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

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

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

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

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

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