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G = Q86M4(2)  order 128 = 27

1st semidirect product of Q8 and M4(2) acting through Inn(Q8)

p-group, metabelian, nilpotent (class 2), monomial

Aliases: Q86M4(2), C42.591C23, (C8×Q8)⋊32C2, C819(C4○D4), C86D438C2, Q82(C8⋊C4), (C4×D4).31C4, (C4×Q8).29C4, C4⋊C8.365C22, (C4×M4(2))⋊39C2, (C2×C8).615C23, (C4×C8).337C22, C42.219(C2×C4), (C2×C4).668C24, C4.34(C2×M4(2)), C42⋊C2.33C4, C42.6C449C2, (C4×D4).297C22, (C4×Q8).332C22, C8⋊C4.165C22, C22⋊C8.144C22, C2.26(Q8○M4(2)), C22.193(C23×C4), C23.105(C22×C4), (C22×C4).938C23, (C2×C42).778C22, C2.17(C22×M4(2)), (C2×M4(2)).370C22, C8⋊C4(C4×Q8), C2.50(C4×C4○D4), C4⋊C4.227(C2×C4), (C4×C4○D4).17C2, (C2×C4○D4).27C4, C4.319(C2×C4○D4), (C2×D4).233(C2×C4), C22⋊C4.75(C2×C4), (C2×Q8).228(C2×C4), (C2×C4).274(C22×C4), (C22×C4).136(C2×C4), SmallGroup(128,1703)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — Q86M4(2)
C1C2C4C2×C4C42C8⋊C4C4×M4(2) — Q86M4(2)
C1C22 — Q86M4(2)
C1C2×C4 — Q86M4(2)
C1C2C2C2×C4 — Q86M4(2)

Generators and relations for Q86M4(2)
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c5 >

Subgroups: 276 in 201 conjugacy classes, 138 normal (16 characteristic)
C1, C2 [×3], C2 [×3], C4 [×8], C4 [×8], C22, C22 [×9], C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×9], C2×C4 [×15], D4 [×6], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], M4(2) [×12], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×4], C4×C8 [×6], C8⋊C4, C8⋊C4 [×3], C22⋊C8 [×6], C4⋊C8 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×M4(2) [×6], C2×C4○D4, C4×M4(2) [×3], C42.6C4 [×3], C86D4 [×6], C8×Q8 [×2], C4×C4○D4, Q86M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C4○D4 [×4], C24, C2×M4(2) [×6], C23×C4, C2×C4○D4 [×2], C4×C4○D4, C22×M4(2), Q8○M4(2), Q86M4(2)

Smallest permutation representation of Q86M4(2)
On 64 points
Generators in S64
(1 47 27 17)(2 18 28 48)(3 41 29 19)(4 20 30 42)(5 43 31 21)(6 22 32 44)(7 45 25 23)(8 24 26 46)(9 55 37 64)(10 57 38 56)(11 49 39 58)(12 59 40 50)(13 51 33 60)(14 61 34 52)(15 53 35 62)(16 63 36 54)
(1 36 27 16)(2 37 28 9)(3 38 29 10)(4 39 30 11)(5 40 31 12)(6 33 32 13)(7 34 25 14)(8 35 26 15)(17 54 47 63)(18 55 48 64)(19 56 41 57)(20 49 42 58)(21 50 43 59)(22 51 44 60)(23 52 45 61)(24 53 46 62)
(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 5)(3 7)(10 14)(12 16)(17 43)(18 48)(19 45)(20 42)(21 47)(22 44)(23 41)(24 46)(25 29)(27 31)(34 38)(36 40)(49 58)(50 63)(51 60)(52 57)(53 62)(54 59)(55 64)(56 61)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4T4U4V4W8A···8H8I···8T
order122222244444···44448···88···8
size111144411112···24442···24···4

50 irreducible representations

dim1111111111224
type++++++
imageC1C2C2C2C2C2C4C4C4C4C4○D4M4(2)Q8○M4(2)
kernelQ86M4(2)C4×M4(2)C42.6C4C86D4C8×Q8C4×C4○D4C42⋊C2C4×D4C4×Q8C2×C4○D4C8Q8C2
# reps1336216622882

Matrix representation of Q86M4(2) in GL4(𝔽17) generated by

4800
01300
0010
0001
,
13000
4400
00160
00016
,
4000
131300
0052
001112
,
1000
161600
00160
0051
G:=sub<GL(4,GF(17))| [4,0,0,0,8,13,0,0,0,0,1,0,0,0,0,1],[13,4,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[4,13,0,0,0,13,0,0,0,0,5,11,0,0,2,12],[1,16,0,0,0,16,0,0,0,0,16,5,0,0,0,1] >;

Q86M4(2) in GAP, Magma, Sage, TeX

Q_8\rtimes_6M_4(2)
% in TeX

G:=Group("Q8:6M4(2)");
// GroupNames label

G:=SmallGroup(128,1703);
// by ID

G=gap.SmallGroup(128,1703);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,723,184,2019,521,80,124]);
// 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=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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