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G = C8.5M4(2)  order 128 = 27

5th non-split extension by C8 of M4(2) acting via M4(2)/C4=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C8.5M4(2), C23.18M4(2), M5(2).10C22, (C4×C8).10C4, C16⋊C4⋊4C2, C8⋊C4.8C4, C8.C8⋊11C2, C8.84(C4○D4), C42.28(C2×C4), (C2×C42).26C4, C23.C8.6C2, (C2×C8).386C23, (C4×C8).158C22, (C4×M4(2)).3C2, (C2×C4).32M4(2), C4.51(C2×M4(2)), (C2×M4(2)).15C4, C8⋊C4.150C22, C4.87(C42⋊C2), C22.25(C2×M4(2)), C2.10(C42.6C4), (C2×M4(2)).330C22, (C2×C8).15(C2×C4), (C22×C4).85(C2×C4), (C2×C4).562(C22×C4), SmallGroup(128,897)

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

Derived series
C1 — C2×C4 — C8.5M4(2)
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C8⋊C4 — C4×M4(2) — C8.5M4(2)
Lower central
C1 — C2 — C2×C4 — C8.5M4(2)
Upper central
C1 — C4 — C8⋊C4 — C8.5M4(2)
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.5M4(2)

Generators and relations for C8.5M4(2)
 G = < a,b,c | a8=1, b8=c2=a4, bab-1=a3, cac-1=a5, cbc-1=a6b5 >

Subgroups: 100 in 64 conjugacy classes, 38 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, M5(2), C2×C42, C2×M4(2), C16⋊C4, C23.C8, C8.C8, C4×M4(2), C8.5M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, C42⋊C2, C2×M4(2), C42.6C4, C8.5M4(2)

Permutation representations of C8.5M4(2)
►On 16 points - transitive group 16T220
Generators in S16
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(1 13 9 5)(2 6 10 14)(3 7 11 15)(4 16 12 8)
 
G:=sub<Sym(16)| (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8)>;
 
G:=Group( (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8) );
 
G=PermutationGroup([[(1,7,13,3,9,15,5,11),(2,4,6,8,10,12,14,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,13,9,5),(2,6,10,14),(3,7,11,15),(4,16,12,8)]])
 
G:=TransitiveGroup(16,220);
 

32 conjugacy classes

class 1 2A2B2C4A4B4C···4G4H4I4J8A8B8C8D8E···8J16A···16H
order1222444···444488888···816···16
size1124112···244422224···48···8

32 irreducible representations

dim11111111122224
type+++++
imageC1C2C2C2C2C4C4C4C4M4(2)C4○D4M4(2)M4(2)C8.5M4(2)
kernelC8.5M4(2)C16⋊C4C23.C8C8.C8C4×M4(2)C4×C8C8⋊C4C2×C42C2×M4(2)C8C8C2×C4C23C1
# reps12221222244224

Matrix representation of C8.5M4(2) ►in GL4(𝔽5) generated by

3410
3402
3014
0432
,
3431
0411
3423
4011
,
3310
0110
0340
0412
G:=sub<GL(4,GF(5))| [3,3,3,0,4,4,0,4,1,0,1,3,0,2,4,2],[3,0,3,4,4,4,4,0,3,1,2,1,1,1,3,1],[3,0,0,0,3,1,3,4,1,1,4,1,0,0,0,2] >;
 

C8.5M4(2) in GAP, Magma, Sage, TeX 

C_8._5M_4(2)
 
% in TeX
 
G:=Group("C8.5M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,897);
 
// by ID
 
G=gap.SmallGroup(128,897);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,758,58,2019,1411,718,102,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^6*b^5>;
 
// generators/relations
 

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