Copied to
clipboard

G = M4(2).10D4order 128 = 27

10th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).10D4, (C2×C8).158D4, C4.73C22≀C2, (C2×D4).112D4, (C2×Q8).103D4, C4.149(C4⋊D4), C22.C429C2, C2.26(D4.3D4), C2.20(D4.4D4), C23.278(C4○D4), C23.36D435C2, (C22×C4).730C23, (C22×C8).168C22, (C22×D4).85C22, C22.236(C4⋊D4), C4.74(C22.D4), C22.12(C4.4D4), C2.24(C23.10D4), (C2×M4(2)).233C22, C22.13(C22.D4), (C2×C8⋊C22).8C2, (C2×D4⋊C4)⋊27C2, (C2×C8.C4)⋊14C2, (C22×C8)⋊C22C2, (C2×C4.D4)⋊25C2, (C2×C4).1371(C2×D4), (C2×C4).346(C4○D4), (C2×C4⋊C4).129C22, (C2×C4○D4).64C22, SmallGroup(128,783)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).10D4
C1C2C22C2×C4C22×C4C2×M4(2)C2×C4.D4 — M4(2).10D4
C1C2C22×C4 — M4(2).10D4
C1C22C22×C4 — M4(2).10D4
C1C2C2C22×C4 — M4(2).10D4

Generators and relations for M4(2).10D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a5b, dad=a-1, bc=cb, dbd=a4b, dcd=a2c-1 >

Subgroups: 352 in 141 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4.D4, D4⋊C4, Q8⋊C4, C8.C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C22.C42, (C22×C8)⋊C2, C2×C4.D4, C2×D4⋊C4, C23.36D4, C2×C8.C4, C2×C8⋊C22, M4(2).10D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4.3D4, D4.4D4, M4(2).10D4

Character table of M4(2).10D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J
 size 11112288822228884444888888
ρ111111111111111111111111111    trivial
ρ2111111-1-1-11111-1-1-1111111-11-11    linear of order 2
ρ3111111-1111111-1111111-1-1-1-1-1-1    linear of order 2
ρ41111111-1-111111-1-11111-1-11-11-1    linear of order 2
ρ5111111-1-1-11111-111-1-1-1-11-1111-1    linear of order 2
ρ611111111111111-1-1-1-1-1-11-1-11-1-1    linear of order 2
ρ71111111-1-11111111-1-1-1-1-11-1-1-11    linear of order 2
ρ8111111-1111111-1-1-1-1-1-1-1-111-111    linear of order 2
ρ92222-2-2-2002-22-22000000000000    orthogonal lifted from D4
ρ102-22-2-2202-222-2-20000000000000    orthogonal lifted from D4
ρ112-22-22-20002-2-220000000-200200    orthogonal lifted from D4
ρ122222-2-2000-22-220002-2-22000000    orthogonal lifted from D4
ρ132-22-22-20002-2-220000000200-200    orthogonal lifted from D4
ρ142-22-2-220-2222-2-20000000000000    orthogonal lifted from D4
ρ152222-2-2000-22-22000-222-2000000    orthogonal lifted from D4
ρ162222-2-22002-22-2-2000000000000    orthogonal lifted from D4
ρ172-22-2-22000-2-222000000002i000-2i    complex lifted from C4○D4
ρ182-22-22-2000-222-20-2i2i0000000000    complex lifted from C4○D4
ρ192-22-2-22000-2-22200000000-2i0002i    complex lifted from C4○D4
ρ20222222000-2-2-2-20000000002i0-2i0    complex lifted from C4○D4
ρ212-22-22-2000-222-202i-2i0000000000    complex lifted from C4○D4
ρ22222222000-2-2-2-2000000000-2i02i0    complex lifted from C4○D4
ρ2344-4-40000000000000-22220000000    orthogonal lifted from D4.4D4
ρ2444-4-4000000000000022-220000000    orthogonal lifted from D4.4D4
ρ254-4-44000000000000-2-2002-2000000    complex lifted from D4.3D4
ρ264-4-440000000000002-200-2-2000000    complex lifted from D4.3D4

Smallest permutation representation of M4(2).10D4
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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 11 25 17)(2 12 26 18)(3 9 27 23)(4 10 28 24)(5 15 29 21)(6 16 30 22)(7 13 31 19)(8 14 32 20)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 32)(26 31)(27 30)(28 29)

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

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

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

Matrix representation of M4(2).10D4 in GL6(𝔽17)

1300000
440000
00001111
000030
00111100
003000
,
1600000
0160000
001000
000100
0000160
0000016
,
120000
16160000
00111100
003600
000006
000030
,
480000
13130000
00001111
000036
00111100
003600

G:=sub<GL(6,GF(17))| [13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,11,3,0,0,0,0,11,0,0,0,11,3,0,0,0,0,11,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,3,0,0,0,0,11,6,0,0,0,0,0,0,0,3,0,0,0,0,6,0],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,0,0,11,3,0,0,0,0,11,6,0,0,11,3,0,0,0,0,11,6,0,0] >;

M4(2).10D4 in GAP, Magma, Sage, TeX

M_4(2)._{10}D_4
% in TeX

G:=Group("M4(2).10D4");
// GroupNames label

G:=SmallGroup(128,783);
// by ID

G=gap.SmallGroup(128,783);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,1411,172,4037,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=a^5*b,d*a*d=a^-1,b*c=c*b,d*b*d=a^4*b,d*c*d=a^2*c^-1>;
// generators/relations

Export

Character table of M4(2).10D4 in TeX

׿
×
𝔽