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G = Q82M4(2)  order 128 = 27

2nd semidirect product of Q8 and M4(2) acting via M4(2)/C8=C2

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

Aliases: Q82M4(2), C42.645C23, C8⋊C86C2, Q8⋊C840C2, (C8×Q8)⋊26C2, C81C817C2, D4⋊C8.14C2, C2.9(C8○D8), (C2×C8).310D4, C4.Q8.10C4, D4⋊C4.5C4, C4.37(C8○D4), C86D4.12C2, Q8⋊C4.5C4, (C2×SD16).6C4, (C4×SD16).2C2, C2.15(C89D4), C4⋊C8.279C22, (C4×C8).317C22, (C4×D4).12C22, C22.136(C4×D4), C4.31(C2×M4(2)), C4.145(C8⋊C22), (C4×Q8).260C22, C2.6(SD16⋊C4), C4.139(C8.C22), (C2×C8).33(C2×C4), C4⋊C4.139(C2×C4), (C2×D4).57(C2×C4), (C2×C4).1481(C2×D4), (C2×Q8).138(C2×C4), (C2×C4).506(C4○D4), (C2×C4).337(C22×C4), SmallGroup(128,320)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q82M4(2)
C1C2C22C2×C4C42C4×C8C8×Q8 — Q82M4(2)
C1C2C2×C4 — Q82M4(2)
C1C2×C4C4×C8 — Q82M4(2)
C1C22C22C42 — Q82M4(2)

Generators and relations for Q82M4(2)
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c5 >

Subgroups: 152 in 83 conjugacy classes, 42 normal (40 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×5], C22, C22 [×3], C8 [×8], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×2], Q8, C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×3], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C2×M4(2), C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C81C8, C86D4, C4×SD16, C8×Q8, Q82M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22, C8.C22, C89D4, SD16⋊C4, C8○D8, Q82M4(2)

Smallest permutation representation of Q82M4(2)
On 64 points
Generators in S64
(1 16 27 43)(2 44 28 9)(3 10 29 45)(4 46 30 11)(5 12 31 47)(6 48 32 13)(7 14 25 41)(8 42 26 15)(17 37 49 60)(18 61 50 38)(19 39 51 62)(20 63 52 40)(21 33 53 64)(22 57 54 34)(23 35 55 58)(24 59 56 36)
(1 20 27 52)(2 64 28 33)(3 22 29 54)(4 58 30 35)(5 24 31 56)(6 60 32 37)(7 18 25 50)(8 62 26 39)(9 21 44 53)(10 34 45 57)(11 23 46 55)(12 36 47 59)(13 17 48 49)(14 38 41 61)(15 19 42 51)(16 40 43 63)
(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 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 53)(18 50)(19 55)(20 52)(21 49)(22 54)(23 51)(24 56)(25 41)(26 46)(27 43)(28 48)(29 45)(30 42)(31 47)(32 44)(33 37)(35 39)(58 62)(60 64)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E···8R8S8T
order12222444444444444488888···888
size11118111122224444822224···488

38 irreducible representations

dim1111111111112222244
type++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4D4C4○D4M4(2)C8○D4C8○D8C8⋊C22C8.C22
kernelQ82M4(2)C8⋊C8D4⋊C8Q8⋊C8C81C8C86D4C4×SD16C8×Q8D4⋊C4Q8⋊C4C4.Q8C2×SD16C2×C8C2×C4Q8C4C2C4C4
# reps1111111122222244811

Matrix representation of Q82M4(2) in GL4(𝔽17) generated by

161500
1100
00160
00016
,
01000
5000
0002
0090
,
8000
9900
0001
0040
,
1200
01600
00160
0001
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,16,0,0,0,0,16],[0,5,0,0,10,0,0,0,0,0,0,9,0,0,2,0],[8,9,0,0,0,9,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,2,16,0,0,0,0,16,0,0,0,0,1] >;

Q82M4(2) in GAP, Magma, Sage, TeX

Q_8\rtimes_2M_4(2)
% in TeX

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

G:=SmallGroup(128,320);
// by ID

G=gap.SmallGroup(128,320);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,2102,268,1123,570,136,172]);
// 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,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^5>;
// generators/relations

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