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## G = D4⋊8SD16order 128 = 27

### 2nd 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⋊8SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — D4×Q8 — D4⋊8SD16
 Lower central C1 — C2 — C2×C4 — D4⋊8SD16
 Upper central C1 — C22 — C4×D4 — D4⋊8SD16
 Jennings C1 — C2 — C2 — C2×C4 — D4⋊8SD16

Generators and relations for D48SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c3 >

Subgroups: 408 in 210 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, 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, C2×Q8, C4○D4, C4×C8, C22⋊C8, 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, C4⋊Q8, C4⋊Q8, C22×C8, C2×SD16, C2×SD16, C22×Q8, C2×C4○D4, C2×Q8⋊C4, C8×D4, C4×SD16, Q8⋊D4, D4.D4, C88D4, Q8⋊Q8, C23.47D4, C4.SD16, D46D4, D4×Q8, D48SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C22×SD16, Q8○D8, D48SD16

Smallest permutation representation of D48SD16
On 64 points
Generators in S64
(1 22 27 42)(2 23 28 43)(3 24 29 44)(4 17 30 45)(5 18 31 46)(6 19 32 47)(7 20 25 48)(8 21 26 41)(9 62 34 53)(10 63 35 54)(11 64 36 55)(12 57 37 56)(13 58 38 49)(14 59 39 50)(15 60 40 51)(16 61 33 52)
(1 63)(2 55)(3 57)(4 49)(5 59)(6 51)(7 61)(8 53)(9 41)(10 22)(11 43)(12 24)(13 45)(14 18)(15 47)(16 20)(17 38)(19 40)(21 34)(23 36)(25 52)(26 62)(27 54)(28 64)(29 56)(30 58)(31 50)(32 60)(33 48)(35 42)(37 44)(39 46)
(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)
(1 5)(2 8)(4 6)(9 36)(10 39)(11 34)(12 37)(13 40)(14 35)(15 38)(16 33)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(41 43)(42 46)(45 47)(49 60)(50 63)(51 58)(52 61)(53 64)(54 59)(55 62)(56 57)

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

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

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

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 SD16 C4○D4 2+ 1+4 Q8○D8 kernel D4⋊8SD16 C2×Q8⋊C4 C8×D4 C4×SD16 Q8⋊D4 D4.D4 C8⋊8D4 Q8⋊Q8 C23.47D4 C4.SD16 D4⋊6D4 D4×Q8 C22⋊C4 C4⋊C4 C2×D4 D4 Q8 C4 C2 # reps 1 2 1 1 2 1 2 1 2 1 1 1 2 1 1 8 4 1 2

Matrix representation of D48SD16 in GL4(𝔽17) generated by

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

D48SD16 in GAP, Magma, Sage, TeX

D_4\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("D4:8SD16");
// GroupNames label

G:=SmallGroup(128,2030);
// by ID

G=gap.SmallGroup(128,2030);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,352,346,4037,1027,124]);
// 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,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations

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