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## G = C2×M5(2)  order 64 = 26

### Direct product of C2 and M5(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×M5(2)
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×M5(2)
 Lower central C1 — C2 — C2×M5(2)
 Upper central C1 — C2×C8 — C2×M5(2)
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×M5(2)

Generators and relations for C2×M5(2)
G = < a,b,c | a2=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Smallest permutation representation of C2×M5(2)
On 32 points
Generators in S32
(1 28)(2 29)(3 30)(4 31)(5 32)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)
(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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)

G:=sub<Sym(32)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)>;

G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31) );

G=PermutationGroup([(1,28),(2,29),(3,30),(4,31),(5,32),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27)], [(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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31)])

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A ··· 8H 8I 8J 8K 8L 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 4 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 1 1 1 1 2 2 1 ··· 1 2 2 2 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 type + + + + image C1 C2 C2 C2 C4 C4 C8 C8 M5(2) kernel C2×M5(2) C2×C16 M5(2) C22×C8 C2×C8 C22×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 12 4 8

Matrix representation of C2×M5(2) in GL3(𝔽17) generated by

 16 0 0 0 16 0 0 0 16
,
 8 0 0 0 15 15 0 15 2
,
 16 0 0 0 1 0 0 15 16
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[8,0,0,0,15,15,0,15,2],[16,0,0,0,1,15,0,0,16] >;

C2×M5(2) in GAP, Magma, Sage, TeX

C_2\times M_5(2)
% in TeX

G:=Group("C2xM5(2)");
// GroupNames label

G:=SmallGroup(64,184);
// by ID

G=gap.SmallGroup(64,184);
# by ID

G:=PCGroup([6,-2,2,2,-2,-2,-2,48,409,69,88]);
// Polycyclic

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

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