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## G = C5×M5(2)  order 160 = 25·5

### Direct product of C5 and M5(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×M5(2)
 Chief series C1 — C2 — C4 — C8 — C40 — C80 — C5×M5(2)
 Lower central C1 — C2 — C5×M5(2)
 Upper central C1 — C40 — C5×M5(2)

Generators and relations for C5×M5(2)
G = < a,b,c | a5=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Smallest permutation representation of C5×M5(2)
On 80 points
Generators in S80
(1 19 69 41 53)(2 20 70 42 54)(3 21 71 43 55)(4 22 72 44 56)(5 23 73 45 57)(6 24 74 46 58)(7 25 75 47 59)(8 26 76 48 60)(9 27 77 33 61)(10 28 78 34 62)(11 29 79 35 63)(12 30 80 36 64)(13 31 65 37 49)(14 32 66 38 50)(15 17 67 39 51)(16 18 68 40 52)
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(66 74)(68 76)(70 78)(72 80)

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

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

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

C5×M5(2) is a maximal subgroup of
C40.9Q8  C80⋊C4  C40.Q8  C8.25D20  D20.4C8  D40.4C4  C20.4D8  D408C4  D20.5C8  D80⋊C2  C16.D10

100 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 16A ··· 16H 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P 40Q ··· 40X 80A ··· 80AF order 1 2 2 4 4 4 5 5 5 5 8 8 8 8 8 8 10 10 10 10 10 10 10 10 16 ··· 16 20 ··· 20 20 20 20 20 40 ··· 40 40 ··· 40 80 ··· 80 size 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C5 C8 C8 C10 C10 C20 C20 C40 C40 M5(2) C5×M5(2) kernel C5×M5(2) C80 C2×C40 C40 C2×C20 M5(2) C20 C2×C10 C16 C2×C8 C8 C2×C4 C4 C22 C5 C1 # reps 1 2 1 2 2 4 4 4 8 4 8 8 16 16 4 16

Matrix representation of C5×M5(2) in GL2(𝔽41) generated by

 16 0 0 16
,
 0 27 1 0
,
 40 0 0 1
G:=sub<GL(2,GF(41))| [16,0,0,16],[0,1,27,0],[40,0,0,1] >;

C5×M5(2) in GAP, Magma, Sage, TeX

C_5\times M_5(2)
% in TeX

G:=Group("C5xM5(2)");
// GroupNames label

G:=SmallGroup(160,60);
// by ID

G=gap.SmallGroup(160,60);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,120,985,69,88]);
// Polycyclic

G:=Group<a,b,c|a^5=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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