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## G = Q8⋊6M4(2)  order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — Q8⋊6M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C8⋊C4 — C4×M4(2) — Q8⋊6M4(2)
 Lower central C1 — C22 — Q8⋊6M4(2)
 Upper central C1 — C2×C4 — Q8⋊6M4(2)
 Jennings C1 — C2 — C2 — C2×C4 — Q8⋊6M4(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, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C4×M4(2), C42.6C4, C86D4, C8×Q8, C4×C4○D4, Q86M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, C24, C2×M4(2), C23×C4, C2×C4○D4, 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 57 38 56)(10 49 39 58)(11 59 40 50)(12 51 33 60)(13 61 34 52)(14 53 35 62)(15 63 36 54)(16 55 37 64)
(1 36 27 15)(2 37 28 16)(3 38 29 9)(4 39 30 10)(5 40 31 11)(6 33 32 12)(7 34 25 13)(8 35 26 14)(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)(9 13)(11 15)(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)(9,57,38,56)(10,49,39,58)(11,59,40,50)(12,51,33,60)(13,61,34,52)(14,53,35,62)(15,63,36,54)(16,55,37,64), (1,36,27,15)(2,37,28,16)(3,38,29,9)(4,39,30,10)(5,40,31,11)(6,33,32,12)(7,34,25,13)(8,35,26,14)(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)(9,13)(11,15)(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)(9,57,38,56)(10,49,39,58)(11,59,40,50)(12,51,33,60)(13,61,34,52)(14,53,35,62)(15,63,36,54)(16,55,37,64), (1,36,27,15)(2,37,28,16)(3,38,29,9)(4,39,30,10)(5,40,31,11)(6,33,32,12)(7,34,25,13)(8,35,26,14)(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)(9,13)(11,15)(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),(9,57,38,56),(10,49,39,58),(11,59,40,50),(12,51,33,60),(13,61,34,52),(14,53,35,62),(15,63,36,54),(16,55,37,64)], [(1,36,27,15),(2,37,28,16),(3,38,29,9),(4,39,30,10),(5,40,31,11),(6,33,32,12),(7,34,25,13),(8,35,26,14),(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),(9,13),(11,15),(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 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4T 4U 4V 4W 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 8 ··· 8 8 ··· 8 size 1 1 1 1 4 4 4 1 1 1 1 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4○D4 M4(2) Q8○M4(2) kernel Q8⋊6M4(2) C4×M4(2) C42.6C4 C8⋊6D4 C8×Q8 C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C8 Q8 C2 # reps 1 3 3 6 2 1 6 6 2 2 8 8 2

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

 4 8 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 13 0 0 0 4 4 0 0 0 0 16 0 0 0 0 16
,
 4 0 0 0 13 13 0 0 0 0 5 2 0 0 11 12
,
 1 0 0 0 16 16 0 0 0 0 16 0 0 0 5 1
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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