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G = M4(2).D4  order 128 = 27

3rd non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).3D4, (C2xC8).1D4, C4.6C22wrC2, (C2xD4).90D4, (C2xQ8).81D4, C22:C4.4D4, C4.19(C4:D4), C23.138(C2xD4), M4(2):4C4:9C2, C42:C22:2C2, C22.38C22wrC2, D8:C22.3C2, C23.38D4:1C2, (C22xC4).40C23, C22.60(C4:D4), C2.14(C23:2D4), C22.10(C4:1D4), (C22xQ8).52C22, C23.38C23:2C2, C23.C23:10C2, C42:C2.51C22, (C2xM4(2)).18C22, (C2xC4).252(C2xD4), (C2xC8.C22):3C2, (C2xC4).338(C4oD4), (C2xC4oD4).50C22, SmallGroup(128,741)

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

Derived series
C1 — C22xC4 — M4(2).D4
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C22xQ8 — C23.38C23 — M4(2).D4
Lower central
C1 — C2 — C22xC4 — M4(2).D4
Upper central
C1 — C2 — C22xC4 — M4(2).D4
Jennings
C1 — C2 — C2 — C22xC4 — M4(2).D4

Generators and relations for M4(2).D4
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=ab, dad=a-1b, bc=cb, dbd=a4b, dcd=a4c3 >

Subgroups: 376 in 174 conjugacy classes, 44 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C23:C4, Q8:C4, C4wrC2, C42:C2, C22:Q8, C22.D4, C4.4D4, C4:Q8, C2xM4(2), C2xSD16, C2xQ16, C4oD8, C8:C22, C8.C22, C22xQ8, C2xC4oD4, M4(2):4C4, C23.C23, C23.38D4, C42:C22, C23.38C23, C2xC8.C22, D8:C22, M4(2).D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C22wrC2, C4:D4, C4:1D4, C23:2D4, M4(2).D4

Character table of M4(2).D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11222882222888888888888
ρ111111111111111111111111    trivial
ρ2111111-11111-11111-1-11-1-1-1-1    linear of order 2
ρ311111111111-1-1-11-11-1-1-11-11    linear of order 2
ρ4111111-111111-1-11-1-11-11-11-1    linear of order 2
ρ511111-11111111-1-1111-1-1-1-1-1    linear of order 2
ρ611111-1-11111-11-1-11-1-1-11111    linear of order 2
ρ711111-111111-1-11-1-11-111-11-1    linear of order 2
ρ811111-1-111111-11-1-1-111-11-11    linear of order 2
ρ922-22-200-222-20020000-20000    orthogonal lifted from D4
ρ10222-2-20022-2-200000000-2020    orthogonal lifted from D4
ρ1122-22-200-222-200-2000020000    orthogonal lifted from D4
ρ1222-2-220-2-22-22000002000000    orthogonal lifted from D4
ρ1322-22-2202-2-22000-200000000    orthogonal lifted from D4
ρ142222200-2-2-2-20-20020000000    orthogonal lifted from D4
ρ1522-2-2202-22-2200000-2000000    orthogonal lifted from D4
ρ16222-2-200-2-22200000000020-2    orthogonal lifted from D4
ρ17222-2-20022-2-20000000020-20    orthogonal lifted from D4
ρ18222-2-200-2-222000000000-202    orthogonal lifted from D4
ρ192222200-2-2-2-20200-20000000    orthogonal lifted from D4
ρ2022-22-2-202-2-22000200000000    orthogonal lifted from D4
ρ2122-2-22002-22-22i00000-2i00000    complex lifted from C4oD4
ρ2222-2-22002-22-2-2i000002i00000    complex lifted from C4oD4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).D4
►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 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)

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

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

Matrix representation of M4(2).D4 ►in GL8(F17)

000001300
00004000
00000010
000000016
016000000
10000000
00400000
000130000
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
00100000
000160000
04000000
130000000
00000004
000000130
00001000
000001600
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000

G:=sub<GL(8,GF(17))| [0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

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

M_4(2).D_4
 
% in TeX
 
G:=Group("M4(2).D4");
 
// GroupNames label
 
G:=SmallGroup(128,741);
 
// by ID
 
G=gap.SmallGroup(128,741);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,352,2019,1018,521,248,2804,718,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=a^4*c^3>;
 
// generators/relations
 

Export

Character table of M4(2).D4 in TeX

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