Copied to
clipboard

G = D86Q8order 128 = 27

The semidirect product of D8 and Q8 acting through Inn(D8)

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

Aliases: D86Q8, C42.521C23, C4.952- 1+4, C4⋊C43D8, (C8×Q8)⋊14C2, D4.9(C2×Q8), C2.38(D4×Q8), C8.35(C2×Q8), C4⋊C4.415D4, C82Q822C2, (C4×D8).10C2, D4.Q811C2, D43Q815C2, D42Q845C2, C2.66(D4○D8), C4.49(C4○D8), (C2×Q8).186D4, C8.5Q810C2, C4.38(C22×Q8), C4⋊C4.269C23, C4⋊C8.327C22, (C2×C8).211C23, (C4×C8).123C22, (C2×C4).572C24, C4⋊Q8.201C22, C2.D8.71C22, (C2×D8).177C22, (C4×D4).210C22, (C2×D4).435C23, (C4×Q8).311C22, C4.Q8.115C22, C22.832(C22×D4), C42.C2.71C22, D4⋊C4.190C22, C2.76(C2×C4○D8), (C2×C4).178(C2×D4), SmallGroup(128,2112)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D86Q8
C1C2C4C2×C4C42C4×D4D43Q8 — D86Q8
C1C2C2×C4 — D86Q8
C1C22C4×Q8 — D86Q8
C1C2C2C2×C4 — D86Q8

Generators and relations for D86Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 328 in 176 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×8], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], D4 [×2], Q8 [×4], C23 [×2], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×8], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×6], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C4×C8, C4×C8 [×2], D4⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×4], C2.D8, C2.D8 [×4], C2×C4⋊C4 [×2], C4×D4 [×6], C4×Q8, C22⋊Q8 [×6], C42.C2 [×4], C4⋊Q8 [×2], C2×D8, C4×D8, C4×D8 [×2], C8×Q8, D42Q8 [×2], D4.Q8 [×4], C8.5Q8 [×2], C82Q8, D43Q8 [×2], D86Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C4○D8 [×2], C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C4○D8, D4○D8, D86Q8

Smallest permutation representation of D86Q8
On 64 points
Generators in S64
(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 36)(10 35)(11 34)(12 33)(13 40)(14 39)(15 38)(16 37)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 48)(49 61)(50 60)(51 59)(52 58)(53 57)(54 64)(55 63)(56 62)
(1 35 29 10)(2 36 30 11)(3 37 31 12)(4 38 32 13)(5 39 25 14)(6 40 26 15)(7 33 27 16)(8 34 28 9)(17 52 45 64)(18 53 46 57)(19 54 47 58)(20 55 48 59)(21 56 41 60)(22 49 42 61)(23 50 43 62)(24 51 44 63)
(1 63 29 51)(2 64 30 52)(3 57 31 53)(4 58 32 54)(5 59 25 55)(6 60 26 56)(7 61 27 49)(8 62 28 50)(9 23 34 43)(10 24 35 44)(11 17 36 45)(12 18 37 46)(13 19 38 47)(14 20 39 48)(15 21 40 41)(16 22 33 42)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L···4Q8A8B8C8D8E···8J
order122222224···44444···488888···8
size111144442···24448···822224···4

35 irreducible representations

dim11111111222244
type+++++++++-+-+
imageC1C2C2C2C2C2C2C2D4Q8D4C4○D82- 1+4D4○D8
kernelD86Q8C4×D8C8×Q8D42Q8D4.Q8C8.5Q8C82Q8D43Q8C4⋊C4D8C2×Q8C4C4C2
# reps13124212341812

Matrix representation of D86Q8 in GL4(𝔽17) generated by

16000
01600
0006
00146
,
16000
01600
0010
00116
,
161600
2100
0010
0001
,
91600
14800
0049
00413
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,14,0,0,6,6],[16,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16],[16,2,0,0,16,1,0,0,0,0,1,0,0,0,0,1],[9,14,0,0,16,8,0,0,0,0,4,4,0,0,9,13] >;

D86Q8 in GAP, Magma, Sage, TeX

D_8\rtimes_6Q_8
% in TeX

G:=Group("D8:6Q8");
// GroupNames label

G:=SmallGroup(128,2112);
// by ID

G=gap.SmallGroup(128,2112);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,346,80,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽