Copied to
clipboard

?

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(C2×D4), C4○D4.21D4, D4.16(C2×D4), Q8.16(C2×D4), D4.3D4⋊2C2, (C2×D4).152D4, D4.5D4⋊4C2, Q8○M4(2)⋊2C2, (C2×C4).20C24, (C2×C8).23C23, (C2×Q8).128D4, C8○D4.5C22, C8.C4⋊4C22, (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.D4⋊16C22, M4(2).C4⋊14C2, C8.C22.4C22, C23.197(C4○D4), C4.10D4⋊16C22, C22.11(C4⋊D4), (C22×C4).288C23, (C2×M4(2)).63C22, (C22×Q8).287C22, M4(2).8C22⋊6C2, (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

Derived series
C1 — C2×C4 — M4(2).38D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).38D4
Lower central
C1 — C2 — C2×C4 — M4(2).38D4
Upper central
C1 — C2 — C22×C4 — M4(2).38D4
Jennings
C1 — C2 — C2 — C2×C4 — M4(2).38D4

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

Generators and relations
 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 >

Smallest permutation representation
►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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(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 26 23 28 21 30 19 32)(18 25 24 27 22 29 20 31)

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

Matrix representation ►G ⊆ 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] >;
 

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
ρ2522-22-20002-2-22000002i00002i2i2i0000    complex lifted from C4○D4
ρ2622222000-2-2-2-2000002i00002i2i2i0000    complex lifted from C4○D4
ρ2722222000-2-2-2-2000002i00002i2i2i0000    complex lifted from C4○D4
ρ2822-22-20002-2-22000002i00002i2i2i0000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

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
 

׿
×
⋊
ℤ
𝔽
○
≀