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## G = D4○SD32order 128 = 27

### Central product of D4 and SD32

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — D4○SD32
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C8○D4 — D4○D8 — D4○SD32
 Lower central C1 — C2 — C4 — C8 — D4○SD32
 Upper central C1 — C2 — C4○D4 — C8○D4 — D4○SD32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — D4○SD32

Generators and relations for D4○SD32
G = < a,b,c,d | a4=d2=1, b2=c8=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 416 in 180 conjugacy classes, 90 normal (15 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C16, C16, C2×C8, M4(2), D8, D8, D8, SD16, Q16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C2×C16, M5(2), D16, SD32, SD32, Q32, C8○D4, C2×D8, C2×Q16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, D4○C16, C2×SD32, C4○D16, C16⋊C22, Q32⋊C2, D4○D8, Q8○D8, D4○SD32
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C22×D8, D4○SD32

Smallest permutation representation of D4○SD32
On 32 points
Generators in S32
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 26 9 18)(2 27 10 19)(3 28 11 20)(4 29 12 21)(5 30 13 22)(6 31 14 23)(7 32 15 24)(8 17 16 25)
(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 27)(2 18)(3 25)(4 32)(5 23)(6 30)(7 21)(8 28)(9 19)(10 26)(11 17)(12 24)(13 31)(14 22)(15 29)(16 20)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 16A 16B 16C 16D 16E ··· 16J order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 16 16 16 16 16 ··· 16 size 1 1 2 2 2 8 8 8 8 2 2 2 2 8 8 8 8 2 2 4 4 4 2 2 2 2 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D8 D8 D4○SD32 kernel D4○SD32 D4○C16 C2×SD32 C4○D16 C16⋊C22 Q32⋊C2 D4○D8 Q8○D8 M4(2) C4○D4 D4 Q8 C1 # reps 1 1 3 3 3 3 1 1 3 1 6 2 4

Matrix representation of D4○SD32 in GL4(𝔽7) generated by

 0 6 5 1 3 0 5 6 3 3 6 1 1 6 3 1
,
 4 2 2 1 1 4 1 3 0 4 5 5 2 4 1 1
,
 4 0 5 1 1 4 2 1 1 6 1 5 5 5 1 0
,
 4 5 6 6 4 3 2 2 2 1 1 0 5 6 5 6
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[4,1,0,2,2,4,4,4,2,1,5,1,1,3,5,1],[4,1,1,5,0,4,6,5,5,2,1,1,1,1,5,0],[4,4,2,5,5,3,1,6,6,2,1,5,6,2,0,6] >;

D4○SD32 in GAP, Magma, Sage, TeX

D_4\circ {\rm SD}_{32}
% in TeX

G:=Group("D4oSD32");
// GroupNames label

G:=SmallGroup(128,2148);
// by ID

G=gap.SmallGroup(128,2148);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,521,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^8=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

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