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G = M4(2).44D4order 128 = 27

8th 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).44D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).44D4
 Lower central C1 — C2 — C2×C4 — M4(2).44D4
 Upper central C1 — C4 — C22×C4 — M4(2).44D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).44D4

Generators and relations for M4(2).44D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=b, bab=a5, cac-1=dad-1=a3b, bc=cb, bd=db, dcd-1=a4bc3 >

Subgroups: 300 in 155 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×4], C22 [×3], C22 [×7], C8 [×7], C2×C4 [×6], C2×C4 [×9], D4 [×2], D4 [×9], Q8 [×2], Q8 [×3], C23, C23 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×9], M4(2) [×4], M4(2) [×8], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], C4○D4 [×6], C22⋊C8 [×2], D4⋊C4, Q8⋊C4, C4≀C2 [×2], C42⋊C2, C22×C8, C2×M4(2) [×3], C2×M4(2) [×2], C8○D4 [×4], C4○D8 [×2], C8⋊C22 [×2], C8⋊C22, C8.C22 [×2], C8.C22, C2×C4○D4 [×2], C4.C42, M4(2)⋊4C4, (C22×C8)⋊C2, C23.24D4, C42⋊C22, Q8○M4(2), D8⋊C22, M4(2).44D4
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).44D4

Smallest permutation representation of M4(2).44D4
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)(25 29)(27 31)
(1 14 23 32 5 10 19 28)(2 9 20 31 6 13 24 27)(3 16 17 26 7 12 21 30)(4 11 22 25 8 15 18 29)
(1 28)(2 31 6 27)(3 30)(4 25 8 29)(5 32)(7 26)(9 24 13 20)(10 23)(11 18 15 22)(12 17)(14 19)(16 21)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A ··· 8L 8M 8N order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 size 1 1 2 2 2 4 4 8 1 1 2 2 2 4 4 8 8 8 4 ··· 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 D4 D4 D4 D4 D4 C4○D4 C4○D4 M4(2).44D4 kernel M4(2).44D4 C4.C42 M4(2)⋊4C4 (C22×C8)⋊C2 C23.24D4 C42⋊C22 Q8○M4(2) D8⋊C22 C8⋊C22 C8.C22 C2×C8 M4(2) C2×D4 C2×Q8 C4○D4 C2×C4 C23 C1 # reps 1 1 1 1 1 1 1 1 4 4 2 2 1 1 2 2 2 4

Matrix representation of M4(2).44D4 in GL4(𝔽17) generated by

 0 16 13 10 0 16 0 10 16 1 0 0 0 2 0 1
,
 1 0 0 1 0 1 0 1 0 0 16 0 0 0 0 16
,
 14 14 12 14 3 14 12 0 0 0 10 12 0 0 10 0
,
 14 3 12 5 3 3 12 8 0 0 10 12 0 0 10 7
G:=sub<GL(4,GF(17))| [0,0,16,0,16,16,1,2,13,0,0,0,10,10,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,1,1,0,16],[14,3,0,0,14,14,0,0,12,12,10,10,14,0,12,0],[14,3,0,0,3,3,0,0,12,12,10,10,5,8,12,7] >;

M4(2).44D4 in GAP, Magma, Sage, TeX

M_4(2)._{44}D_4
% in TeX

G:=Group("M4(2).44D4");
// GroupNames label

G:=SmallGroup(128,613);
// by ID

G=gap.SmallGroup(128,613);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,248,2804,718,172,2028]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=b,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*b*c^3>;
// generators/relations

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