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G = M5(2)  order 32 = 25

Modular maximal-cyclic group

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: M5(2), C4.C8, C163C2, C8.2C4, C22.C8, C8.8C22, (C2×C8).8C2, (C2×C4).5C4, C2.3(C2×C8), C4.12(C2×C4), SmallGroup(32,17)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — M5(2)
C1C2C4C8C2×C8 — M5(2)
C1C2 — M5(2)
C1C8 — M5(2)
C1C2C2C2C2C4C4C8 — M5(2)

Generators and relations for M5(2)
 G = < a,b | a16=b2=1, bab=a9 >

2C2

Character table of M5(2)

 class 12A2B4A4B4C8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 11211211112222222222
ρ111111111111111111111    trivial
ρ211-111-11111-1-1-11-11-111-1    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-111-11111-1-11-11-11-1-11    linear of order 2
ρ511-111-1-1-1-1-111-iii-ii-ii-i    linear of order 4
ρ6111111-1-1-1-1-1-1ii-i-i-i-iii    linear of order 4
ρ711-111-1-1-1-1-111i-i-ii-ii-ii    linear of order 4
ρ8111111-1-1-1-1-1-1-i-iiiii-i-i    linear of order 4
ρ9111-1-1-1-ii-ii-iiζ85ζ85ζ87ζ87ζ83ζ83ζ8ζ8    linear of order 8
ρ1011-1-1-11i-ii-i-iiζ87ζ83ζ85ζ8ζ8ζ85ζ87ζ83    linear of order 8
ρ1111-1-1-11-ii-iii-iζ8ζ85ζ83ζ87ζ87ζ83ζ8ζ85    linear of order 8
ρ1211-1-1-11i-ii-i-iiζ83ζ87ζ8ζ85ζ85ζ8ζ83ζ87    linear of order 8
ρ13111-1-1-1i-ii-ii-iζ83ζ83ζ8ζ8ζ85ζ85ζ87ζ87    linear of order 8
ρ14111-1-1-1-ii-ii-iiζ8ζ8ζ83ζ83ζ87ζ87ζ85ζ85    linear of order 8
ρ1511-1-1-11-ii-iii-iζ85ζ8ζ87ζ83ζ83ζ87ζ85ζ8    linear of order 8
ρ16111-1-1-1i-ii-ii-iζ87ζ87ζ85ζ85ζ8ζ8ζ83ζ83    linear of order 8
ρ172-202i-2i088385870000000000    complex faithful
ρ182-20-2i2i083887850000000000    complex faithful
ρ192-202i-2i085878830000000000    complex faithful
ρ202-20-2i2i087858380000000000    complex faithful

Permutation representations of M5(2)
On 16 points - transitive group 16T22
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 10)(4 12)(6 14)(8 16)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,10)(4,12)(6,14)(8,16)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,10)(4,12)(6,14)(8,16) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,10),(4,12),(6,14),(8,16)])

G:=TransitiveGroup(16,22);

Matrix representation of M5(2) in GL2(𝔽17) generated by

915
148
,
10
916
G:=sub<GL(2,GF(17))| [9,14,15,8],[1,9,0,16] >;

M5(2) in GAP, Magma, Sage, TeX

M_{5(2})
% in TeX

G:=Group("M5(2)");
// GroupNames label

G:=SmallGroup(32,17);
// by ID

G=gap.SmallGroup(32,17);
# by ID

G:=PCGroup([5,-2,2,-2,-2,-2,20,181,42,58]);
// Polycyclic

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

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