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## G = Q8⋊8SD16order 128 = 27

### 2nd semidirect product of Q8 and SD16 acting through Inn(Q8)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — Q8⋊8SD16
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — Q82 — Q8⋊8SD16
 Lower central C1 — C2 — C2×C4 — Q8⋊8SD16
 Upper central C1 — C22 — C4×Q8 — Q8⋊8SD16
 Jennings C1 — C2 — C2 — C2×C4 — Q8⋊8SD16

Generators and relations for Q88SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c3 >

Subgroups: 408 in 196 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C41D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×C4○D4, C4×SD16, C8×Q8, C4⋊SD16, D42Q8, C4.4D8, Q86D4, Q82, Q88SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, 2- 1+4, Q85D4, C22×SD16, D4○D8, Q88SD16

Smallest permutation representation of Q88SD16
On 64 points
Generators in S64
(1 51 59 25)(2 52 60 26)(3 53 61 27)(4 54 62 28)(5 55 63 29)(6 56 64 30)(7 49 57 31)(8 50 58 32)(9 20 33 46)(10 21 34 47)(11 22 35 48)(12 23 36 41)(13 24 37 42)(14 17 38 43)(15 18 39 44)(16 19 40 45)
(1 46 59 20)(2 21 60 47)(3 48 61 22)(4 23 62 41)(5 42 63 24)(6 17 64 43)(7 44 57 18)(8 19 58 45)(9 51 33 25)(10 26 34 52)(11 53 35 27)(12 28 36 54)(13 55 37 29)(14 30 38 56)(15 49 39 31)(16 32 40 50)
(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 33)(2 36)(3 39)(4 34)(5 37)(6 40)(7 35)(8 38)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(41 52)(42 55)(43 50)(44 53)(45 56)(46 51)(47 54)(48 49)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4H 4I ··· 4O 4P 4Q 4R 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 8 8 8 8 8 ··· 8 size 1 1 1 1 8 8 8 2 ··· 2 4 ··· 4 8 8 8 2 2 2 2 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 SD16 C4○D4 2- 1+4 D4○D8 kernel Q8⋊8SD16 C4×SD16 C8×Q8 C4⋊SD16 D4⋊2Q8 C4.4D8 Q8⋊6D4 Q82 C4⋊C4 C2×Q8 Q8 Q8 C4 C2 # reps 1 3 1 3 3 3 1 1 3 1 8 4 1 2

Matrix representation of Q88SD16 in GL4(𝔽17) generated by

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

Q88SD16 in GAP, Magma, Sage, TeX

Q_8\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("Q8:8SD16");
// GroupNames label

G:=SmallGroup(128,2094);
// by ID

G=gap.SmallGroup(128,2094);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=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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