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G = M7(2)  order 128 = 27

Modular maximal-cyclic group

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

Aliases: M7(2), C4.C32, C643C2, C32.2C4, C16.5C8, C8.3C16, C22.C32, C32.8C22, C8.29(C2×C8), C2.3(C2×C32), (C2×C4).5C16, (C2×C8).16C8, (C2×C32).8C2, C4.13(C2×C16), (C2×C16).18C4, C16.25(C2×C4), SmallGroup(128,160)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — M7(2)
C1C2C4C8C16C32C2×C32 — M7(2)
C1C2 — M7(2)
C1C32 — M7(2)
C1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4C4C8C8C8C8C16C16C32 — M7(2)

Generators and relations for M7(2)
 G = < a,b | a64=b2=1, bab=a33 >

2C2

Smallest permutation representation of M7(2)
On 64 points
Generators in S64
(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)
(2 34)(4 36)(6 38)(8 40)(10 42)(12 44)(14 46)(16 48)(18 50)(20 52)(22 54)(24 56)(26 58)(28 60)(30 62)(32 64)

G:=sub<Sym(64)| (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), (2,34)(4,36)(6,38)(8,40)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64)>;

G:=Group( (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), (2,34)(4,36)(6,38)(8,40)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64) );

G=PermutationGroup([(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)], [(2,34),(4,36),(6,38),(8,40),(10,42),(12,44),(14,46),(16,48),(18,50),(20,52),(22,54),(24,56),(26,58),(28,60),(30,62),(32,64)])

80 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F16A···16H16I16J16K16L32A···32P32Q···32X64A···64AF
order12244488888816···161616161632···3232···3264···64
size1121121111221···122221···12···22···2

80 irreducible representations

dim111111111112
type+++
imageC1C2C2C4C4C8C8C16C16C32C32M7(2)
kernelM7(2)C64C2×C32C32C2×C16C16C2×C8C8C2×C4C4C22C1
# reps121224488161616

Matrix representation of M7(2) in GL2(𝔽193) generated by

9132
46184
,
18
0192
G:=sub<GL(2,GF(193))| [9,46,132,184],[1,0,8,192] >;

M7(2) in GAP, Magma, Sage, TeX

M_7(2)
% in TeX

G:=Group("M7(2)");
// GroupNames label

G:=SmallGroup(128,160);
// by ID

G=gap.SmallGroup(128,160);
# by ID

G:=PCGroup([7,-2,2,-2,-2,-2,-2,-2,28,925,58,80,102,124]);
// Polycyclic

G:=Group<a,b|a^64=b^2=1,b*a*b=a^33>;
// generators/relations

Export

Subgroup lattice of M7(2) in TeX

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