metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.6Q8, C8.2Dic10, C20.36SD16, C5⋊2C16⋊2C4, C5⋊4(C8.Q8), C8.29(C4×D5), C40.74(C2×C4), (C2×C20).94D4, (C2×C8).42D10, C20.36(C4⋊C4), C40⋊6C4.12C2, (C2×C10).3SD16, C10.8(C4.Q8), C8.C4.2D5, C20.4C8.4C2, C4.11(D4.D5), C22.2(Q8⋊D5), (C2×C40).153C22, C4.5(C10.D4), C2.4(C20.Q8), (C5×C8.C4).3C2, (C2×C4).17(C5⋊D4), SmallGroup(320,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.6Q8
G = < a,b,c | a40=1, b4=a20, c2=a5b2, bab-1=a31, cac-1=a29, cbc-1=a15b3 >
Smallest permutation representation of C40.6Q8
►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 65 11 55 21 45 31 75)(2 56 12 46 22 76 32 66)(3 47 13 77 23 67 33 57)(4 78 14 68 24 58 34 48)(5 69 15 59 25 49 35 79)(6 60 16 50 26 80 36 70)(7 51 17 41 27 71 37 61)(8 42 18 72 28 62 38 52)(9 73 19 63 29 53 39 43)(10 64 20 54 30 44 40 74)
(1 75 16 70 31 65 6 60 21 55 36 50 11 45 26 80)(2 64 17 59 32 54 7 49 22 44 37 79 12 74 27 69)(3 53 18 48 33 43 8 78 23 73 38 68 13 63 28 58)(4 42 19 77 34 72 9 67 24 62 39 57 14 52 29 47)(5 71 20 66 35 61 10 56 25 51 40 46 15 41 30 76)
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), (1,65,11,55,21,45,31,75)(2,56,12,46,22,76,32,66)(3,47,13,77,23,67,33,57)(4,78,14,68,24,58,34,48)(5,69,15,59,25,49,35,79)(6,60,16,50,26,80,36,70)(7,51,17,41,27,71,37,61)(8,42,18,72,28,62,38,52)(9,73,19,63,29,53,39,43)(10,64,20,54,30,44,40,74), (1,75,16,70,31,65,6,60,21,55,36,50,11,45,26,80)(2,64,17,59,32,54,7,49,22,44,37,79,12,74,27,69)(3,53,18,48,33,43,8,78,23,73,38,68,13,63,28,58)(4,42,19,77,34,72,9,67,24,62,39,57,14,52,29,47)(5,71,20,66,35,61,10,56,25,51,40,46,15,41,30,76)>;
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), (1,65,11,55,21,45,31,75)(2,56,12,46,22,76,32,66)(3,47,13,77,23,67,33,57)(4,78,14,68,24,58,34,48)(5,69,15,59,25,49,35,79)(6,60,16,50,26,80,36,70)(7,51,17,41,27,71,37,61)(8,42,18,72,28,62,38,52)(9,73,19,63,29,53,39,43)(10,64,20,54,30,44,40,74), (1,75,16,70,31,65,6,60,21,55,36,50,11,45,26,80)(2,64,17,59,32,54,7,49,22,44,37,79,12,74,27,69)(3,53,18,48,33,43,8,78,23,73,38,68,13,63,28,58)(4,42,19,77,34,72,9,67,24,62,39,57,14,52,29,47)(5,71,20,66,35,61,10,56,25,51,40,46,15,41,30,76) );
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)], [(1,65,11,55,21,45,31,75),(2,56,12,46,22,76,32,66),(3,47,13,77,23,67,33,57),(4,78,14,68,24,58,34,48),(5,69,15,59,25,49,35,79),(6,60,16,50,26,80,36,70),(7,51,17,41,27,71,37,61),(8,42,18,72,28,62,38,52),(9,73,19,63,29,53,39,43),(10,64,20,54,30,44,40,74)], [(1,75,16,70,31,65,6,60,21,55,36,50,11,45,26,80),(2,64,17,59,32,54,7,49,22,44,37,79,12,74,27,69),(3,53,18,48,33,43,8,78,23,73,38,68,13,63,28,58),(4,42,19,77,34,72,9,67,24,62,39,57,14,52,29,47),(5,71,20,66,35,61,10,56,25,51,40,46,15,41,30,76)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | SD16 | SD16 | D10 | Dic10 | C4×D5 | C5⋊D4 | C8.Q8 | D4.D5 | Q8⋊D5 | C40.6Q8 |
| kernel | C40.6Q8 | C20.4C8 | C40⋊6C4 | C5×C8.C4 | C5⋊2C16 | C40 | C2×C20 | C8.C4 | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C40.6Q8 ►in GL4(𝔽241) generated by
| 37 | 68 | 0 | 0 |
| 173 | 116 | 0 | 0 |
| 61 | 22 | 75 | 116 |
| 219 | 122 | 125 | 68 |
| 56 | 77 | 239 | 0 |
| 164 | 149 | 0 | 239 |
| 70 | 223 | 185 | 164 |
| 18 | 42 | 77 | 92 |
| 104 | 108 | 91 | 144 |
| 2 | 137 | 232 | 150 |
| 218 | 225 | 153 | 167 |
| 216 | 23 | 164 | 88 |
G:=sub<GL(4,GF(241))| [37,173,61,219,68,116,22,122,0,0,75,125,0,0,116,68],[56,164,70,18,77,149,223,42,239,0,185,77,0,239,164,92],[104,2,218,216,108,137,225,23,91,232,153,164,144,150,167,88] >;
C40.6Q8 in GAP, Magma, Sage, TeX
C_{40}._6Q_8 % in TeX
G:=Group("C40.6Q8"); // GroupNames label
G:=SmallGroup(320,52);
// by ID
G=gap.SmallGroup(320,52);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,365,36,758,184,346,80,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=a^20,c^2=a^5*b^2,b*a*b^-1=a^31,c*a*c^-1=a^29,c*b*c^-1=a^15*b^3>;
// generators/relations
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