Copied to
clipboard

## G = C10×C5⋊C8order 400 = 24·52

### Direct product of C10 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C10×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5×Dic5 — C5×C5⋊C8 — C10×C5⋊C8
 Lower central C5 — C10×C5⋊C8
 Upper central C1 — C2×C10

Generators and relations for C10×C5⋊C8
G = < a,b,c | a10=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C10×C5⋊C8
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 3 5 7 9)(2 4 6 8 10)(11 19 17 15 13)(12 20 18 16 14)(21 27 23 29 25)(22 28 24 30 26)(31 33 35 37 39)(32 34 36 38 40)(41 49 47 45 43)(42 50 48 46 44)(51 55 59 53 57)(52 56 60 54 58)(61 67 63 69 65)(62 68 64 70 66)(71 75 79 73 77)(72 76 80 74 78)
(1 26 45 72 37 65 13 58)(2 27 46 73 38 66 14 59)(3 28 47 74 39 67 15 60)(4 29 48 75 40 68 16 51)(5 30 49 76 31 69 17 52)(6 21 50 77 32 70 18 53)(7 22 41 78 33 61 19 54)(8 23 42 79 34 62 20 55)(9 24 43 80 35 63 11 56)(10 25 44 71 36 64 12 57)

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), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14)(21,27,23,29,25)(22,28,24,30,26)(31,33,35,37,39)(32,34,36,38,40)(41,49,47,45,43)(42,50,48,46,44)(51,55,59,53,57)(52,56,60,54,58)(61,67,63,69,65)(62,68,64,70,66)(71,75,79,73,77)(72,76,80,74,78), (1,26,45,72,37,65,13,58)(2,27,46,73,38,66,14,59)(3,28,47,74,39,67,15,60)(4,29,48,75,40,68,16,51)(5,30,49,76,31,69,17,52)(6,21,50,77,32,70,18,53)(7,22,41,78,33,61,19,54)(8,23,42,79,34,62,20,55)(9,24,43,80,35,63,11,56)(10,25,44,71,36,64,12,57)>;

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), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14)(21,27,23,29,25)(22,28,24,30,26)(31,33,35,37,39)(32,34,36,38,40)(41,49,47,45,43)(42,50,48,46,44)(51,55,59,53,57)(52,56,60,54,58)(61,67,63,69,65)(62,68,64,70,66)(71,75,79,73,77)(72,76,80,74,78), (1,26,45,72,37,65,13,58)(2,27,46,73,38,66,14,59)(3,28,47,74,39,67,15,60)(4,29,48,75,40,68,16,51)(5,30,49,76,31,69,17,52)(6,21,50,77,32,70,18,53)(7,22,41,78,33,61,19,54)(8,23,42,79,34,62,20,55)(9,24,43,80,35,63,11,56)(10,25,44,71,36,64,12,57) );

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)], [(1,3,5,7,9),(2,4,6,8,10),(11,19,17,15,13),(12,20,18,16,14),(21,27,23,29,25),(22,28,24,30,26),(31,33,35,37,39),(32,34,36,38,40),(41,49,47,45,43),(42,50,48,46,44),(51,55,59,53,57),(52,56,60,54,58),(61,67,63,69,65),(62,68,64,70,66),(71,75,79,73,77),(72,76,80,74,78)], [(1,26,45,72,37,65,13,58),(2,27,46,73,38,66,14,59),(3,28,47,74,39,67,15,60),(4,29,48,75,40,68,16,51),(5,30,49,76,31,69,17,52),(6,21,50,77,32,70,18,53),(7,22,41,78,33,61,19,54),(8,23,42,79,34,62,20,55),(9,24,43,80,35,63,11,56),(10,25,44,71,36,64,12,57)]])

100 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5I 8A ··· 8H 10A ··· 10L 10M ··· 10AA 20A ··· 20P 40A ··· 40AF order 1 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 5 5 5 5 1 1 1 1 4 ··· 4 5 ··· 5 1 ··· 1 4 ··· 4 5 ··· 5 5 ··· 5

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 F5 C5⋊C8 C2×F5 C5×F5 C5×C5⋊C8 C10×F5 kernel C10×C5⋊C8 C5×C5⋊C8 C10×Dic5 C5×Dic5 C102 C2×C5⋊C8 C5×C10 C5⋊C8 C2×Dic5 Dic5 C2×C10 C10 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 2 2 4 8 8 4 8 8 32 1 2 1 4 8 4

Matrix representation of C10×C5⋊C8 in GL5(𝔽41)

 37 0 0 0 0 0 31 0 0 0 0 0 31 0 0 0 0 0 31 0 0 0 0 0 31
,
 1 0 0 0 0 0 18 0 0 0 0 19 16 0 0 0 18 0 37 0 0 4 0 0 10
,
 3 0 0 0 0 0 16 0 26 0 0 9 0 1 40 0 17 40 25 0 0 38 0 13 0

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[1,0,0,0,0,0,18,19,18,4,0,0,16,0,0,0,0,0,37,0,0,0,0,0,10],[3,0,0,0,0,0,16,9,17,38,0,0,0,40,0,0,26,1,25,13,0,0,40,0,0] >;

C10×C5⋊C8 in GAP, Magma, Sage, TeX

C_{10}\times C_5\rtimes C_8
% in TeX

G:=Group("C10xC5:C8");
// GroupNames label

G:=SmallGroup(400,139);
// by ID

G=gap.SmallGroup(400,139);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,69,5765,599]);
// Polycyclic

G:=Group<a,b,c|a^10=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

׿
×
𝔽