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G = M5(2).1C4order 128 = 27

1st non-split extension by M5(2) of C4 acting via C4/C2=C2

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

Aliases: M5(2).1C4, C23.10Q16, C16.7(C2×C4), C4.91(C2×D8), C8.11(C4⋊C4), (C2×C8).21Q8, C8.25(C2×Q8), (C2×C4).146D8, (C2×C8).129D4, C8.4Q83C2, (C2×C4).25Q16, C4.7(C2.D8), C8.56(C22×C4), C22.2(C2×Q16), (C2×C8).579C23, (C2×C16).18C22, (C22×C4).342D4, (C2×M5(2)).2C2, C22.7(C2.D8), C8.C4.15C22, (C22×C8).241C22, C4.55(C2×C4⋊C4), (C2×C8).92(C2×C4), C2.16(C2×C2.D8), (C2×C4).59(C4⋊C4), (C2×C4).767(C2×D4), (C2×C8.C4).25C2, SmallGroup(128,893)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — M5(2).1C4
C1C2C4C2×C4C2×C8C22×C8C2×M5(2) — M5(2).1C4
C1C2C4C8 — M5(2).1C4
C1C4C22×C4C22×C8 — M5(2).1C4
C1C2C2C2C2C4C4C2×C8 — M5(2).1C4

Generators and relations for M5(2).1C4
 G = < a,b,c | a16=b2=1, c4=a8, bab=a9, cac-1=a7, cbc-1=a8b >

Subgroups: 108 in 70 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C22, C8 [×2], C8 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C16 [×4], C2×C8 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×6], C22×C4, C8.C4 [×4], C8.C4 [×2], C2×C16 [×2], M5(2) [×4], C22×C8, C2×M4(2) [×2], C8.4Q8 [×4], C2×C8.C4 [×2], C2×M5(2), M5(2).1C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, C2.D8 [×4], C2×C4⋊C4, C2×D8, C2×Q16, C2×C2.D8, M5(2).1C4

Smallest permutation representation of M5(2).1C4
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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
(1 32 13 20 9 24 5 28)(2 23 14 27 10 31 6 19)(3 30 15 18 11 22 7 26)(4 21 16 25 12 29 8 17)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111112222224
type+++++-++--
imageC1C2C2C2C4D4Q8D4D8Q16Q16M5(2).1C4
kernelM5(2).1C4C8.4Q8C2×C8.C4C2×M5(2)M5(2)C2×C8C2×C8C22×C4C2×C4C2×C4C23C1
# reps142181214224

Matrix representation of M5(2).1C4 in GL4(𝔽17) generated by

2161614
13151515
0008
0040
,
1110
01600
00160
0001
,
401010
0001
9131313
0400
G:=sub<GL(4,GF(17))| [2,13,0,0,16,15,0,0,16,15,0,4,14,15,8,0],[1,0,0,0,1,16,0,0,1,0,16,0,0,0,0,1],[4,0,9,0,0,0,13,4,10,0,13,0,10,1,13,0] >;

M5(2).1C4 in GAP, Magma, Sage, TeX

M_5(2)._1C_4
% in TeX

G:=Group("M5(2).1C4");
// GroupNames label

G:=SmallGroup(128,893);
// by ID

G=gap.SmallGroup(128,893);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,288,723,1123,360,172,4037,124]);
// 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,c*b*c^-1=a^8*b>;
// generators/relations

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