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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

Aliases: M4(2).51D4, C42.284C23, M4(2).33C23, C8○D86C2, D86(C2×C4), C8⋊C226C4, M4(2)C4≀C2, Q166(C2×C4), C8.88(C2×D4), C4.79(C4×D4), C8.26D42C2, (C4×C8)⋊23C22, SD165(C2×C4), C8○D49C22, C8.C226C4, C8.5(C22×C4), C4≀C221C22, (C4×M4(2))⋊3C2, C4.33(C23×C4), C22.18(C4×D4), C8⋊C442C22, Q8○M4(2)⋊14C2, (C2×C8).421C23, (C2×C4).213C24, C4○D4.25C23, C4○D8.25C22, D4.15(C22×C4), C4.204(C22×D4), Q8.15(C22×C4), C8.C413C22, D8⋊C22.7C2, M4(2).14(C2×C4), M4(2).C412C2, C23.195(C4○D4), (C22×C4).932C23, (C2×C42).769C22, (C2×M4(2)).244C22, C2.73(C2×C4×D4), (C2×C4≀C2)⋊32C2, C4○D4.11(C2×C4), C22.4(C2×C4○D4), (C2×D4).139(C2×C4), (C2×C4).1087(C2×D4), (C2×C4).73(C22×C4), (C2×Q8).116(C2×C4), (C2×C4).476(C4○D4), (C2×C4○D4).91C22, SmallGroup(128,1688)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2).51D4
C1C2C4C2×C4C22×C4C2×M4(2)Q8○M4(2) — M4(2).51D4
C1C2C4 — M4(2).51D4
C1C4C2×M4(2) — M4(2).51D4
C1C2C2C2×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 2A2B2C2D2E2F2G2H4A4B4C···4I4J···4O8A···8H8I···8T
order122222222444···44···48···88···8
size112224444112···24···42···24···4

44 irreducible representations

dim11111111112224
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4D4C4○D4C4○D4M4(2).51D4
kernelM4(2).51D4C4×M4(2)C2×C4≀C2M4(2).C4C8○D8C8.26D4Q8○M4(2)D8⋊C22C8⋊C22C8.C22M4(2)C2×C4C23C1
# reps11214421884224

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

0300
1000
0002
0040
,
1000
0400
0010
0004
,
4000
0400
0030
0003
,
0010
0004
2000
0300
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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