direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C10, C20.20C22, C10.12C23, (C2×C20).9C2, (C2×C4).3C10, C4.4(C2×C10), C2.2(C22×C10), C22.4(C2×C10), (C2×C10).15C22, SmallGroup(80,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C10
G = < a,b,c | a10=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C10
►Regular action 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 41 24 33)(2 42 25 34)(3 43 26 35)(4 44 27 36)(5 45 28 37)(6 46 29 38)(7 47 30 39)(8 48 21 40)(9 49 22 31)(10 50 23 32)(11 63 80 55)(12 64 71 56)(13 65 72 57)(14 66 73 58)(15 67 74 59)(16 68 75 60)(17 69 76 51)(18 70 77 52)(19 61 78 53)(20 62 79 54)
(1 61 24 53)(2 62 25 54)(3 63 26 55)(4 64 27 56)(5 65 28 57)(6 66 29 58)(7 67 30 59)(8 68 21 60)(9 69 22 51)(10 70 23 52)(11 35 80 43)(12 36 71 44)(13 37 72 45)(14 38 73 46)(15 39 74 47)(16 40 75 48)(17 31 76 49)(18 32 77 50)(19 33 78 41)(20 34 79 42)
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), (1,41,24,33)(2,42,25,34)(3,43,26,35)(4,44,27,36)(5,45,28,37)(6,46,29,38)(7,47,30,39)(8,48,21,40)(9,49,22,31)(10,50,23,32)(11,63,80,55)(12,64,71,56)(13,65,72,57)(14,66,73,58)(15,67,74,59)(16,68,75,60)(17,69,76,51)(18,70,77,52)(19,61,78,53)(20,62,79,54), (1,61,24,53)(2,62,25,54)(3,63,26,55)(4,64,27,56)(5,65,28,57)(6,66,29,58)(7,67,30,59)(8,68,21,60)(9,69,22,51)(10,70,23,52)(11,35,80,43)(12,36,71,44)(13,37,72,45)(14,38,73,46)(15,39,74,47)(16,40,75,48)(17,31,76,49)(18,32,77,50)(19,33,78,41)(20,34,79,42)>;
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), (1,41,24,33)(2,42,25,34)(3,43,26,35)(4,44,27,36)(5,45,28,37)(6,46,29,38)(7,47,30,39)(8,48,21,40)(9,49,22,31)(10,50,23,32)(11,63,80,55)(12,64,71,56)(13,65,72,57)(14,66,73,58)(15,67,74,59)(16,68,75,60)(17,69,76,51)(18,70,77,52)(19,61,78,53)(20,62,79,54), (1,61,24,53)(2,62,25,54)(3,63,26,55)(4,64,27,56)(5,65,28,57)(6,66,29,58)(7,67,30,59)(8,68,21,60)(9,69,22,51)(10,70,23,52)(11,35,80,43)(12,36,71,44)(13,37,72,45)(14,38,73,46)(15,39,74,47)(16,40,75,48)(17,31,76,49)(18,32,77,50)(19,33,78,41)(20,34,79,42) );
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)], [(1,41,24,33),(2,42,25,34),(3,43,26,35),(4,44,27,36),(5,45,28,37),(6,46,29,38),(7,47,30,39),(8,48,21,40),(9,49,22,31),(10,50,23,32),(11,63,80,55),(12,64,71,56),(13,65,72,57),(14,66,73,58),(15,67,74,59),(16,68,75,60),(17,69,76,51),(18,70,77,52),(19,61,78,53),(20,62,79,54)], [(1,61,24,53),(2,62,25,54),(3,63,26,55),(4,64,27,56),(5,65,28,57),(6,66,29,58),(7,67,30,59),(8,68,21,60),(9,69,22,51),(10,70,23,52),(11,35,80,43),(12,36,71,44),(13,37,72,45),(14,38,73,46),(15,39,74,47),(16,40,75,48),(17,31,76,49),(18,32,77,50),(19,33,78,41),(20,34,79,42)]])
Q8×C10 is a maximal subgroup of
Q8⋊Dic5 C20.10D4 C20.C23 Dic5⋊Q8 D10⋊3Q8 C20.23D4 Q8.10D10
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | Q8 | C5×Q8 |
| kernel | Q8×C10 | C2×C20 | C5×Q8 | C2×Q8 | C2×C4 | Q8 | C10 | C2 |
| # reps | 1 | 3 | 4 | 4 | 12 | 16 | 2 | 8 |
Matrix representation of Q8×C10 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 23 |
| 40 | 0 | 0 |
| 0 | 40 | 39 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 30 | 30 |
| 0 | 26 | 11 |
G:=sub<GL(3,GF(41))| [40,0,0,0,23,0,0,0,23],[40,0,0,0,40,1,0,39,1],[1,0,0,0,30,26,0,30,11] >;
Q8×C10 in GAP, Magma, Sage, TeX
Q_8\times C_{10} % in TeX
G:=Group("Q8xC10"); // GroupNames label
G:=SmallGroup(80,47);
// by ID
G=gap.SmallGroup(80,47);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-2,200,421,206]);
// Polycyclic
G:=Group<a,b,c|a^10=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export