Copied to
clipboard

G = M4(2).D4order 128 = 27

3rd non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).3D4, (C2×C8).1D4, C4.6C22≀C2, (C2×D4).90D4, (C2×Q8).81D4, C22⋊C4.4D4, C4.19(C4⋊D4), C23.138(C2×D4), M4(2)⋊4C49C2, C42⋊C222C2, C22.38C22≀C2, D8⋊C22.3C2, C23.38D41C2, (C22×C4).40C23, C22.60(C4⋊D4), C2.14(C232D4), C22.10(C41D4), (C22×Q8).52C22, C23.38C232C2, C23.C2310C2, C42⋊C2.51C22, (C2×M4(2)).18C22, (C2×C4).252(C2×D4), (C2×C8.C22)⋊3C2, (C2×C4).338(C4○D4), (C2×C4○D4).50C22, SmallGroup(128,741)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).D4
C1C2C22C2×C4C22×C4C22×Q8C23.38C23 — M4(2).D4
C1C2C22×C4 — M4(2).D4
C1C2C22×C4 — M4(2).D4
C1C2C2C22×C4 — M4(2).D4

Generators and relations for M4(2).D4
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=ab, dad=a-1b, bc=cb, dbd=a4b, dcd=a4c3 >

Subgroups: 376 in 174 conjugacy classes, 44 normal (36 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×7], C22 [×3], C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×13], D4 [×8], Q8 [×10], C23, C23 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×6], C2×C8 [×2], C2×C8, M4(2) [×2], M4(2) [×3], D8 [×2], SD16 [×8], Q16 [×6], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×8], C23⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C42⋊C2 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4⋊Q8, C2×M4(2) [×2], C2×SD16, C2×Q16, C4○D8 [×2], C8⋊C22 [×2], C8.C22 [×6], C22×Q8, C2×C4○D4 [×2], M4(2)⋊4C4, C23.C23, C23.38D4, C42⋊C22, C23.38C23, C2×C8.C22, D8⋊C22, M4(2).D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, M4(2).D4

Character table of M4(2).D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11222882222888888888888
ρ111111111111111111111111    trivial
ρ2111111-11111-11111-1-11-1-1-1-1    linear of order 2
ρ311111111111-1-1-11-11-1-1-11-11    linear of order 2
ρ4111111-111111-1-11-1-11-11-11-1    linear of order 2
ρ511111-11111111-1-1111-1-1-1-1-1    linear of order 2
ρ611111-1-11111-11-1-11-1-1-11111    linear of order 2
ρ711111-111111-1-11-1-11-111-11-1    linear of order 2
ρ811111-1-111111-11-1-1-111-11-11    linear of order 2
ρ922-22-200-222-20020000-20000    orthogonal lifted from D4
ρ10222-2-20022-2-200000000-2020    orthogonal lifted from D4
ρ1122-22-200-222-200-2000020000    orthogonal lifted from D4
ρ1222-2-220-2-22-22000002000000    orthogonal lifted from D4
ρ1322-22-2202-2-22000-200000000    orthogonal lifted from D4
ρ142222200-2-2-2-20-20020000000    orthogonal lifted from D4
ρ1522-2-2202-22-2200000-2000000    orthogonal lifted from D4
ρ16222-2-200-2-22200000000020-2    orthogonal lifted from D4
ρ17222-2-20022-2-20000000020-20    orthogonal lifted from D4
ρ18222-2-200-2-222000000000-202    orthogonal lifted from D4
ρ192222200-2-2-2-20200-20000000    orthogonal lifted from D4
ρ2022-22-2-202-2-22000200000000    orthogonal lifted from D4
ρ2122-2-22002-22-22i00000-2i00000    complex lifted from C4○D4
ρ2222-2-22002-22-2-2i000002i00000    complex lifted from C4○D4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

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

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

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

Matrix representation of M4(2).D4 in GL8(𝔽17)

000001300
00004000
00000010
000000016
016000000
10000000
00400000
000130000
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
00100000
000160000
04000000
130000000
00000004
000000130
00001000
000001600
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000

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

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

M_4(2).D_4
% in TeX

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

G:=SmallGroup(128,741);
// by ID

G=gap.SmallGroup(128,741);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,352,2019,1018,521,248,2804,718,172,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=a^4*c^3>;
// generators/relations

Export

Character table of M4(2).D4 in TeX

׿
×
𝔽