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G = M4(2).38D4order 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(C2×D4), C4○D4.21D4, D4.16(C2×D4), Q8.16(C2×D4), D4.3D42C2, (C2×D4).152D4, D4.5D44C2, Q8○M4(2)⋊2C2, (C2×C4).20C24, (C2×C8).23C23, (C2×Q8).128D4, C8○D4.5C22, C8.C44C22, (C2×Q16)⋊19C22, C4○D4.32C23, (C2×D4).74C23, C4.167(C22×D4), C8⋊C22.5C22, (C2×Q8).62C23, C4.178(C4⋊D4), (C2×SD16)⋊14C22, D8⋊C22.8C2, C4.D416C22, M4(2).C414C2, C8.C22.4C22, C23.197(C4○D4), C4.10D416C22, C22.11(C4⋊D4), (C22×C4).288C23, (C2×M4(2)).63C22, (C22×Q8).287C22, M4(2).8C226C2, (C2×C4).478(C2×D4), C2.91(C2×C4⋊D4), (C2×C8.C22)⋊21C2, C22.23(C2×C4○D4), (C2×C4).480(C4○D4), (C2×C4.10D4)⋊12C2, (C2×C4○D4).133C22, SmallGroup(128,1801)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).38D4
C1C2C4C2×C4C22×C4C2×C4○D4Q8○M4(2) — M4(2).38D4
C1C2C2×C4 — M4(2).38D4
C1C2C22×C4 — M4(2).38D4
C1C2C2C2×C4 — 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, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.D4, C4.10D4, C8.C4, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C2×C4.10D4, M4(2).8C22, M4(2).C4, D4.3D4, D4.5D4, Q8○M4(2), C2×C8.C22, D8⋊C22, M4(2).38D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊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 C4○D4
ρ2622222000-2-2-2-2000002i0000-2i-2i2i0000    complex lifted from C4○D4
ρ2722222000-2-2-2-200000-2i00002i2i-2i0000    complex lifted from C4○D4
ρ2822-22-20002-2-22000002i00002i-2i-2i0000    complex lifted from C4○D4
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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