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## G = C10×C5⋊2C8order 400 = 24·52

### Direct product of C10 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C10×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×C5⋊2C8 — C10×C5⋊2C8
 Lower central C5 — C10×C5⋊2C8
 Upper central C1 — C2×C20

Generators and relations for C10×C52C8
G = < a,b,c | a10=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 100 in 60 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2, C4, C22, C5, C5, C8, C2×C4, C10, C10, C10, C2×C8, C20, C20, C2×C10, C2×C10, C52, C52C8, C40, C2×C20, C2×C20, C5×C10, C5×C10, C2×C52C8, C2×C40, C5×C20, C102, C5×C52C8, C10×C20, C10×C52C8
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D5, C10, C2×C8, Dic5, C20, D10, C2×C10, C52C8, C40, C2×Dic5, C2×C20, C5×D5, C2×C52C8, C2×C40, C5×Dic5, D5×C10, C5×C52C8, C10×Dic5, C10×C52C8

Smallest permutation representation of C10×C52C8
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 13 15 17 19)(12 14 16 18 20)(21 29 27 25 23)(22 30 28 26 24)(31 33 35 37 39)(32 34 36 38 40)(41 43 45 47 49)(42 44 46 48 50)(51 59 57 55 53)(52 60 58 56 54)(61 69 67 65 63)(62 70 68 66 64)(71 79 77 75 73)(72 80 78 76 74)
(1 21 45 77 37 70 13 53)(2 22 46 78 38 61 14 54)(3 23 47 79 39 62 15 55)(4 24 48 80 40 63 16 56)(5 25 49 71 31 64 17 57)(6 26 50 72 32 65 18 58)(7 27 41 73 33 66 19 59)(8 28 42 74 34 67 20 60)(9 29 43 75 35 68 11 51)(10 30 44 76 36 69 12 52)

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

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

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

160 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5N 8A ··· 8H 10A ··· 10L 10M ··· 10AP 20A ··· 20P 20Q ··· 20BD 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 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 1 1 1 1 2 ··· 2 5 ··· 5 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 D5 Dic5 D10 Dic5 C5⋊2C8 C5×D5 C5×Dic5 D5×C10 C5×Dic5 C5×C5⋊2C8 kernel C10×C5⋊2C8 C5×C5⋊2C8 C10×C20 C5×C20 C102 C2×C5⋊2C8 C5×C10 C5⋊2C8 C2×C20 C20 C2×C10 C10 C2×C20 C20 C20 C2×C10 C10 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 4 8 8 4 8 8 32 2 2 2 2 8 8 8 8 8 32

Matrix representation of C10×C52C8 in GL3(𝔽41) generated by

 23 0 0 0 16 0 0 0 16
,
 1 0 0 0 10 0 0 19 37
,
 1 0 0 0 39 36 0 19 2
G:=sub<GL(3,GF(41))| [23,0,0,0,16,0,0,0,16],[1,0,0,0,10,19,0,0,37],[1,0,0,0,39,19,0,36,2] >;

C10×C52C8 in GAP, Magma, Sage, TeX

C_{10}\times C_5\rtimes_2C_8
% in TeX

G:=Group("C10xC5:2C8");
// GroupNames label

G:=SmallGroup(400,81);
// by ID

G=gap.SmallGroup(400,81);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,69,11525]);
// 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^-1>;
// generators/relations

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