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G = M4(2).11D4order 128 = 27

11st non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).11D4, (C2×C8).48D4, C4.74C22≀C2, (C2×D4).113D4, (C2×Q8).104D4, C4.150(C4⋊D4), C22.C4210C2, C2.20(D4.5D4), C2.27(D4.3D4), C23.279(C4○D4), (C22×C4).731C23, (C22×C8).169C22, (C22×Q8).71C22, C22.237(C4⋊D4), C23.36D4.11C2, C4.75(C22.D4), C22.13(C4.4D4), C2.25(C23.10D4), (C2×M4(2)).234C22, C22.14(C22.D4), (C2×C8.C4)⋊15C2, (C2×Q8⋊C4)⋊27C2, (C2×C4).1372(C2×D4), (C2×C8.C22).8C2, (C2×C4).347(C4○D4), (C2×C4.10D4)⋊25C2, (C2×C4⋊C4).130C22, (C22×C8)⋊C2.1C2, (C2×C4○D4).65C22, SmallGroup(128,784)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).11D4
C1C2C22C2×C4C22×C4C2×M4(2)C2×C4.10D4 — M4(2).11D4
C1C2C22×C4 — M4(2).11D4
C1C22C22×C4 — M4(2).11D4
C1C2C2C22×C4 — M4(2).11D4

Generators and relations for M4(2).11D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=ab, dad=a3, bc=cb, dbd=a4b, dcd=a6c-1 >

Subgroups: 272 in 131 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C8 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×4], Q8 [×8], C23, C23, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×2], M4(2) [×5], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C22⋊C8 [×2], C4.10D4 [×2], D4⋊C4, Q8⋊C4 [×3], C8.C4 [×2], C2×C4⋊C4, C22×C8, C2×M4(2) [×3], C2×SD16, C2×Q16, C8.C22 [×4], C22×Q8, C2×C4○D4, C22.C42, (C22×C8)⋊C2, C2×C4.10D4, C2×Q8⋊C4, C23.36D4, C2×C8.C4, C2×C8.C22, M4(2).11D4
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, D4.5D4, M4(2).11D4

Character table of M4(2).11D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J
 size 11112282222888884444888888
ρ111111111111111111111111111    trivial
ρ2111111111111-111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ3111111-1111111-1111111-1-1-1-1-1-1    linear of order 2
ρ4111111-111111-1-11-1-1-1-1-11-1111-1    linear of order 2
ρ5111111-11111-11-1-11-1-1-1-1-111-111    linear of order 2
ρ6111111-11111-1-1-1-1-1111111-11-11    linear of order 2
ρ711111111111-111-11-1-1-1-11-1-11-1-1    linear of order 2
ρ811111111111-1-11-1-11111-1-11-11-1    linear of order 2
ρ92-22-22-20-22-22000000000-200200    orthogonal lifted from D4
ρ102-22-22-20-22-22000000000200-200    orthogonal lifted from D4
ρ112222-2-22-222-200-2000000000000    orthogonal lifted from D4
ρ122-22-2-22022-2-20-20020000000000    orthogonal lifted from D4
ρ132222-2-202-2-2200000-222-2000000    orthogonal lifted from D4
ρ142-22-2-22022-2-20200-20000000000    orthogonal lifted from D4
ρ152222-2-202-2-22000002-2-22000000    orthogonal lifted from D4
ρ162222-2-2-2-222-2002000000000000    orthogonal lifted from D4
ρ172-22-2-220-2-2220000000000-2i0002i    complex lifted from C4○D4
ρ182-22-2-220-2-22200000000002i000-2i    complex lifted from C4○D4
ρ192222220-2-2-2-2000000000002i0-2i0    complex lifted from C4○D4
ρ202-22-22-202-22-22i00-2i00000000000    complex lifted from C4○D4
ρ212222220-2-2-2-200000000000-2i02i0    complex lifted from C4○D4
ρ222-22-22-202-22-2-2i002i00000000000    complex lifted from C4○D4
ρ234-4-44000000000000-220022000000    symplectic lifted from D4.5D4, Schur index 2
ρ244-4-440000000000002200-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ2544-4-40000000000000-2-22-20000000    complex lifted from D4.3D4
ρ2644-4-400000000000002-2-2-20000000    complex lifted from D4.3D4

Smallest permutation representation of M4(2).11D4
On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 48)(10 45)(11 42)(12 47)(13 44)(14 41)(15 46)(16 43)(25 53)(26 50)(27 55)(28 52)(29 49)(30 54)(31 51)(32 56)(33 58)(34 63)(35 60)(36 57)(37 62)(38 59)(39 64)(40 61)
(1 57 53 43)(2 33 54 13)(3 63 55 41)(4 39 56 11)(5 61 49 47)(6 37 50 9)(7 59 51 45)(8 35 52 15)(10 17 38 31)(12 23 40 29)(14 21 34 27)(16 19 36 25)(18 64 32 42)(20 62 26 48)(22 60 28 46)(24 58 30 44)
(2 4)(3 7)(6 8)(9 37)(10 40)(11 35)(12 38)(13 33)(14 36)(15 39)(16 34)(18 20)(19 23)(22 24)(25 29)(26 32)(28 30)(41 61)(42 64)(43 59)(44 62)(45 57)(46 60)(47 63)(48 58)(50 52)(51 55)(54 56)

G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,48)(10,45)(11,42)(12,47)(13,44)(14,41)(15,46)(16,43)(25,53)(26,50)(27,55)(28,52)(29,49)(30,54)(31,51)(32,56)(33,58)(34,63)(35,60)(36,57)(37,62)(38,59)(39,64)(40,61), (1,57,53,43)(2,33,54,13)(3,63,55,41)(4,39,56,11)(5,61,49,47)(6,37,50,9)(7,59,51,45)(8,35,52,15)(10,17,38,31)(12,23,40,29)(14,21,34,27)(16,19,36,25)(18,64,32,42)(20,62,26,48)(22,60,28,46)(24,58,30,44), (2,4)(3,7)(6,8)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(41,61)(42,64)(43,59)(44,62)(45,57)(46,60)(47,63)(48,58)(50,52)(51,55)(54,56)>;

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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,48)(10,45)(11,42)(12,47)(13,44)(14,41)(15,46)(16,43)(25,53)(26,50)(27,55)(28,52)(29,49)(30,54)(31,51)(32,56)(33,58)(34,63)(35,60)(36,57)(37,62)(38,59)(39,64)(40,61), (1,57,53,43)(2,33,54,13)(3,63,55,41)(4,39,56,11)(5,61,49,47)(6,37,50,9)(7,59,51,45)(8,35,52,15)(10,17,38,31)(12,23,40,29)(14,21,34,27)(16,19,36,25)(18,64,32,42)(20,62,26,48)(22,60,28,46)(24,58,30,44), (2,4)(3,7)(6,8)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(41,61)(42,64)(43,59)(44,62)(45,57)(46,60)(47,63)(48,58)(50,52)(51,55)(54,56) );

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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,19),(2,24),(3,21),(4,18),(5,23),(6,20),(7,17),(8,22),(9,48),(10,45),(11,42),(12,47),(13,44),(14,41),(15,46),(16,43),(25,53),(26,50),(27,55),(28,52),(29,49),(30,54),(31,51),(32,56),(33,58),(34,63),(35,60),(36,57),(37,62),(38,59),(39,64),(40,61)], [(1,57,53,43),(2,33,54,13),(3,63,55,41),(4,39,56,11),(5,61,49,47),(6,37,50,9),(7,59,51,45),(8,35,52,15),(10,17,38,31),(12,23,40,29),(14,21,34,27),(16,19,36,25),(18,64,32,42),(20,62,26,48),(22,60,28,46),(24,58,30,44)], [(2,4),(3,7),(6,8),(9,37),(10,40),(11,35),(12,38),(13,33),(14,36),(15,39),(16,34),(18,20),(19,23),(22,24),(25,29),(26,32),(28,30),(41,61),(42,64),(43,59),(44,62),(45,57),(46,60),(47,63),(48,58),(50,52),(51,55),(54,56)])

Matrix representation of M4(2).11D4 in GL6(𝔽17)

16150000
110000
0040413
00013413
0015204
00152130
,
1600000
0160000
000100
001000
000001
000010
,
400000
13130000
000606
006060
00014011
00140110
,
100000
16160000
0016000
000100
001010
00016016

G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,4,0,15,15,0,0,0,13,2,2,0,0,4,4,0,13,0,0,13,13,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,6,0,14,0,0,6,0,14,0,0,0,0,6,0,11,0,0,6,0,11,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,1,0,0,0,0,1,0,16,0,0,0,0,1,0,0,0,0,0,0,16] >;

M4(2).11D4 in GAP, Magma, Sage, TeX

M_4(2)._{11}D_4
% in TeX

G:=Group("M4(2).11D4");
// GroupNames label

G:=SmallGroup(128,784);
// by ID

G=gap.SmallGroup(128,784);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,58,2019,1018,248,2804,1411,172,4037,1027]);
// 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*b,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=a^6*c^-1>;
// generators/relations

Export

Character table of M4(2).11D4 in TeX

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