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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

Aliases: C2×M5(2), C4M5(2), C8M5(2), C164C22, C23.3C8, C8.16C23, (C2×C16)⋊8C2, (C2×C4).6C8, C8.21(C2×C4), C4.10(C2×C8), (C2×C8).14C4, C2.6(C22×C8), C22.11(C2×C8), C4.35(C22×C4), (C22×C8).16C2, (C22×C4).17C4, (C2×C8).104C22, (C2×C4).76(C2×C4), SmallGroup(64,184)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×M5(2)
C1C2C4C8C2×C8C22×C8 — C2×M5(2)
C1C2 — C2×M5(2)
C1C2×C8 — C2×M5(2)
C1C2C2C2C2C4C4C8 — C2×M5(2)

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

2C2
2C2
2C22
2C22

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)])

C2×M5(2) is a maximal subgroup of
C42.2C8  C42.7C8  M5(2)⋊C4  M4(2).C8  M5(2)⋊7C4  C8.11C42  C23.9D8  C8.13C42  C8.C42  C8.2C42  M5(2).C4  C8.4C42  C23.C16  C162M5(2)  C8.23C42  C24.5C8  (C2×D4).5C8  M5(2).19C22  M5(2)⋊12C22  C23.39D8  C23.40D8  C23.41D8  C23.20SD16  C23.13D8  C23.21SD16  C4⋊M5(2)  C4⋊C4.7C8  M4(2).1C8  M5(2)⋊3C4  M5(2)⋊1C4  M5(2).1C4  C169D4  C166D4  C16⋊D4  C16.D4  C162D4  Q8○M5(2)  D16⋊C22  D5⋊M5(2)
C2×M5(2) is a maximal quotient of
C24.5C8  C4⋊M5(2)  C42.13C8  C42.6C8  C169D4  C166D4  C164Q8  D5⋊M5(2)

40 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F8A···8H8I8J8K8L16A···16P
order1222224444448···8888816···16
size1111221111221···122222···2

40 irreducible representations

dim111111112
type++++
imageC1C2C2C2C4C4C8C8M5(2)
kernelC2×M5(2)C2×C16M5(2)C22×C8C2×C8C22×C4C2×C4C23C2
# reps1241621248

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

1600
0160
0016
,
800
01515
0152
,
1600
010
01516
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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Subgroup lattice of C2×M5(2) in TeX

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