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## G = M4(2).46D4order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).46D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).46D4
 Lower central C1 — C2 — C2×C4 — M4(2).46D4
 Upper central C1 — C2 — C22×C4 — M4(2).46D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).46D4

Generators and relations for M4(2).46D4
G = < a,b,c,d | a8=b2=c4=1, d2=a4, bab=a5, cac-1=a5b, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=a6c-1 >

Subgroups: 284 in 152 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×5], C22 [×3], C22 [×4], C8 [×7], C2×C4 [×6], C2×C4 [×10], D4 [×2], D4 [×5], Q8 [×2], Q8 [×7], C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8 [×4], C2×C8 [×6], M4(2) [×2], M4(2) [×11], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×2], C4.10D4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C42⋊C2, C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×4], C22×Q8, C2×C4○D4, C4.10C42, M4(2)⋊4C4, C2×C4.10D4, C23.38D4, C42⋊C22, Q8○M4(2), C2×C8.C22, M4(2).46D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, M4(2).46D4

Character table of M4(2).46D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 2 2 2 4 4 2 2 2 2 4 4 8 8 8 8 4 4 4 4 4 4 4 4 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ9 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 i -1 -i i i -i -i i -i i -i -i i 1 -1 linear of order 4 ρ10 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -i -1 i i -i i -i i -i -i i -i i -1 1 linear of order 4 ρ11 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -i -1 i -i -i i i -i i -i i i -i 1 -1 linear of order 4 ρ12 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 i -1 -i -i i -i i -i i i -i i -i -1 1 linear of order 4 ρ13 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 i 1 -i i -i i -i i -i -i i i -i 1 -1 linear of order 4 ρ14 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -i 1 i i i -i -i i -i i -i i -i -1 1 linear of order 4 ρ15 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 -i 1 i -i i -i i -i i i -i -i i 1 -1 linear of order 4 ρ16 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 i 1 -i -i -i i i -i i -i i -i i -1 1 linear of order 4 ρ17 2 2 -2 2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 2 2 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 -2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 -2 2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 -2 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 0 2 0 0 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 0 -2 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 -2 -2 0 0 0 2 2 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 -2i 2i 0 0 0 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 2i 0 0 2i -2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 -2i 0 0 -2i 2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 2i -2i 0 0 0 -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).46D4
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)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(1 26 21 11)(2 27 18 16)(3 32 23 9)(4 25 20 14)(5 30 17 15)(6 31 22 12)(7 28 19 13)(8 29 24 10)
(1 14 5 10)(2 11 6 15)(3 12 7 16)(4 9 8 13)(17 29 21 25)(18 26 22 30)(19 27 23 31)(20 32 24 28)

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

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

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

Matrix representation of M4(2).46D4 in GL8(𝔽17)

 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 4 0 0 0 0 0 0 0 0 13 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 13 0 0 0 0 0 0 0 0 4 0 0
,
 1 0 0 0 0 0 0 0 0 1 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 16 0 0 0 0 0 0 0 0 1 0 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 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
,
 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0

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

M4(2).46D4 in GAP, Magma, Sage, TeX

M_4(2)._{46}D_4
% in TeX

G:=Group("M4(2).46D4");
// GroupNames label

G:=SmallGroup(128,634);
// by ID

G=gap.SmallGroup(128,634);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,1411,718,172,1027]);
// Polycyclic

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

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