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## G = M4(2).51D4order 128 = 27

### 1st non-split extension by M4(2) of D4 acting through Inn(M4(2))

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — M4(2).51D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — Q8○M4(2) — M4(2).51D4
 Lower central C1 — C2 — C4 — M4(2).51D4
 Upper central C1 — C4 — C2×M4(2) — M4(2).51D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).51D4

Generators and relations for M4(2).51D4
G = < a,b,c,d | a8=b2=c4=1, d2=a6, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a6c-1 >

Subgroups: 348 in 229 conjugacy classes, 138 normal (18 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×7], C8 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×2], C42, C2×C8 [×4], C2×C8 [×10], M4(2) [×12], M4(2) [×10], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], C4≀C2 [×8], C8.C4 [×4], C2×C42, C2×M4(2) [×2], C2×M4(2) [×2], C2×M4(2) [×4], C8○D4 [×8], C8○D4 [×4], C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C4×M4(2), C2×C4≀C2 [×2], M4(2).C4, C8○D8 [×4], C8.26D4 [×4], Q8○M4(2) [×2], D8⋊C22, M4(2).51D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, M4(2).51D4

Permutation representations of M4(2).51D4
On 16 points - transitive group 16T205
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(9 15 13 11)(10 16 14 12)
(1 11 7 9 5 15 3 13)(2 12 8 10 6 16 4 14)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (9,15,13,11)(10,16,14,12), (1,11,7,9,5,15,3,13)(2,12,8,10,6,16,4,14)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (9,15,13,11)(10,16,14,12), (1,11,7,9,5,15,3,13)(2,12,8,10,6,16,4,14) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(9,15,13,11),(10,16,14,12)], [(1,11,7,9,5,15,3,13),(2,12,8,10,6,16,4,14)])

G:=TransitiveGroup(16,205);

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C ··· 4I 4J ··· 4O 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 2 2 2 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 2 2 2 4 4 4 4 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 D4 C4○D4 C4○D4 M4(2).51D4 kernel M4(2).51D4 C4×M4(2) C2×C4≀C2 M4(2).C4 C8○D8 C8.26D4 Q8○M4(2) D8⋊C22 C8⋊C22 C8.C22 M4(2) C2×C4 C23 C1 # reps 1 1 2 1 4 4 2 1 8 8 4 2 2 4

Matrix representation of M4(2).51D4 in GL4(𝔽5) generated by

 0 3 0 0 1 0 0 0 0 0 0 2 0 0 4 0
,
 1 0 0 0 0 4 0 0 0 0 1 0 0 0 0 4
,
 4 0 0 0 0 4 0 0 0 0 3 0 0 0 0 3
,
 0 0 1 0 0 0 0 4 2 0 0 0 0 3 0 0
G:=sub<GL(4,GF(5))| [0,1,0,0,3,0,0,0,0,0,0,4,0,0,2,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3],[0,0,2,0,0,0,0,3,1,0,0,0,0,4,0,0] >;

M4(2).51D4 in GAP, Magma, Sage, TeX

M_4(2)._{51}D_4
% in TeX

G:=Group("M4(2).51D4");
// GroupNames label

G:=SmallGroup(128,1688);
// by ID

G=gap.SmallGroup(128,1688);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,2804,1411,172,124]);
// Polycyclic

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

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