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G = Q85D8order 128 = 27

2nd semidirect product of Q8 and D8 acting through Inn(Q8)

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

Aliases: Q85D8, C42.525C23, C4.1422+ 1+4, (C8×Q8)⋊9C2, (C4×D8)⋊18C2, C4.48(C2×D8), C817(C4○D4), C4⋊D816C2, C84D413C2, C4⋊C4.283D4, Q82(C2.D8), Q86D412C2, C2.70(D4○D8), (C4×C8).97C22, (C2×Q8).273D4, C2.23(C22×D8), C4⋊C4.442C23, C4⋊C8.306C22, (C2×C8).218C23, (C2×C4).583C24, (C2×D8).40C22, C2.37(Q86D4), (C2×D4).278C23, (C4×D4).217C22, (C4×Q8).314C22, C2.D8.236C22, C41D4.104C22, C22.843(C22×D4), D4⋊C4.174C22, C4.161(C2×C4○D4), (C2×C4).1105(C2×D4), SmallGroup(128,2123)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q85D8
C1C2C4C2×C4C42C4×D4Q86D4 — Q85D8
C1C2C2×C4 — Q85D8
C1C22C4×Q8 — Q85D8
C1C2C2C2×C4 — Q85D8

Generators and relations for Q85D8
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 552 in 230 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×6], C4 [×5], C22, C22 [×18], C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×30], Q8 [×4], C23 [×6], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×5], C2×C8, C2×C8 [×3], D8 [×12], C22×C4 [×6], C2×D4 [×6], C2×D4 [×12], C2×Q8, C4○D4 [×8], C4×C8 [×3], D4⋊C4 [×6], C4⋊C8 [×3], C2.D8, C4×D4 [×6], C4×Q8, C4⋊D4 [×6], C41D4 [×6], C2×D8 [×9], C2×C4○D4 [×2], C4×D8 [×3], C8×Q8, C4⋊D8 [×6], C84D4 [×3], Q86D4 [×2], Q85D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×D8 [×6], C22×D4, C2×C4○D4, 2+ 1+4, Q86D4, C22×D8, D4○D8, Q85D8

Smallest permutation representation of Q85D8
On 64 points
Generators in S64
(1 32 62 55)(2 56 63 25)(3 26 64 49)(4 50 57 27)(5 28 58 51)(6 52 59 29)(7 30 60 53)(8 54 61 31)(9 20 38 45)(10 46 39 21)(11 22 40 47)(12 48 33 23)(13 24 34 41)(14 42 35 17)(15 18 36 43)(16 44 37 19)
(1 36 62 15)(2 16 63 37)(3 38 64 9)(4 10 57 39)(5 40 58 11)(6 12 59 33)(7 34 60 13)(8 14 61 35)(17 31 42 54)(18 55 43 32)(19 25 44 56)(20 49 45 26)(21 27 46 50)(22 51 47 28)(23 29 48 52)(24 53 41 30)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 48)(8 47)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 58)(18 57)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 40)(32 39)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I···4O8A8B8C8D8E···8J
order12222···24···44···488888···8
size11118···82···24···422224···4

35 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2D4D4C4○D4D82+ 1+4D4○D8
kernelQ85D8C4×D8C8×Q8C4⋊D8C84D4Q86D4C4⋊C4C2×Q8C8Q8C4C2
# reps131632314812

Matrix representation of Q85D8 in GL4(𝔽17) generated by

1000
0100
0040
00413
,
16000
01600
00115
00116
,
31400
3300
0049
00013
,
31400
141400
0010
00116
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,4,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,1,1,0,0,15,16],[3,3,0,0,14,3,0,0,0,0,4,0,0,0,9,13],[3,14,0,0,14,14,0,0,0,0,1,1,0,0,0,16] >;

Q85D8 in GAP, Magma, Sage, TeX

Q_8\rtimes_5D_8
% in TeX

G:=Group("Q8:5D8");
// GroupNames label

G:=SmallGroup(128,2123);
// by ID

G=gap.SmallGroup(128,2123);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,100,346,80,4037,1027,124]);
// Polycyclic

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

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