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## G = M4(2)⋊16D4order 128 = 27

### 3rd semidirect product of M4(2) and 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)⋊16D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2)⋊16D4
 Lower central C1 — C2 — C2×C4 — M4(2)⋊16D4
 Upper central C1 — C22 — C2×C4○D4 — M4(2)⋊16D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2)⋊16D4

Generators and relations for M4(2)⋊16D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a3, dad=a-1, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 548 in 252 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×21], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×22], Q8 [×4], C23, C23 [×2], C23 [×7], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], M4(2) [×4], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×D4 [×12], C2×Q8 [×2], C4○D4 [×8], C24, D4⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C22≀C2 [×2], C4⋊D4 [×6], C4⋊D4 [×3], C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C41D4, C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C2×D4⋊C4, C23.24D4, M4(2)⋊C4, C88D4 [×2], C87D4 [×2], C8⋊D4 [×2], C82D4 [×2], C22.29C24, C22.31C24, C2×C8○D4, C2×C8⋊C22, M4(2)⋊16D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○D8, D4○SD16, M4(2)⋊16D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D4○D8 D4○SD16 kernel M4(2)⋊16D4 C2×D4⋊C4 C23.24D4 M4(2)⋊C4 C8⋊8D4 C8⋊7D4 C8⋊D4 C8⋊2D4 C22.29C24 C22.31C24 C2×C8○D4 C2×C8⋊C22 M4(2) C2×D4 C2×Q8 C2×C4 C2 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 4 3 1 4 2 2

Matrix representation of M4(2)⋊16D4 in GL6(𝔽17)

 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 6 11 0 0 0 0 3 0 0 0 11 6 0 0 0 0 14 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 1 0 0 1 0 0 0 0 0 1 16 0 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 1 16 0 0 0 0 0 0 16 0 0 0 0 0 16 1

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,11,14,0,0,0,0,6,0,0,0,6,3,0,0,0,0,11,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,16,16,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,1] >;

M4(2)⋊16D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{16}D_4
% in TeX

G:=Group("M4(2):16D4");
// GroupNames label

G:=SmallGroup(128,1794);
// by ID

G=gap.SmallGroup(128,1794);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,521,2804,172]);
// Polycyclic

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

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