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G = C2xM5(2)  order 64 = 26

Direct product of C2 and M5(2)

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

Aliases: C2xM5(2), C4oM5(2), C8oM5(2), C16:4C22, C23.3C8, C8.16C23, (C2xC16):8C2, (C2xC4).6C8, C8.21(C2xC4), C4.10(C2xC8), (C2xC8).14C4, C2.6(C22xC8), C22.11(C2xC8), C4.35(C22xC4), (C22xC8).16C2, (C22xC4).17C4, (C2xC8).104C22, (C2xC4).76(C2xC4), SmallGroup(64,184)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2xM5(2)
C1C2C4C8C2xC8C22xC8 — C2xM5(2)
C1C2 — C2xM5(2)
C1C2xC8 — C2xM5(2)
C1C2C2C2C2C4C4C8 — C2xM5(2)

Generators and relations for C2xM5(2)
 G = < a,b,c | a2=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Subgroups: 49 in 45 conjugacy classes, 41 normal (15 characteristic)
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C2xC8, C22xC4, M5(2), C22xC8, C2xM5(2)
2C2
2C2
2C22
2C22

Smallest permutation representation of C2xM5(2)
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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)(18 26)(20 28)(22 30)(24 32)

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

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

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

C2xM5(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  C16o2M5(2)  C8.23C42  C24.5C8  (C2xD4).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  C16:9D4  C16:6D4  C16:D4  C16.D4  C16:2D4  Q8oM5(2)  D16:C22  D5:M5(2)
C2xM5(2) is a maximal quotient of
C24.5C8  C4:M5(2)  C42.13C8  C42.6C8  C16:9D4  C16:6D4  C16:4Q8  D5:M5(2)

40 conjugacy classes

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

40 irreducible representations

dim111111112
type++++
imageC1C2C2C2C4C4C8C8M5(2)
kernelC2xM5(2)C2xC16M5(2)C22xC8C2xC8C22xC4C2xC4C23C2
# reps1241621248

Matrix representation of C2xM5(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] >;

C2xM5(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

Export

Subgroup lattice of C2xM5(2) in TeX

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