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G = M4(2).51D4  order 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, C8oD8:6C2, D8:6(C2xC4), C8:C22:6C4, M4(2)oC4wrC2, Q16:6(C2xC4), C8.88(C2xD4), C4.79(C4xD4), C8.26D4:2C2, (C4xC8):23C22, SD16:5(C2xC4), C8oD4:9C22, C8.C22:6C4, C8.5(C22xC4), C4wrC2:21C22, (C4xM4(2)):3C2, C4.33(C23xC4), C22.18(C4xD4), C8:C4:42C22, Q8oM4(2):14C2, (C2xC8).421C23, (C2xC4).213C24, C4oD4.25C23, C4oD8.25C22, D4.15(C22xC4), C4.204(C22xD4), Q8.15(C22xC4), C8.C4:13C22, D8:C22.7C2, M4(2).14(C2xC4), M4(2).C4:12C2, C23.195(C4oD4), (C22xC4).932C23, (C2xC42).769C22, (C2xM4(2)).244C22, C2.73(C2xC4xD4), (C2xC4wrC2):32C2, C4oD4.11(C2xC4), C22.4(C2xC4oD4), (C2xD4).139(C2xC4), (C2xC4).1087(C2xD4), (C2xC4).73(C22xC4), (C2xQ8).116(C2xC4), (C2xC4).476(C4oD4), (C2xC4oD4).91C22, SmallGroup(128,1688)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — M4(2).51D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xM4(2) — Q8oM4(2) — M4(2).51D4
Lower central
C1 — C2 — C4 — M4(2).51D4
Upper central
C1 — C4 — C2xM4(2) — M4(2).51D4
Jennings
C1 — C2 — C2 — C2xC4 — 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, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C4xC8, C8:C4, C4wrC2, C8.C4, C2xC42, C2xM4(2), C2xM4(2), C2xM4(2), C8oD4, C8oD4, C4oD8, C8:C22, C8.C22, C2xC4oD4, C4xM4(2), C2xC4wrC2, M4(2).C4, C8oD8, C8.26D4, Q8oM4(2), D8:C22, M4(2).51D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, 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+++++++++
imageC1C2C2C2C2C2C2C2C4C4D4C4oD4C4oD4M4(2).51D4
kernelM4(2).51D4C4xM4(2)C2xC4wrC2M4(2).C4C8oD8C8.26D4Q8oM4(2)D8:C22C8:C22C8.C22M4(2)C2xC4C23C1
# 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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