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G = Q8oD16order 128 = 27

Central product of Q8 and D16

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

Aliases: Q8oD16, D4oQ32, D4.14D8, Q8.14D8, C8.18C24, C16.5C23, D8.7C23, SD32.C22, D16.4C22, Q32.4C22, Q16.7C23, M4(2).23D4, M5(2).14C22, Q8oD8:6C2, C4oD16:6C2, D4oC16:5C2, C8.17(C2xD4), C4.51(C2xD8), (C2xQ32):13C2, C4oD4.37D4, Q32:C2:6C2, C22.8(C2xD8), C2.33(C22xD8), C4.24(C22xD4), (C2xC8).296C23, (C2xC16).35C22, C8oD4.14C22, C4oD8.11C22, (C2xQ16).97C22, (C2xC4).186(C2xD4), SmallGroup(128,2149)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — Q8oD16
C1C2C4C8C2xC8C8oD4Q8oD8 — Q8oD16
C1C2C4C8 — Q8oD16
C1C2C4oD4C8oD4 — Q8oD16
C1C2C2C2C2C4C4C8 — Q8oD16

Generators and relations for Q8oD16
 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, C4, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C16, C16, C2xC8, M4(2), D8, SD16, Q16, Q16, C2xQ8, C4oD4, C4oD4, C2xC16, M5(2), D16, SD32, Q32, C8oD4, C2xQ16, C4oD8, C8.C22, 2- 1+4, D4oC16, C2xQ32, C4oD16, Q32:C2, Q8oD8, Q8oD16
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C24, C2xD8, C22xD4, C22xD8, Q8oD16

Smallest permutation representation of Q8oD16
On 64 points
Generators in S64
(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)(33 63 41 55)(34 64 42 56)(35 49 43 57)(36 50 44 58)(37 51 45 59)(38 52 46 60)(39 53 47 61)(40 54 48 62)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 33)(32 34)
(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 27 9 19)(2 26 10 18)(3 25 11 17)(4 24 12 32)(5 23 13 31)(6 22 14 30)(7 21 15 29)(8 20 16 28)(33 62 41 54)(34 61 42 53)(35 60 43 52)(36 59 44 51)(37 58 45 50)(38 57 46 49)(39 56 47 64)(40 55 48 63)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111122224
type++++++++++-
imageC1C2C2C2C2C2D4D4D8D8Q8oD16
kernelQ8oD16D4oC16C2xQ32C4oD16Q32:C2Q8oD8M4(2)C4oD4D4Q8C1
# reps11336231624

Matrix representation of Q8oD16 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] >;

Q8oD16 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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