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G = Q8○M4(2)  order 64 = 26

Central product of Q8 and M4(2)

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

Aliases: Q8M4(2), D4M4(2), C4.17C24, C8.15C23, M4(2)M4(2), M4(2)⋊12C22, C8○D48C2, (C2×C8)⋊9C22, C4○D4.2C4, D4.9(C2×C4), (C2×Q8).8C4, (C2×D4).10C4, Q8.10(C2×C4), M4(2)(C4○D4), C2.12(C23×C4), C23.13(C2×C4), C4.23(C22×C4), (C2×M4(2))⋊16C2, (C2×C4).134C23, C4○D4.15C22, C22.5(C22×C4), (C22×C4).78C22, (C2×C4).32(C2×C4), (C2×C4○D4).11C2, SmallGroup(64,249)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — Q8○M4(2)
C1C2C4C2×C4C22×C4C2×C4○D4 — Q8○M4(2)
C1C2 — Q8○M4(2)
C1C4 — Q8○M4(2)
C1C2C2C4 — Q8○M4(2)

Generators and relations for Q8○M4(2)
 G = < a,b,c,d | a4=d2=1, b2=c4=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c >

Subgroups: 145 in 129 conjugacy classes, 119 normal (9 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, Q8○M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, Q8○M4(2)

Permutation representations of Q8○M4(2)
On 16 points - transitive group 16T70
Generators in S16
(1 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 13)(11 15)

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

G:=Group( (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15) );

G=PermutationGroup([(1,14,5,10),(2,15,6,11),(3,16,7,12),(4,9,8,13)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,13),(11,15)])

G:=TransitiveGroup(16,70);

Q8○M4(2) is a maximal subgroup of
M4(2).40D4  M4(2).41D4  (C2×D4).Q8  M4(2).44D4  M4(2)⋊19D4  (C2×C8)⋊D4  M4(2).46D4  M4(2).47D4  C4.(C4×D4)  (C2×C8)⋊4D4  M4(2)⋊21D4  M4(2).50D4  M4(2).24C23  M4(2).25C23  2- 1+45C4  M4(2).29C23  M4(2).51D4  M4(2)○D8  M4(2).37D4  M4(2).38D4  M4(2).A4  Dic5.C24  Dic5.20C24  Dic5.22C24
 C4p.C24: C4.22C25  D8⋊C23  C4.C25  M4(2)⋊26D6  M4(2)⋊28D6  C12.76C24  C40.47C23  C20.72C24 ...
Q8○M4(2) is a maximal quotient of
M4(2)○2M4(2)  D4.5C42  C24.73(C2×C4)  C42.257C23  C42.259C23  C42.261C23  C42.262C23  C42.678C23  C42.265C23  C42.266C23  D4×M4(2)  C42.287C23  M4(2)⋊9Q8  Q8×M4(2)  C42.292C23  C42.293C23  C42.294C23  D46M4(2)  Q86M4(2)  C42.691C23  C233M4(2)  C42.693C23  C42.297C23  C42.299C23  C42.300C23  C42.301C23  C42.695C23  C42.302C23  C42.696C23  C42.305C23  C42.697C23  C42.698C23  C42.307C23  C42.308C23  C42.310C23  Dic5.C24  Dic5.20C24  Dic5.22C24
 M4(2)⋊D2p: M4(2)⋊22D4  M4(2)⋊23D4  M4(2)⋊26D6  M4(2)⋊28D6  C40.47C23  C20.72C24  C28.70C24  C56.49C23 ...
 C4○D4.D2p: D4○(C22⋊C8)  C42.674C23  C12.76C24  C20.76C24  C28.76C24 ...

34 conjugacy classes

class 1 2A2B···2H4A4B4C···4I8A···8P
order122···2444···48···8
size112···2112···22···2

34 irreducible representations

dim11111114
type++++
imageC1C2C2C2C4C4C4Q8○M4(2)
kernelQ8○M4(2)C2×M4(2)C8○D4C2×C4○D4C2×D4C2×Q8C4○D4C1
# reps16816282

Matrix representation of Q8○M4(2) in GL4(𝔽5) generated by

2000
0200
0030
0003
,
0010
0004
4000
0100
,
0300
1000
0002
0040
,
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0100
0040
0001
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3],[0,0,4,0,0,0,0,1,1,0,0,0,0,4,0,0],[0,1,0,0,3,0,0,0,0,0,0,4,0,0,2,0],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1] >;

Q8○M4(2) in GAP, Magma, Sage, TeX

Q_8\circ M_4(2)
% in TeX

G:=Group("Q8oM4(2)");
// GroupNames label

G:=SmallGroup(64,249);
// by ID

G=gap.SmallGroup(64,249);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,96,332,963,88]);
// Polycyclic

G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^4=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c>;
// generators/relations

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