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## G = D4⋊9SD16order 128 = 27

### 3rd semidirect product of D4 and SD16 acting through Inn(D4)

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4⋊9SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — D4⋊6D4 — D4⋊9SD16
 Lower central C1 — C2 — C2×C4 — D4⋊9SD16
 Upper central C1 — C22 — C4×D4 — D4⋊9SD16
 Jennings C1 — C2 — C2 — C2×C4 — D4⋊9SD16

Generators and relations for D49SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c3 >

Subgroups: 368 in 194 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C22×C8, C2×SD16, C2×C4○D4, C2×C4.Q8, C8×D4, C4×SD16, C88D4, Q8⋊Q8, D42Q8, C23.46D4, C23.47D4, C83Q8, D46D4, D43Q8, D49SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, 2- 1+4, D46D4, C22×SD16, D4○SD16, D49SD16

Smallest permutation representation of D49SD16
On 64 points
Generators in S64
(1 14 62 34)(2 15 63 35)(3 16 64 36)(4 9 57 37)(5 10 58 38)(6 11 59 39)(7 12 60 40)(8 13 61 33)(17 55 43 31)(18 56 44 32)(19 49 45 25)(20 50 46 26)(21 51 47 27)(22 52 48 28)(23 53 41 29)(24 54 42 30)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 49)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 43)(18 46)(19 41)(20 44)(21 47)(22 42)(23 45)(24 48)(25 53)(26 56)(27 51)(28 54)(29 49)(30 52)(31 55)(32 50)(33 39)(35 37)(36 40)(57 63)(59 61)(60 64)

G:=sub<Sym(64)| (1,14,62,34)(2,15,63,35)(3,16,64,36)(4,9,57,37)(5,10,58,38)(6,11,59,39)(7,12,60,40)(8,13,61,33)(17,55,43,31)(18,56,44,32)(19,49,45,25)(20,50,46,26)(21,51,47,27)(22,52,48,28)(23,53,41,29)(24,54,42,30), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,49)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,43)(18,46)(19,41)(20,44)(21,47)(22,42)(23,45)(24,48)(25,53)(26,56)(27,51)(28,54)(29,49)(30,52)(31,55)(32,50)(33,39)(35,37)(36,40)(57,63)(59,61)(60,64)>;

G:=Group( (1,14,62,34)(2,15,63,35)(3,16,64,36)(4,9,57,37)(5,10,58,38)(6,11,59,39)(7,12,60,40)(8,13,61,33)(17,55,43,31)(18,56,44,32)(19,49,45,25)(20,50,46,26)(21,51,47,27)(22,52,48,28)(23,53,41,29)(24,54,42,30), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,49)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,43)(18,46)(19,41)(20,44)(21,47)(22,42)(23,45)(24,48)(25,53)(26,56)(27,51)(28,54)(29,49)(30,52)(31,55)(32,50)(33,39)(35,37)(36,40)(57,63)(59,61)(60,64) );

G=PermutationGroup([[(1,14,62,34),(2,15,63,35),(3,16,64,36),(4,9,57,37),(5,10,58,38),(6,11,59,39),(7,12,60,40),(8,13,61,33),(17,55,43,31),(18,56,44,32),(19,49,45,25),(20,50,46,26),(21,51,47,27),(22,52,48,28),(23,53,41,29),(24,54,42,30)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,49),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,43),(18,46),(19,41),(20,44),(21,47),(22,42),(23,45),(24,48),(25,53),(26,56),(27,51),(28,54),(29,49),(30,52),(31,55),(32,50),(33,39),(35,37),(36,40),(57,63),(59,61),(60,64)]])

35 conjugacy classes

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

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 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 SD16 2- 1+4 D4○SD16 kernel D4⋊9SD16 C2×C4.Q8 C8×D4 C4×SD16 C8⋊8D4 Q8⋊Q8 D4⋊2Q8 C23.46D4 C23.47D4 C8⋊3Q8 D4⋊6D4 D4⋊3Q8 C22⋊C4 C4⋊C4 C2×D4 C8 D4 C4 C2 # reps 1 2 1 1 2 1 1 2 2 1 1 1 2 1 1 4 8 1 2

Matrix representation of D49SD16 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 4 0 0 0 0 13
,
 1 0 0 0 0 1 0 0 0 0 0 13 0 0 4 0
,
 12 12 0 0 5 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,13,0],[12,5,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

D49SD16 in GAP, Magma, Sage, TeX

D_4\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("D4:9SD16");
// GroupNames label

G:=SmallGroup(128,2067);
// by ID

G=gap.SmallGroup(128,2067);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,100,346,4037,1027,124]);
// Polycyclic

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

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