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## G = Q8×D8order 128 = 27

### Direct product of Q8 and D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — Q8×D8
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — D4×Q8 — Q8×D8
 Lower central C1 — C2 — C2×C4 — Q8×D8
 Upper central C1 — C22 — C4×Q8 — Q8×D8
 Jennings C1 — C2 — C2 — C2×C4 — Q8×D8

Generators and relations for Q8×D8
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 392 in 202 conjugacy classes, 104 normal (14 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C2×D8, C22×Q8, C4×D8, C8×Q8, D4⋊Q8, C82Q8, D4×Q8, Q8×D8
Quotients: C1, C2, C22, D4, Q8, C23, D8, C2×D4, C2×Q8, C24, C2×D8, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C22×D8, Q8○D8, Q8×D8

Smallest permutation representation of Q8×D8
On 64 points
Generators in S64
(1 39 29 9)(2 40 30 10)(3 33 31 11)(4 34 32 12)(5 35 25 13)(6 36 26 14)(7 37 27 15)(8 38 28 16)(17 46 49 60)(18 47 50 61)(19 48 51 62)(20 41 52 63)(21 42 53 64)(22 43 54 57)(23 44 55 58)(24 45 56 59)
(1 57 29 43)(2 58 30 44)(3 59 31 45)(4 60 32 46)(5 61 25 47)(6 62 26 48)(7 63 27 41)(8 64 28 42)(9 22 39 54)(10 23 40 55)(11 24 33 56)(12 17 34 49)(13 18 35 50)(14 19 36 51)(15 20 37 52)(16 21 38 53)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(17 51)(18 50)(19 49)(20 56)(21 55)(22 54)(23 53)(24 52)(41 59)(42 58)(43 57)(44 64)(45 63)(46 62)(47 61)(48 60)

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

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

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

35 conjugacy classes

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

35 irreducible representations

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

Matrix representation of Q8×D8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 13 0 0 0 10 4
,
 16 0 0 0 0 16 0 0 0 0 11 2 0 0 7 6
,
 14 3 0 0 14 14 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,10,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,11,7,0,0,2,6],[14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;

Q8×D8 in GAP, Magma, Sage, TeX

Q_8\times D_8
% in TeX

G:=Group("Q8xD8");
// GroupNames label

G:=SmallGroup(128,2110);
// by ID

G=gap.SmallGroup(128,2110);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,346,80,4037,1027,124]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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