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G = M4(2).40D4order 128 = 27

4th non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).40D4, C4.52(C4xD4), (C2xD4).6Q8, C23.3(C2xQ8), C23.5(C4:C4), C4.D4.2C4, (C22xC4).61D4, C4.117C22wrC2, C4.C42:2C2, C4.10D4.2C4, Q8oM4(2).1C2, M4(2).2(C2xC4), M4(2).C4:1C2, (C22xC8).27C22, (C22xC4).674C23, C22.15(C22:Q8), C2.14(C23.8Q8), C4.106(C22.D4), (C2xM4(2)).172C22, M4(2).8C22.1C2, (C2xC4).6(C4:C4), (C2xD4).67(C2xC4), (C2xC4).984(C2xD4), C22.24(C2xC4:C4), (C2xC4).8(C22xC4), (C2xQ8).60(C2xC4), (C2xC4).317(C4oD4), (C2xC4oD4).10C22, (C22xC8):C2.13C2, SmallGroup(128,590)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — M4(2).40D4
C1C2C4C2xC4C22xC4C2xC4oD4Q8oM4(2) — M4(2).40D4
C1C2C2xC4 — M4(2).40D4
C1C4C22xC4 — M4(2).40D4
C1C2C2C22xC4 — M4(2).40D4

Generators and relations for M4(2).40D4
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6, bab=a5, cac-1=a3b, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=a2c3 >

Subgroups: 212 in 126 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C22:C8, C4.D4, C4.10D4, C8.C4, C22xC8, C2xM4(2), C2xM4(2), C2xM4(2), C8oD4, C2xC4oD4, C4.C42, (C22xC8):C2, M4(2).8C22, M4(2).C4, Q8oM4(2), M4(2).40D4
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C4:C4, C22xC4, C2xD4, C2xQ8, C4oD4, C2xC4:C4, C4xD4, C22wrC2, C22:Q8, C22.D4, C23.8Q8, M4(2).40D4

Smallest permutation representation of M4(2).40D4
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 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)
(1 28 7 26 5 32 3 30)(2 10 4 12 6 14 8 16)(9 17 15 23 13 21 11 19)(18 31 20 25 22 27 24 29)
(1 28 7 26 5 32 3 30)(2 10 8 16 6 14 4 12)(9 21 15 19 13 17 11 23)(18 27 24 25 22 31 20 29)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G8A···8L8M···8R
order122222244444448···88···8
size112224411222444···48···8

32 irreducible representations

dim1111111122224
type++++++++-
imageC1C2C2C2C2C2C4C4D4D4Q8C4oD4M4(2).40D4
kernelM4(2).40D4C4.C42(C22xC8):C2M4(2).8C22M4(2).C4Q8oM4(2)C4.D4C4.10D4M4(2)C22xC4C2xD4C2xC4C1
# reps1211214442244

Matrix representation of M4(2).40D4 in GL4(F17) generated by

16008
16044
10101
11001
,
13800
13400
16104
01130
,
9000
0900
0009
15080
,
9000
9800
0009
0090
G:=sub<GL(4,GF(17))| [16,16,10,11,0,0,1,0,0,4,0,0,8,4,1,1],[13,13,16,0,8,4,1,1,0,0,0,13,0,0,4,0],[9,0,0,15,0,9,0,0,0,0,0,8,0,0,9,0],[9,9,0,0,0,8,0,0,0,0,0,9,0,0,9,0] >;

M4(2).40D4 in GAP, Magma, Sage, TeX

M_4(2)._{40}D_4
% in TeX

G:=Group("M4(2).40D4");
// GroupNames label

G:=SmallGroup(128,590);
// by ID

G=gap.SmallGroup(128,590);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,1411,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^3>;
// generators/relations

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