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G = Q8○M5(2)  order 128 = 27

Central product of Q8 and M5(2)

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

Aliases: Q8M5(2), D4M5(2), C8.25C24, M5(2)M5(2), M4(2)M5(2), C16.15C23, M5(2)⋊17C22, D4○C168C2, C8○D4.5C4, C4○D4.4C8, D4.9(C2×C8), C4○D4M5(2), (C2×D4).11C8, (C2×Q8).10C8, Q8.10(C2×C8), (C2×C16)⋊13C22, M5(2)(C8○D4), C23.12(C2×C8), C4.23(C22×C8), C2.12(C23×C8), C4.64(C23×C4), C8.64(C22×C4), (C2×M5(2))⋊21C2, (C2×C8).618C23, C8○D4.20C22, C22.5(C22×C8), M4(2).36(C2×C4), (C2×M4(2)).35C4, (C22×C8).463C22, (C2×C4).33(C2×C8), (C2×C8).153(C2×C4), C4○D4.39(C2×C4), (C2×C8○D4).22C2, (C2×C4○D4).32C4, (C2×C4).477(C22×C4), (C22×C4).371(C2×C4), SmallGroup(128,2139)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — Q8○M5(2)
C1C2C4C8C2×C8C22×C8C2×C8○D4 — Q8○M5(2)
C1C2 — Q8○M5(2)
C1C8 — Q8○M5(2)
C1C2C2C2C2C4C4C8 — Q8○M5(2)

Generators and relations for Q8○M5(2)
 G = < a,b,c,d | a4=b2=d2=1, c8=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c >

Subgroups: 196 in 180 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C8 [×2], C8 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C16 [×8], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C16 [×12], M5(2) [×16], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C2×M5(2) [×6], D4○C16 [×8], C2×C8○D4, Q8○M5(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, C23×C8, Q8○M5(2)

Smallest permutation representation of Q8○M5(2)
On 32 points
Generators in S32
(1 32 9 24)(2 17 10 25)(3 18 11 26)(4 19 12 27)(5 20 13 28)(6 21 14 29)(7 22 15 30)(8 23 16 31)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(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)(17 25)(19 27)(21 29)(23 31)

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

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

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

68 conjugacy classes

class 1 2A2B···2H4A4B4C···4I8A8B8C8D8E···8R16A···16AF
order122···2444···488888···816···16
size112···2112···211112···22···2

68 irreducible representations

dim11111111114
type++++
imageC1C2C2C2C4C4C4C8C8C8Q8○M5(2)
kernelQ8○M5(2)C2×M5(2)D4○C16C2×C8○D4C2×M4(2)C8○D4C2×C4○D4C2×D4C2×Q8C4○D4C1
# reps1681682124164

Matrix representation of Q8○M5(2) in GL4(𝔽17) generated by

130159
241512
00016
0010
,
161305
0100
00160
0001
,
41355
0001
48137
0200
,
101512
0100
00160
00016
G:=sub<GL(4,GF(17))| [13,2,0,0,0,4,0,0,15,15,0,1,9,12,16,0],[16,0,0,0,13,1,0,0,0,0,16,0,5,0,0,1],[4,0,4,0,13,0,8,2,5,0,13,0,5,1,7,0],[1,0,0,0,0,1,0,0,15,0,16,0,12,0,0,16] >;

Q8○M5(2) in GAP, Magma, Sage, TeX

Q_8\circ M_5(2)
% in TeX

G:=Group("Q8oM5(2)");
// GroupNames label

G:=SmallGroup(128,2139);
// by ID

G=gap.SmallGroup(128,2139);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,723,2019,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^8=a^2,b*a*b=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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