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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 [×6], C4 [×4], C4 [×5], C22 [×3], C22 [×6], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×12], D4 [×2], D4 [×9], Q8 [×2], Q8 [×9], C23, C23 [×2], C2×C8 [×2], C2×C8 [×7], M4(2) [×10], M4(2) [×7], D8 [×2], SD16 [×12], Q16 [×10], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×6], C4.D4 [×2], C4.10D4 [×6], C8.C4 [×4], C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C8○D4 [×2], C2×SD16 [×2], C2×SD16, C2×Q16 [×2], C2×Q16, C4○D8 [×2], C8⋊C22 [×2], C8⋊C22, C8.C22 [×6], C8.C22 [×7], C22×Q8, C2×C4○D4 [×2], C2×C4.10D4, M4(2).8C22, M4(2).C4, D4.3D4 [×4], D4.5D4 [×4], Q8○M4(2), C2×C8.C22 [×2], D8⋊C22, M4(2).38D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], 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 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
(1 9 3 15 5 13 7 11)(2 16 4 14 6 12 8 10)(17 30 23 32 21 26 19 28)(18 29 24 31 22 25 20 27)
(1 26 5 30)(2 25 6 29)(3 32 7 28)(4 31 8 27)(9 17 13 21)(10 24 14 20)(11 23 15 19)(12 22 16 18)

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

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

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

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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