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G = Q8oM5(2)  order 128 = 27

Central product of Q8 and M5(2)

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

Aliases: Q8oM5(2), D4oM5(2), C8.25C24, M5(2)oM5(2), M4(2)oM5(2), C16.15C23, M5(2):17C22, D4oC16:8C2, C8oD4.5C4, C4oD4.4C8, D4.9(C2xC8), C4oD4oM5(2), (C2xD4).11C8, (C2xQ8).10C8, Q8.10(C2xC8), (C2xC16):13C22, M5(2)o(C8oD4), C23.12(C2xC8), C4.23(C22xC8), C2.12(C23xC8), C4.64(C23xC4), C8.64(C22xC4), (C2xM5(2)):21C2, (C2xC8).618C23, C8oD4.20C22, C22.5(C22xC8), M4(2).36(C2xC4), (C2xM4(2)).35C4, (C22xC8).463C22, (C2xC4).33(C2xC8), (C2xC8).153(C2xC4), C4oD4.39(C2xC4), (C2xC8oD4).22C2, (C2xC4oD4).32C4, (C2xC4).477(C22xC4), (C22xC4).371(C2xC4), SmallGroup(128,2139)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — Q8oM5(2)
C1C2C4C8C2xC8C22xC8C2xC8oD4 — Q8oM5(2)
C1C2 — Q8oM5(2)
C1C8 — Q8oM5(2)
C1C2C2C2C2C4C4C8 — Q8oM5(2)

Generators and relations for Q8oM5(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, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, C23, C16, C2xC8, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4oD4, C2xC16, M5(2), C22xC8, C2xM4(2), C8oD4, C2xC4oD4, C2xM5(2), D4oC16, C2xC8oD4, Q8oM5(2)
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C2xC8, C22xC4, C24, C22xC8, C23xC4, C23xC8, Q8oM5(2)

Smallest permutation representation of Q8oM5(2)
On 32 points
Generators in S32
(1 27 9 19)(2 28 10 20)(3 29 11 21)(4 30 12 22)(5 31 13 23)(6 32 14 24)(7 17 15 25)(8 18 16 26)
(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)(18 26)(20 28)(22 30)(24 32)

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

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

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

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++++
imageC1C2C2C2C4C4C4C8C8C8Q8oM5(2)
kernelQ8oM5(2)C2xM5(2)D4oC16C2xC8oD4C2xM4(2)C8oD4C2xC4oD4C2xD4C2xQ8C4oD4C1
# reps1681682124164

Matrix representation of Q8oM5(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] >;

Q8oM5(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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