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## G = C10×M4(2)  order 160 = 25·5

### Direct product of C10 and M4(2)

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C10×M4(2)
 Chief series C1 — C2 — C4 — C20 — C40 — C5×M4(2) — C10×M4(2)
 Lower central C1 — C2 — C10×M4(2)
 Upper central C1 — C2×C20 — C10×M4(2)

Generators and relations for C10×M4(2)
G = < a,b,c | a10=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C2×C40, C5×M4(2), C22×C20, C10×M4(2)
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, M4(2), C22×C4, C20, C2×C10, C2×M4(2), C2×C20, C22×C10, C5×M4(2), C22×C20, C10×M4(2)

Smallest permutation representation of C10×M4(2)
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 20 50 38 70 28 52 79)(2 11 41 39 61 29 53 80)(3 12 42 40 62 30 54 71)(4 13 43 31 63 21 55 72)(5 14 44 32 64 22 56 73)(6 15 45 33 65 23 57 74)(7 16 46 34 66 24 58 75)(8 17 47 35 67 25 59 76)(9 18 48 36 68 26 60 77)(10 19 49 37 69 27 51 78)
(11 29)(12 30)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 71)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20X 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 M4(2) C5×M4(2) kernel C10×M4(2) C2×C40 C5×M4(2) C22×C20 C2×C20 C22×C10 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C10 C2 # reps 1 2 4 1 6 2 4 8 16 4 24 8 4 16

Matrix representation of C10×M4(2) in GL3(𝔽41) generated by

 40 0 0 0 25 0 0 0 25
,
 32 0 0 0 9 39 0 4 32
,
 40 0 0 0 1 0 0 9 40
G:=sub<GL(3,GF(41))| [40,0,0,0,25,0,0,0,25],[32,0,0,0,9,4,0,39,32],[40,0,0,0,1,9,0,0,40] >;

C10×M4(2) in GAP, Magma, Sage, TeX

C_{10}\times M_4(2)
% in TeX

G:=Group("C10xM4(2)");
// GroupNames label

G:=SmallGroup(160,191);
// by ID

G=gap.SmallGroup(160,191);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,985,88]);
// Polycyclic

G:=Group<a,b,c|a^10=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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