Copied to
clipboard

G = Q8.4M4(2)  order 128 = 27

The non-split extension by Q8 of M4(2) acting through Inn(Q8)

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

Aliases: Q8.4M4(2), C42.303C23, C4.1192- 1+4, (C8×Q8)⋊33C2, C8⋊4Q8⋊40C2, Q8○3(C22⋊C8), (C4×Q8).32C4, C4⋊C8.368C22, (C2×C8).441C23, (C2×C4).681C24, (C4×C8).340C22, C42.229(C2×C4), C4.36(C2×M4(2)), (C22×Q8).34C4, C22.17(C8○D4), (C4×Q8).334C22, C8⋊C4.100C22, C22⋊C8.237C22, C42.6C4.33C2, C22.204(C23×C4), C23.232(C22×C4), (C2×C42).788C22, C2.22(C22×M4(2)), C42.12C4.46C2, (C22×C4).1285C23, C2.24(C23.32C23), (C2×C4⋊C4).81C4, (C2×C4×Q8).47C2, C2.32(C2×C8○D4), C4⋊C4.232(C2×C4), (C2×Q8).212(C2×C4), (C22×C4).360(C2×C4), (C2×C4).279(C22×C4), SmallGroup(128,1716)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — Q8.4M4(2)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×Q8 — Q8.4M4(2)
Lower central
C1 — C22 — Q8.4M4(2)
Upper central
C1 — C2×C4 — Q8.4M4(2)
Jennings
C1 — C2 — C2 — C2×C4 — Q8.4M4(2)

Generators and relations for Q8.4M4(2)
 G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2c5 >

Subgroups: 228 in 182 conjugacy classes, 136 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, C42.12C4, C42.6C4, C8×Q8, C8⋊4Q8, C2×C4×Q8, Q8.4M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C24, C2×M4(2), C8○D4, C23×C4, 2- 1+4, C23.32C23, C22×M4(2), C2×C8○D4, Q8.4M4(2)

Smallest permutation representation of Q8.4M4(2)
►On 64 points
Generators in S64
(1 52 61 23)(2 24 62 53)(3 54 63 17)(4 18 64 55)(5 56 57 19)(6 20 58 49)(7 50 59 21)(8 22 60 51)(9 46 26 35)(10 36 27 47)(11 48 28 37)(12 38 29 41)(13 42 30 39)(14 40 31 43)(15 44 32 33)(16 34 25 45)

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4R4S···4X8A···8H8I···8T
order12222244444···44···48···88···8
size11112211112···24···42···24···4

50 irreducible representations

dim111111111224
type++++++-
imageC1C2C2C2C2C2C4C4C4M4(2)C8○D42- 1+4
kernelQ8.4M4(2)C42.12C4C42.6C4C8×Q8C8⋊4Q8C2×C4×Q8C2×C4⋊C4C4×Q8C22×Q8Q8C22C4
# reps133261682882

Matrix representation of Q8.4M4(2) ►in GL4(𝔽17) generated by

4000
01300
00160
00016
,
01300
13000
00160
00016
,
0900
9000
00816
0099
,
01300
13000
00160
0011
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,13,0,0,0,0,0,16,0,0,0,0,16],[0,9,0,0,9,0,0,0,0,0,8,9,0,0,16,9],[0,13,0,0,13,0,0,0,0,0,16,1,0,0,0,1] >;
 

Q8.4M4(2) in GAP, Magma, Sage, TeX 

Q_8._4M_4(2)
 
% in TeX
 
G:=Group("Q8.4M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,1716);
 
// by ID
 
G=gap.SmallGroup(128,1716);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,891,100,675,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^5>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁