,61),(4,77,19,72,34,67,9,62,24,57,39,52,14,47,29,42),(5,58,20,53,35,48,10,43,25,78,40,73,15,68,30,63)], [(1,6,11,16,21,26,31,36),(2,15,12,25,22,35,32,5),(3,24,13,34,23,4,33,14),(7,20,17,30,27,40,37,10),(8,29,18,39,28,9,38,19),(41,72,71,62,61,52,51,42),(43,50,73,80,63,70,53,60),(44,59,74,49,64,79,54,69),(45,68,75,58,65,48,55,78),(46,77,76,67,66,57,56,47)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D5 | M4(2) | D10 | Dic10 | C5⋊D4 | C4×D5 | C8.C8 | C8×D5 | C8⋊D5 | C40.9Q8 |
| kernel | C40.9Q8 | C20.4C8 | C8×Dic5 | C5×M5(2) | C2×C5⋊2C8 | C4×Dic5 | C5⋊2C8 | C40 | C40 | M5(2) | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
Matrix representation of C40.9Q8 ►in GL4(𝔽241) generated by
| 211 | 36 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 52 |
| 0 | 0 | 190 | 190 |
| 211 | 201 | 0 | 0 |
| 191 | 30 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 30 | 170 | 0 | 0 |
| 0 | 233 | 0 | 0 |
| 0 | 0 | 190 | 189 |
| 0 | 0 | 50 | 51 |
G:=sub<GL(4,GF(241))| [211,0,0,0,36,30,0,0,0,0,0,190,0,0,52,190],[211,191,0,0,201,30,0,0,0,0,177,0,0,0,0,177],[30,0,0,0,170,233,0,0,0,0,190,50,0,0,189,51] >;
C40.9Q8 in GAP, Magma, Sage, TeX
C_{40}._9Q_8 % in TeX
G:=Group("C40.9Q8"); // GroupNames label
G:=SmallGroup(320,69);
// by ID
G=gap.SmallGroup(320,69);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,141,36,100,570,136,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=a^30,c^2=a^35*b^2,b*a*b^-1=a^21,c*a*c^-1=a^9,c*b*c^-1=a^35*b^3>;
// generators/relations
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