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G = M4(2).38D4  order 128 = 27

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

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

Aliases: M4(2).38D4, M4(2).12C23, C8.26(C2xD4), C4oD4.21D4, D4.16(C2xD4), Q8.16(C2xD4), D4.3D4:2C2, (C2xD4).152D4, D4.5D4:4C2, Q8oM4(2):2C2, (C2xC4).20C24, (C2xC8).23C23, (C2xQ8).128D4, C8oD4.5C22, C8.C4:4C22, (C2xQ16):19C22, C4oD4.32C23, (C2xD4).74C23, C4.167(C22xD4), C8:C22.5C22, (C2xQ8).62C23, C4.178(C4:D4), (C2xSD16):14C22, D8:C22.8C2, C4.D4:16C22, M4(2).C4:14C2, C8.C22.4C22, C23.197(C4oD4), C4.10D4:16C22, C22.11(C4:D4), (C22xC4).288C23, (C2xM4(2)).63C22, (C22xQ8).287C22, M4(2).8C22:6C2, (C2xC4).478(C2xD4), C2.91(C2xC4:D4), (C2xC8.C22):21C2, C22.23(C2xC4oD4), (C2xC4).480(C4oD4), (C2xC4.10D4):12C2, (C2xC4oD4).133C22, SmallGroup(128,1801)

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

Derived series
C1 — C2xC4 — M4(2).38D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC4oD4 — Q8oM4(2) — M4(2).38D4
Lower central
C1 — C2 — C2xC4 — M4(2).38D4
Upper central
C1 — C2 — C22xC4 — M4(2).38D4
Jennings
C1 — C2 — C2 — C2xC4 — M4(2).38D4

Generators and relations for M4(2).38D4
 G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 396 in 223 conjugacy classes, 98 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C4.D4, C4.10D4, C8.C4, C2xM4(2), C2xM4(2), C8oD4, C8oD4, C2xSD16, C2xSD16, C2xQ16, C2xQ16, C4oD8, C8:C22, C8:C22, C8.C22, C8.C22, C22xQ8, C2xC4oD4, C2xC4.10D4, M4(2).8C22, M4(2).C4, D4.3D4, D4.5D4, Q8oM4(2), C2xC8.C22, D8:C22, M4(2).38D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, C2xC4:D4, M4(2).38D4

Character table of M4(2).38D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222448222244888444444448888
ρ111111111111111111111111111111    trivial
ρ211-11-11-1-1-111-1-1111-1-11-11-11-111-11-1    linear of order 2
ρ311-11-1-11-1-111-11-111-111-11-1-11-1-11-11    linear of order 2
ρ411111-1-111111-1-1111-11111-1-1-1-1-1-1-1    linear of order 2
ρ511-11-1-111-111-11-1-11-1-1-11-111-11-1-111    linear of order 2
ρ611111-1-1-11111-1-1-1111-1-1-1-1111-111-1    linear of order 2
ρ71111111-1111111-111-1-1-1-1-1-1-1-11-1-11    linear of order 2
ρ811-11-11-11-111-1-11-11-11-11-11-11-111-1-1    linear of order 2
ρ911-11-11-1-1-111-1-111-111-11-11-11-1-1-111    linear of order 2
ρ10111111111111111-1-1-1-1-1-1-1-1-1-1-111-1    linear of order 2
ρ1111111-1-111111-1-11-1-11-1-1-1-11111-1-11    linear of order 2
ρ1211-11-1-11-1-111-11-11-11-1-11-111-1111-1-1    linear of order 2
ρ1311111-1-1-11111-1-1-1-1-1-11111-1-1-11111    linear of order 2
ρ1411-11-1-111-111-11-1-1-1111-11-1-11-11-11-1    linear of order 2
ρ1511-11-11-11-111-1-11-1-11-11-11-11-11-11-11    linear of order 2
ρ161111111-1111111-1-1-111111111-1-1-1-1    linear of order 2
ρ17222-2-2-2-20-22-2222000000000000000    orthogonal lifted from D4
ρ18222-2-2220-22-22-2-2000000000000000    orthogonal lifted from D4
ρ1922-2-22000-2-2220000002-2-220000000    orthogonal lifted from D4
ρ2022-2-222-2022-2-22-2000000000000000    orthogonal lifted from D4
ρ21222-2-20002-22-2000000-2-2220000000    orthogonal lifted from D4
ρ22222-2-20002-22-200000022-2-20000000    orthogonal lifted from D4
ρ2322-2-22-22022-2-2-22000000000000000    orthogonal lifted from D4
ρ2422-2-22000-2-222000000-222-20000000    orthogonal lifted from D4
ρ2522-22-20002-2-2200000-2i0000-2i2i2i0000    complex lifted from C4oD4
ρ2622222000-2-2-2-2000002i0000-2i-2i2i0000    complex lifted from C4oD4
ρ2722222000-2-2-2-200000-2i00002i2i-2i0000    complex lifted from C4oD4
ρ2822-22-20002-2-22000002i00002i-2i-2i0000    complex lifted from C4oD4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).38D4
►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 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)

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

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

Matrix representation of M4(2).38D4 ►in GL8(F17)

215220000
221520000
15151520000
21515150000
000021522
000022152
00001515152
00002151515
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
000015222
000022215
000022152
000021522
222150000
21515150000
215220000
15152150000
,
222150000
21515150000
215220000
15152150000
000015222
000022215
000022152
000021522

G:=sub<GL(8,GF(17))| [2,2,15,2,0,0,0,0,15,2,15,15,0,0,0,0,2,15,15,15,0,0,0,0,2,2,2,15,0,0,0,0,0,0,0,0,2,2,15,2,0,0,0,0,15,2,15,15,0,0,0,0,2,15,15,15,0,0,0,0,2,2,2,15],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,2,2,2,15,0,0,0,0,2,15,15,15,0,0,0,0,2,15,2,2,0,0,0,0,15,15,2,15,15,2,2,2,0,0,0,0,2,2,2,15,0,0,0,0,2,2,15,2,0,0,0,0,2,15,2,2,0,0,0,0],[2,2,2,15,0,0,0,0,2,15,15,15,0,0,0,0,2,15,2,2,0,0,0,0,15,15,2,15,0,0,0,0,0,0,0,0,15,2,2,2,0,0,0,0,2,2,2,15,0,0,0,0,2,2,15,2,0,0,0,0,2,15,2,2] >;
 

M4(2).38D4 in GAP, Magma, Sage, TeX 

M_4(2)._{38}D_4
 
% in TeX
 
G:=Group("M4(2).38D4");
 
// GroupNames label
 
G:=SmallGroup(128,1801);
 
// by ID
 
G=gap.SmallGroup(128,1801);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,2804,172,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Character table of M4(2).38D4 in TeX

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