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G = C8○2M4(2)  order 64 = 26

Central product of C8 and M4(2)

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

Aliases: C8○2M4(2), C4.5C42, M4(2)⋊5C4, C22.5C42, C42.60C22, C8○(C4⋊C4), (C2×C8)⋊9C4, (C4×C8)⋊14C2, C8○(C8⋊C4), C4⋊C4.10C4, C8○(C22⋊C4), C8.12(C2×C4), C8⋊C4⋊13C2, (C2×C8)○M4(2), C8○(C2×M4(2)), C2.1(C8○D4), C22⋊C4.6C4, C2.7(C2×C42), C8○(C42⋊C2), C4.34(C22×C4), (C22×C8).15C2, (C2×C8).99C22, C23.15(C2×C4), (C2×C4).143C23, C42⋊C2.14C2, (C2×M4(2)).17C2, C22.19(C22×C4), (C22×C4).106C22, (C2×C8)○(C8⋊C4), (C2×C4).34(C2×C4), (C2×C8)○(C42⋊C2), SmallGroup(64,86)

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

Derived series
C1 — C2 — C8○2M4(2)
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C8○2M4(2)
Lower central
C1 — C2 — C8○2M4(2)
Upper central
C1 — C2×C8 — C8○2M4(2)
Jennings
C1 — C2 — C2 — C2×C4 — C8○2M4(2)

Generators and relations for C8○2M4(2)
 G = < a,b,c | a8=c2=1, b4=a4, ab=ba, ac=ca, cbc=a4b >

Subgroups: 73 in 65 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C8○2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, C8○D4, C8○2M4(2)

Smallest permutation representation of C8○2M4(2)
►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 23 25 16 5 19 29 12)(2 24 26 9 6 20 30 13)(3 17 27 10 7 21 31 14)(4 18 28 11 8 22 32 15)

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

C8○2M4(2) is a maximal subgroup of
 M4(2).C8  M5(2)⋊7C4  C23.5C42  C8.(C4⋊C4)  M4(2).3Q8  M4(2).24D4  C4.10D4⋊3C4  C4.D4⋊3C4  M4(2).5Q8  M4(2).6Q8  M4(2).27D4  M4(2).30D4  M4(2).31D4  M4(2).32D4  M4(2).33D4  C4⋊C4.7C8  M4(2).1C8  C8.12M4(2)  C8.19M4(2)  M4(2)○2M4(2)  C42.262C23  C42.678C23  C42.264C23  C42.265C23  M4(2)⋊22D4  C42.275C23  C42.276C23  C42.277C23  C42.278C23  C42.279C23  C42.280C23  C42.281C23  C42.283C23  M4(2)○D8  C42.286C23  C42.287C23  M4(2)⋊9Q8  C42.291C23  C42.292C23  C42.293C23  C42.294C23  C42.366C23  C42.367C23  M4(2).20D4  M4(2)⋊3Q8  M4(2)⋊4Q8  C42.385C23  C42.387C23  C42.389C23  C42.390C23  Dic5.C42
 D2p.C42: Q8.C42  D4.3C42  C4×C8○D4  D4.5C42  D6.C42  D6.4C42  D10.5C42  D10.7C42 ...
 C4p.C42: C8.16C42  C8.14C42  C8.5C42  C16○2M5(2)  C8.23C42  C12.5C42  C12.12C42  C12.7C42 ...
 C8⋊pD4⋊C2: M4(2)⋊10D4  M4(2)⋊11D4  C42.386C23  C42.388C23  C42.391C23 ...
C8○2M4(2) is a maximal quotient of
 C8×M4(2)  C82⋊C2  C8⋊9M4(2)  C23.27C42  C82⋊15C2  C82⋊2C2  C23.29C42  (C4×C8)⋊12C4  C8×C22⋊C4  C23.36C42  C23.17C42  C8×C4⋊C4  C4⋊C8⋊13C4  C4⋊C8⋊14C4  Dic5.C42
 C4p.C42: C2.C43  C12.5C42  C12.12C42  C12.7C42  C20.35C42  C20.42C42  C20.37C42  D10.C42 ...
 C42.D2p: C42.379D4  C42.45Q8  D6.C42  D6.4C42  D10.5C42  D10.7C42  D14.C42  D14.4C42 ...

40 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8H8I···8T
order12222244444···48···88···8
size11112211112···21···12···2

40 irreducible representations

dim11111111112
type++++++
imageC1C2C2C2C2C2C4C4C4C4C8○D4
kernelC8○2M4(2)C4×C8C8⋊C4C42⋊C2C22×C8C2×M4(2)C22⋊C4C4⋊C4C2×C8M4(2)C2
# reps12211144888

Matrix representation of C8○2M4(2) ►in GL3(𝔽17) generated by

1300
0150
0015
,
100
008
080
,
1600
010
0016
G:=sub<GL(3,GF(17))| [13,0,0,0,15,0,0,0,15],[1,0,0,0,0,8,0,8,0],[16,0,0,0,1,0,0,0,16] >;
 

C8○2M4(2) in GAP, Magma, Sage, TeX 

C_8\circ_2M_4(2)
 
% in TeX
 
G:=Group("C8o2M4(2)");
 
// GroupNames label
 
G:=SmallGroup(64,86);
 
// by ID
 
G=gap.SmallGroup(64,86);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,332,117]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=c^2=1,b^4=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b>;
 
// generators/relations
 

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