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G = Q87M4(2)  order 128 = 27

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

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

Aliases: Q87M4(2), C42.699C23, C4.1792+ 1+4, C4.1262- 1+4, Q83(C4⋊C8), (C8×Q8)⋊36C2, C86D446C2, (C4×D4).39C4, (C4×Q8).36C4, C4.50(C8○D4), C4⋊C8.371C22, (C2×C4).688C24, (C2×C8).484C23, (C4×C8).345C22, C42.236(C2×C4), C4.38(C2×M4(2)), C4⋊M4(2)⋊38C2, C42⋊C2.37C4, (C4×D4).305C22, (C4×Q8).336C22, C42.12C458C2, C22⋊C8.241C22, C22.210(C23×C4), C23.110(C22×C4), (C2×C42).795C22, (C22×C4).949C23, C2.25(C22×M4(2)), (C2×M4(2)).251C22, C2.46(C23.33C23), C2.36(C2×C8○D4), C4⋊C4.235(C2×C4), (C2×C4○D4).31C4, (C4×C4○D4).21C2, (C2×D4).239(C2×C4), C22⋊C4.81(C2×C4), (C2×Q8).231(C2×C4), (C22×C4).366(C2×C4), (C2×C4).299(C22×C4), SmallGroup(128,1723)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — Q87M4(2)
C1C2C4C2×C4C42C2×C42C4×C4○D4 — Q87M4(2)
C1C22 — Q87M4(2)
C1C2×C4 — Q87M4(2)
C1C2C2C2×C4 — Q87M4(2)

Generators and relations for Q87M4(2)
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c5 >

Subgroups: 276 in 197 conjugacy classes, 136 normal (18 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×8], C4 [×7], C22, C22 [×9], C8 [×8], C2×C4 [×3], C2×C4 [×9], C2×C4 [×18], D4 [×6], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], M4(2) [×6], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×4], C4×C8 [×6], C22⋊C8 [×6], C4⋊C8, C4⋊C8 [×9], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×M4(2) [×6], C2×C4○D4, C4⋊M4(2) [×3], C42.12C4 [×3], C86D4 [×6], C8×Q8 [×2], C4×C4○D4, Q87M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C8○D4 [×2], C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, C22×M4(2), C2×C8○D4, Q87M4(2)

Smallest permutation representation of Q87M4(2)
On 64 points
Generators in S64
(1 51 60 26)(2 52 61 27)(3 53 62 28)(4 54 63 29)(5 55 64 30)(6 56 57 31)(7 49 58 32)(8 50 59 25)(9 21 37 46)(10 22 38 47)(11 23 39 48)(12 24 40 41)(13 17 33 42)(14 18 34 43)(15 19 35 44)(16 20 36 45)
(1 18 60 43)(2 44 61 19)(3 20 62 45)(4 46 63 21)(5 22 64 47)(6 48 57 23)(7 24 58 41)(8 42 59 17)(9 54 37 29)(10 30 38 55)(11 56 39 31)(12 32 40 49)(13 50 33 25)(14 26 34 51)(15 52 35 27)(16 28 36 53)
(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 38)(2 35)(3 40)(4 37)(5 34)(6 39)(7 36)(8 33)(9 63)(10 60)(11 57)(12 62)(13 59)(14 64)(15 61)(16 58)(17 50)(18 55)(19 52)(20 49)(21 54)(22 51)(23 56)(24 53)(25 42)(26 47)(27 44)(28 41)(29 46)(30 43)(31 48)(32 45)

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim11111111112244
type+++++++-
imageC1C2C2C2C2C2C4C4C4C4M4(2)C8○D42+ 1+42- 1+4
kernelQ87M4(2)C4⋊M4(2)C42.12C4C86D4C8×Q8C4×C4○D4C42⋊C2C4×D4C4×Q8C2×C4○D4Q8C4C4C4
# reps13362166228811

Matrix representation of Q87M4(2) in GL4(𝔽17) generated by

1000
0100
00130
00134
,
1000
0100
0049
00013
,
16200
10100
0020
00215
,
16000
16100
00115
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,13,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,9,13],[16,10,0,0,2,1,0,0,0,0,2,2,0,0,0,15],[16,16,0,0,0,1,0,0,0,0,1,0,0,0,15,16] >;

Q87M4(2) in GAP, Magma, Sage, TeX

Q_8\rtimes_7M_4(2)
% in TeX

G:=Group("Q8:7M4(2)");
// GroupNames label

G:=SmallGroup(128,1723);
// by ID

G=gap.SmallGroup(128,1723);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,675,80,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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