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## G = Q8×C10order 80 = 24·5

### Direct product of C10 and Q8

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

 Derived series C1 — C2 — Q8×C10
 Chief series C1 — C2 — C10 — C20 — C5×Q8 — Q8×C10
 Lower central C1 — C2 — Q8×C10
 Upper central C1 — C2×C10 — Q8×C10

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 30 33)(2 42 21 34)(3 43 22 35)(4 44 23 36)(5 45 24 37)(6 46 25 38)(7 47 26 39)(8 48 27 40)(9 49 28 31)(10 50 29 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 30 53)(2 62 21 54)(3 63 22 55)(4 64 23 56)(5 65 24 57)(6 66 25 58)(7 67 26 59)(8 68 27 60)(9 69 28 51)(10 70 29 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), (1,41,30,33)(2,42,21,34)(3,43,22,35)(4,44,23,36)(5,45,24,37)(6,46,25,38)(7,47,26,39)(8,48,27,40)(9,49,28,31)(10,50,29,32)(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), (1,61,30,53)(2,62,21,54)(3,63,22,55)(4,64,23,56)(5,65,24,57)(6,66,25,58)(7,67,26,59)(8,68,27,60)(9,69,28,51)(10,70,29,52)(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), (1,41,30,33)(2,42,21,34)(3,43,22,35)(4,44,23,36)(5,45,24,37)(6,46,25,38)(7,47,26,39)(8,48,27,40)(9,49,28,31)(10,50,29,32)(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), (1,61,30,53)(2,62,21,54)(3,63,22,55)(4,64,23,56)(5,65,24,57)(6,66,25,58)(7,67,26,59)(8,68,27,60)(9,69,28,51)(10,70,29,52)(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)], [(1,41,30,33),(2,42,21,34),(3,43,22,35),(4,44,23,36),(5,45,24,37),(6,46,25,38),(7,47,26,39),(8,48,27,40),(9,49,28,31),(10,50,29,32),(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)], [(1,61,30,53),(2,62,21,54),(3,63,22,55),(4,64,23,56),(5,65,24,57),(6,66,25,58),(7,67,26,59),(8,68,27,60),(9,69,28,51),(10,70,29,52),(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  D103Q8  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

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