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## G = M5(2).C4order 128 = 27

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

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

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 [×3], C4 [×4], C22 [×3], C22, C8 [×4], C8 [×4], C2×C4 [×6], C23, C16 [×2], C2×C8 [×6], C2×C8 [×2], M4(2) [×6], C22×C4, C8.C4 [×4], C8.C4 [×2], C2×C16, M5(2) [×2], M5(2), C22×C8, C2×M4(2) [×2], C2×C8.C4 [×2], C2×M5(2), M5(2).C4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], 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 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G ··· 8N 16A ··· 16H order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 8 8 8 ··· 8 16 ··· 16 size 1 1 2 2 2 1 1 2 2 2 2 2 2 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + - + + - image C1 C2 C2 C4 C4 D4 Q8 D4 D8 SD16 Q16 M5(2).C4 kernel M5(2).C4 C2×C8.C4 C2×M5(2) C8.C4 M5(2) C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C1 # reps 1 2 1 8 4 2 1 1 2 4 2 4

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

 0 16 0 0 15 0 0 0 0 0 0 8 0 0 1 0
,
 16 0 0 0 0 1 0 0 0 0 1 0 0 0 0 16
,
 0 0 1 0 0 0 0 1 13 0 0 0 0 13 0 0
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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