metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊1C4, C40.1Q8, C16⋊1Dic5, C20.6SD16, C8.3Dic10, M5(2).1D5, C5⋊5(C8.Q8), (C2×C4).8D20, (C2×C20).98D4, (C2×C8).45D10, C20.57(C4⋊C4), C40⋊6C4.1C2, C40.113(C2×C4), (C2×C10).7SD16, C8.18(C2×Dic5), C2.3(C40⋊6C4), C40.6C4.5C2, C4.11(C40⋊C2), (C2×C40).49C22, C4.11(C4⋊Dic5), C10.11(C4.Q8), (C5×M5(2)).1C2, C22.5(C40⋊C2), SmallGroup(320,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.Q8
G = < a,b,c | a40=1, b4=a10, c2=a15b2, bab-1=a21, cac-1=a19, cbc-1=a20b3 >
Smallest permutation representation of C40.Q8
►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 49 26 54 11 59 36 64 21 69 6 74 31 79 16 44)(2 70 27 75 12 80 37 45 22 50 7 55 32 60 17 65)(3 51 28 56 13 61 38 66 23 71 8 76 33 41 18 46)(4 72 29 77 14 42 39 47 24 52 9 57 34 62 19 67)(5 53 30 58 15 63 40 68 25 73 10 78 35 43 20 48)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 62 61 42)(43 60 63 80)(44 79 64 59)(45 58 65 78)(46 77 66 57)(47 56 67 76)(48 75 68 55)(49 54 69 74)(50 73 70 53)(51 52 71 72)
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), (1,49,26,54,11,59,36,64,21,69,6,74,31,79,16,44)(2,70,27,75,12,80,37,45,22,50,7,55,32,60,17,65)(3,51,28,56,13,61,38,66,23,71,8,76,33,41,18,46)(4,72,29,77,14,42,39,47,24,52,9,57,34,62,19,67)(5,53,30,58,15,63,40,68,25,73,10,78,35,43,20,48), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,62,61,42)(43,60,63,80)(44,79,64,59)(45,58,65,78)(46,77,66,57)(47,56,67,76)(48,75,68,55)(49,54,69,74)(50,73,70,53)(51,52,71,72)>;
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), (1,49,26,54,11,59,36,64,21,69,6,74,31,79,16,44)(2,70,27,75,12,80,37,45,22,50,7,55,32,60,17,65)(3,51,28,56,13,61,38,66,23,71,8,76,33,41,18,46)(4,72,29,77,14,42,39,47,24,52,9,57,34,62,19,67)(5,53,30,58,15,63,40,68,25,73,10,78,35,43,20,48), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,62,61,42)(43,60,63,80)(44,79,64,59)(45,58,65,78)(46,77,66,57)(47,56,67,76)(48,75,68,55)(49,54,69,74)(50,73,70,53)(51,52,71,72) );
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)], [(1,49,26,54,11,59,36,64,21,69,6,74,31,79,16,44),(2,70,27,75,12,80,37,45,22,50,7,55,32,60,17,65),(3,51,28,56,13,61,38,66,23,71,8,76,33,41,18,46),(4,72,29,77,14,42,39,47,24,52,9,57,34,62,19,67),(5,53,30,58,15,63,40,68,25,73,10,78,35,43,20,48)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,62,61,42),(43,60,63,80),(44,79,64,59),(45,58,65,78),(46,77,66,57),(47,56,67,76),(48,75,68,55),(49,54,69,74),(50,73,70,53),(51,52,71,72)]])
56 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 | 40J | 40K | 40L | 80A | ··· | 80P |
| 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 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | SD16 | SD16 | Dic5 | D10 | Dic10 | D20 | C40⋊C2 | C40⋊C2 | C8.Q8 | C40.Q8 |
| kernel | C40.Q8 | C40⋊6C4 | C40.6C4 | C5×M5(2) | C80 | C40 | C2×C20 | M5(2) | C20 | C2×C10 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of C40.Q8 ►in GL4(𝔽241) generated by
| 200 | 225 | 19 | 61 |
| 125 | 184 | 184 | 4 |
| 0 | 0 | 166 | 125 |
| 0 | 0 | 116 | 173 |
| 117 | 44 | 82 | 0 |
| 78 | 161 | 8 | 8 |
| 204 | 171 | 202 | 78 |
| 107 | 118 | 163 | 2 |
| 190 | 50 | 46 | 121 |
| 189 | 51 | 165 | 119 |
| 0 | 0 | 173 | 204 |
| 0 | 0 | 125 | 68 |
G:=sub<GL(4,GF(241))| [200,125,0,0,225,184,0,0,19,184,166,116,61,4,125,173],[117,78,204,107,44,161,171,118,82,8,202,163,0,8,78,2],[190,189,0,0,50,51,0,0,46,165,173,125,121,119,204,68] >;
C40.Q8 in GAP, Magma, Sage, TeX
C_{40}.Q_8 % in TeX
G:=Group("C40.Q8"); // GroupNames label
G:=SmallGroup(320,71);
// by ID
G=gap.SmallGroup(320,71);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,387,675,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=a^10,c^2=a^15*b^2,b*a*b^-1=a^21,c*a*c^-1=a^19,c*b*c^-1=a^20*b^3>;
// generators/relations
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