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G = M4(2).13D4order 128 = 27

13rd non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).13D4, (C2×Q8).7Q8, C4⋊C4.105D4, (C2×C8).161D4, C4.152(C4⋊D4), C4.101(C22⋊Q8), C22.6(C22⋊Q8), C2.29(D4.3D4), C2.21(D4.5D4), C23.281(C4○D4), M4(2)⋊C4.7C2, (C22×C8).171C22, (C22×C4).736C23, C23.38D4.9C2, C2.5(C23.Q8), (C22×Q8).74C22, C22.242(C4⋊D4), C22.C42.15C2, C42⋊C2.67C22, C22.9(C422C2), (C2×M4(2)).28C22, C42.6C22.1C2, (C2×C4).21(C2×Q8), (C2×C4).1054(C2×D4), (C2×C8.C4).20C2, (C2×C4).351(C4○D4), (C2×C4⋊C4).140C22, (C2×Q8⋊C4).25C2, (C2×C4.10D4).3C2, SmallGroup(128,796)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).13D4
C1C2C4C2×C4C22×C4C22×Q8C2×Q8⋊C4 — M4(2).13D4
C1C2C22×C4 — M4(2).13D4
C1C22C22×C4 — M4(2).13D4
C1C2C2C22×C4 — M4(2).13D4

Generators and relations for M4(2).13D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a6b, bab=a5, cac-1=a-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a2bc-1 >

Subgroups: 200 in 104 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×5], C22 [×3], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×8], Q8 [×6], C23, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×2], M4(2) [×5], C22×C4, C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×5], C4.10D4 [×2], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8, C2.D8, C8.C4 [×2], C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2) [×3], C22×Q8, C22.C42, C2×C4.10D4, C2×Q8⋊C4, C23.38D4, C42.6C22, M4(2)⋊C4, C2×C8.C4, M4(2).13D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C23.Q8, D4.3D4, D4.5D4, M4(2).13D4

Character table of M4(2).13D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11112222228888884444888888
ρ111111111111111111111111111    trivial
ρ211111111111-111-111111-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-111111-1111-1    linear of order 2
ρ41111111111-11-1-11-11111-11-1-1-11    linear of order 2
ρ51111111111-111-111-1-1-1-11-11-1-1-1    linear of order 2
ρ61111111111-1-11-1-11-1-1-1-1-11-1111    linear of order 2
ρ711111111111-1-11-1-1-1-1-1-1111-1-11    linear of order 2
ρ8111111111111-111-1-1-1-1-1-1-1-111-1    linear of order 2
ρ92222-2-2-222-2000000-222-2000000    orthogonal lifted from D4
ρ102222-2-22-2-220-200200000000000    orthogonal lifted from D4
ρ112-2-22-2222-2-20000000000000-220    orthogonal lifted from D4
ρ122222-2-2-222-20000002-2-22000000    orthogonal lifted from D4
ρ132222-2-22-2-220200-200000000000    orthogonal lifted from D4
ρ142-2-22-2222-2-200000000000002-20    orthogonal lifted from D4
ρ152-2-22-22-2-22200-20020000000000    symplectic lifted from Q8, Schur index 2
ρ162-2-22-22-2-22200200-20000000000    symplectic lifted from Q8, Schur index 2
ρ172-2-222-22-22-2-2i002i000000000000    complex lifted from C4○D4
ρ182-2-222-2-22-2200000000002i0-2i000    complex lifted from C4○D4
ρ192-2-222-2-22-220000000000-2i02i000    complex lifted from C4○D4
ρ20222222-2-2-2-200000000000-2i0002i    complex lifted from C4○D4
ρ21222222-2-2-2-2000000000002i000-2i    complex lifted from C4○D4
ρ222-2-222-22-22-22i00-2i000000000000    complex lifted from C4○D4
ρ2344-4-40000000000002200-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ2444-4-4000000000000-220022000000    symplectic lifted from D4.5D4, Schur index 2
ρ254-44-400000000000002-2-2-20000000    complex lifted from D4.3D4
ρ264-44-40000000000000-2-22-20000000    complex lifted from D4.3D4

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

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

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

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

Matrix representation of M4(2).13D4 in GL6(𝔽17)

1300000
040000
0032150
00516015
001101415
001414121
,
100000
010000
001000
000100
0032160
00516016
,
010000
1600000
0014300
003300
0012633
001110314
,
0160000
100000
005500
0012500
00165512
00121655

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,3,5,1,14,0,0,2,16,10,14,0,0,15,0,14,12,0,0,0,15,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,3,5,0,0,0,1,2,16,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,14,3,12,11,0,0,3,3,6,10,0,0,0,0,3,3,0,0,0,0,3,14],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,5,12,16,12,0,0,5,5,5,16,0,0,0,0,5,5,0,0,0,0,12,5] >;

M4(2).13D4 in GAP, Magma, Sage, TeX

M_4(2)._{13}D_4
% in TeX

G:=Group("M4(2).13D4");
// GroupNames label

G:=SmallGroup(128,796);
// by ID

G=gap.SmallGroup(128,796);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,352,2019,521,248,2804,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b*c^-1>;
// generators/relations

Export

Character table of M4(2).13D4 in TeX

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