Copied to
clipboard

G = M4(2).6Q8  order 128 = 27

4th non-split extension by M4(2) of Q8 acting via Q8/C4=C2

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

Aliases: M4(2).6Q8, C8.8(C4⋊C4), (C2×C8).16Q8, C8.C4⋊8C4, C4⋊C4.221D4, (C2×C8).349D4, C4.148(C4×D4), C4.20(C4⋊Q8), C22.9(C4×Q8), C4.79(C22⋊Q8), C2.4(D4.4D4), C2.4(D4.5D4), C8○2M4(2).3C2, M4(2).11(C2×C4), C23.268(C4○D4), C22.C42.2C2, M4(2)⋊C4.4C2, (C22×C4).697C23, (C22×C8).223C22, C22.142(C4⋊D4), C22.10(C42.C2), C42⋊C2.276C22, (C2×M4(2)).324C22, C2.19(C23.65C23), C4.46(C2×C4⋊C4), (C2×C8).69(C2×C4), (C2×C4).117(C2×Q8), (C2×C2.D8).36C2, (C2×C4).62(C4○D4), (C2×C4).1017(C2×D4), (C2×C4⋊C4).82C22, (C2×C8.C4).17C2, (C2×C4).198(C22×C4), SmallGroup(128,684)

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

Derived series
C1 — C2×C4 — M4(2).6Q8
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).6Q8
Lower central
C1 — C2 — C2×C4 — M4(2).6Q8
Upper central
C1 — C22 — C22×C4 — M4(2).6Q8
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).6Q8

Generators and relations for M4(2).6Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a5b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 172 in 98 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C4.Q8, C2.D8, C8.C4, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C22.C42, C8○2M4(2), C2×C2.D8, M4(2)⋊C4, C2×C8.C4, M4(2).6Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, D4.4D4, D4.5D4, M4(2).6Q8

Smallest permutation representation of M4(2).6Q8
►On 64 points
Generators in S64
(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 21)(19 23)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E···8J8K8L8M8N
order12222244444444444488888···88888
size11112222224444888822224···48888

32 irreducible representations

dim111111122222244
type++++++++--+-
imageC1C2C2C2C2C2C4D4D4Q8Q8C4○D4C4○D4D4.4D4D4.5D4
kernelM4(2).6Q8C22.C42C8○2M4(2)C2×C2.D8M4(2)⋊C4C2×C8.C4C8.C4C4⋊C4C2×C8C2×C8M4(2)C2×C4C23C2C2
# reps121121822222222

Matrix representation of M4(2).6Q8 ►in GL6(𝔽17)

1600000
010000
0000314
000033
0031400
003300
,
1600000
0160000
0016000
0001600
000010
000001
,
1300000
0130000
0000160
0000016
001000
000100
,
090000
1500000
00121200
0012500
0000125
000055

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,3,3,0,0,0,0,14,3,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[0,15,0,0,0,0,9,0,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,12,5,0,0,0,0,5,5] >;
 

M4(2).6Q8 in GAP, Magma, Sage, TeX 

M_4(2)._6Q_8
 
% in TeX
 
G:=Group("M4(2).6Q8");
 
// GroupNames label
 
G:=SmallGroup(128,684);
 
// by ID
 
G=gap.SmallGroup(128,684);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,436,2019,1018,248,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^5,d*a*d^-1=a^5*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁