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G = Q8×D8order 128 = 27

Direct product of Q8 and D8

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

Aliases: Q8×D8, C42.519C23, C4.932- 1+4, C88(C2×Q8), D44(C2×Q8), (C8×Q8)⋊8C2, (D4×Q8)⋊13C2, (C4×D8).9C2, C4.47(C2×D8), C2.36(D4×Q8), C4⋊C4.280D4, C82Q821C2, D4⋊Q817C2, (C4×C8).95C22, (C2×Q8).270D4, C2.66(Q8○D8), C2.22(C22×D8), C4.36(C22×Q8), C4⋊C4.267C23, C4⋊C8.304C22, (C2×C4).570C24, (C2×C8).210C23, C4⋊Q8.199C22, C2.D8.70C22, (C2×D4).433C23, (C4×D4).208C22, (C2×D8).176C22, (C4×Q8).310C22, C22.830(C22×D4), D4⋊C4.172C22, (C2×C4).1102(C2×D4), SmallGroup(128,2110)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q8×D8
C1C2C4C2×C4C42C4×D4D4×Q8 — Q8×D8
C1C2C2×C4 — Q8×D8
C1C22C4×Q8 — Q8×D8
C1C2C2C2×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 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×9], C22, C22 [×8], C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×18], D4 [×4], D4 [×2], Q8 [×4], Q8 [×12], C23 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×9], C4⋊C4 [×6], C2×C8, C2×C8 [×3], D8 [×4], C22×C4 [×6], C2×D4 [×2], C2×Q8, C2×Q8 [×14], C4×C8 [×3], D4⋊C4 [×6], C4⋊C8 [×3], C2.D8 [×9], C4×D4 [×6], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×6], C2×D8, C22×Q8 [×2], C4×D8 [×3], C8×Q8, D4⋊Q8 [×6], C82Q8 [×3], D4×Q8 [×2], Q8×D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C2×Q8 [×6], C24, C2×D8 [×6], 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 2A2B2C2D2E2F2G4A···4H4I4J4K4L···4Q8A8B8C8D8E···8J
order122222224···44444···488888···8
size111144442···24448···822224···4

35 irreducible representations

dim111111222244
type+++++++-++--
imageC1C2C2C2C2C2D4Q8D4D82- 1+4Q8○D8
kernelQ8×D8C4×D8C8×Q8D4⋊Q8C82Q8D4×Q8C4⋊C4D8C2×Q8Q8C4C2
# reps131632341812

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

16000
01600
00130
00104
,
16000
01600
00112
0076
,
14300
141400
0010
0001
,
1000
01600
00160
00016
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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