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G = M5(2).C4  order 128 = 27

2nd non-split extension by M5(2) of C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C8.3C42, C23.6Q16, M5(2).2C4, (C2×C8).7Q8, C8.14(C4⋊C4), (C2×C4).124D8, (C2×C8).354D4, C4.4(C4.Q8), C8.C4.5C4, (C2×C4).23SD16, C8.43(C22⋊C4), (C22×C4).194D4, C4.51(D4⋊C4), C22.4(C2.D8), (C2×M5(2)).16C2, C4.9(C2.C42), (C22×C8).207C22, C22.4(Q8⋊C4), C2.18(C22.4Q16), (C2×C8).54(C2×C4), (C2×C4).30(C4⋊C4), (C2×C8.C4).6C2, (C2×C4).232(C22⋊C4), SmallGroup(128,120)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

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

Generators and relations for M5(2).C4
 G = < a,b,c | a16=b2=1, c4=a8, bab=a9, cac-1=a7b, cbc-1=a8b >

Subgroups: 104 in 62 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2, C4, C22, C22, C8, C8, C2×C4, C23, C16, C2×C8, C2×C8, M4(2), C22×C4, C8.C4, C8.C4, C2×C16, M5(2), M5(2), C22×C8, C2×M4(2), C2×C8.C4, C2×M5(2), M5(2).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C22.4Q16, M5(2).C4

Smallest permutation representation of M5(2).C4
►On 32 points
Generators in S32
(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)

(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G···8N16A···16H
order12222444448888888···816···16
size11222112222222448···84···4

32 irreducible representations

dim111112222224
type++++-++-
imageC1C2C2C4C4D4Q8D4D8SD16Q16M5(2).C4
kernelM5(2).C4C2×C8.C4C2×M5(2)C8.C4M5(2)C2×C8C2×C8C22×C4C2×C4C2×C4C23C1
# reps121842112424

Matrix representation of M5(2).C4 ►in GL4(𝔽17) generated by

01600
15000
0008
0010
,
16000
0100
0010
00016
,
0010
0001
13000
01300
G:=sub<GL(4,GF(17))| [0,15,0,0,16,0,0,0,0,0,0,1,0,0,8,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[0,0,13,0,0,0,0,13,1,0,0,0,0,1,0,0] >;
 

M5(2).C4 in GAP, Magma, Sage, TeX 

M_5(2).C_4
 
% in TeX
 
G:=Group("M5(2).C4");
 
// GroupNames label
 
G:=SmallGroup(128,120);
 
// by ID
 
G=gap.SmallGroup(128,120);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,248,1684,102,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^2=1,c^4=a^8,b*a*b=a^9,c*a*c^-1=a^7*b,c*b*c^-1=a^8*b>;
 
// generators/relations
 

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