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G = Q8○D16order 128 = 27

Central product of Q8 and D16

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

Aliases: Q8D16, D4Q32, D4.14D8, Q8.14D8, C8.18C24, C16.5C23, D8.7C23, SD32.C22, D16.4C22, Q32.4C22, Q16.7C23, M4(2).23D4, M5(2).14C22, Q8○D86C2, C4○D166C2, D4○C165C2, C8.17(C2×D4), C4.51(C2×D8), (C2×Q32)⋊13C2, C4○D4.37D4, Q32⋊C26C2, C22.8(C2×D8), C2.33(C22×D8), C4.24(C22×D4), (C2×C8).296C23, (C2×C16).35C22, C8○D4.14C22, C4○D8.11C22, (C2×Q16).97C22, (C2×C4).186(C2×D4), SmallGroup(128,2149)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — Q8○D16
C1C2C4C8C2×C8C8○D4Q8○D8 — Q8○D16
C1C2C4C8 — Q8○D16
C1C2C4○D4C8○D4 — Q8○D16
C1C2C2C2C2C4C4C8 — Q8○D16

Generators and relations for Q8○D16
 G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c7 >

Subgroups: 352 in 175 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×8], Q8, Q8 [×12], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], D8 [×2], SD16 [×6], Q16 [×6], Q16 [×6], C2×Q8 [×8], C4○D4, C4○D4 [×12], C2×C16 [×3], M5(2) [×3], D16, SD32 [×6], Q32 [×9], C8○D4, C2×Q16 [×6], C4○D8 [×6], C8.C22 [×6], 2- 1+4 [×2], D4○C16, C2×Q32 [×3], C4○D16 [×3], Q32⋊C2 [×6], Q8○D8 [×2], Q8○D16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, Q8○D16

Smallest permutation representation of Q8○D16
On 64 points
Generators in S64
(1 50 9 58)(2 51 10 59)(3 52 11 60)(4 53 12 61)(5 54 13 62)(6 55 14 63)(7 56 15 64)(8 57 16 49)(17 37 25 45)(18 38 26 46)(19 39 27 47)(20 40 28 48)(21 41 29 33)(22 42 30 34)(23 43 31 35)(24 44 32 36)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 62)(18 63)(19 64)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)
(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 51 9 59)(2 50 10 58)(3 49 11 57)(4 64 12 56)(5 63 13 55)(6 62 14 54)(7 61 15 53)(8 60 16 52)(17 38 25 46)(18 37 26 45)(19 36 27 44)(20 35 28 43)(21 34 29 42)(22 33 30 41)(23 48 31 40)(24 47 32 39)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4J8A8B8C8D8E16A16B16C16D16E···16J
order122222244444···4888881616161616···16
size112228822228···82244422224···4

32 irreducible representations

dim11111122224
type++++++++++-
imageC1C2C2C2C2C2D4D4D8D8Q8○D16
kernelQ8○D16D4○C16C2×Q32C4○D16Q32⋊C2Q8○D8M4(2)C4○D4D4Q8C1
# reps11336231624

Matrix representation of Q8○D16 in GL4(𝔽17) generated by

70162
07161
115100
116010
,
115100
116010
70162
07161
,
15900
4700
00159
0047
,
15525
921315
25155
131592
G:=sub<GL(4,GF(17))| [7,0,1,1,0,7,15,16,16,16,10,0,2,1,0,10],[1,1,7,0,15,16,0,7,10,0,16,16,0,10,2,1],[15,4,0,0,9,7,0,0,0,0,15,4,0,0,9,7],[15,9,2,13,5,2,5,15,2,13,15,9,5,15,5,2] >;

Q8○D16 in GAP, Magma, Sage, TeX

Q_8\circ D_{16}
% in TeX

G:=Group("Q8oD16");
// GroupNames label

G:=SmallGroup(128,2149);
// by ID

G=gap.SmallGroup(128,2149);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,456,521,1684,851,242,4037,2028,124]);
// Polycyclic

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

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