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G = M4(2).2Q8  order 128 = 27

2nd non-split extension by M4(2) of Q8 acting via Q8/C2=C22

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

Aliases: M4(2).2Q8, C4⋊C4.114D4, (C2×C8).167D4, C4.39(C4⋊Q8), C4.15(C22⋊Q8), C4.C42.5C2, C2.31(D4.3D4), C23.285(C4○D4), C22.C42.6C2, M4(2)⋊C4.9C2, (C22×C8).329C22, (C22×C4).741C23, C22.263(C4⋊D4), C42⋊C2.72C22, (C2×M4(2)).34C22, C42.6C22.7C2, C22.12(C42.C2), C4.119(C22.D4), C2.12(C23.81C23), (C2×C4).23(C2×Q8), (C2×C4.Q8).28C2, (C2×C4).1383(C2×D4), (C2×C4).357(C4○D4), (C2×C4⋊C4).161C22, SmallGroup(128,822)

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

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

Generators and relations for M4(2).2Q8
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2bc2, bab=a5, cac-1=a5b, dad-1=a-1, cbc-1=dbd-1=a4b, dcd-1=a6bc3 >

Subgroups: 168 in 88 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C4.C42, C22.C42, C42.6C22, C2×C4.Q8, M4(2)⋊C4, M4(2).2Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C23.81C23, D4.3D4, M4(2).2Q8

Character table of M4(2).2Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11112222228888884444888888
ρ111111111111111111111111111    trivial
ρ211111111111-1-11-1-111111-1-11-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-11111-111-111    linear of order 2
ρ41111111111-111-1111111-1-1-1-1-1-1    linear of order 2
ρ511111111111-111-11-1-1-1-1-1-11-1-11    linear of order 2
ρ6111111111111-111-1-1-1-1-1-11-1-11-1    linear of order 2
ρ71111111111-11-1-11-1-1-1-1-11-111-11    linear of order 2
ρ81111111111-1-11-1-11-1-1-1-111-111-1    linear of order 2
ρ92222-2-222-2-200000022-2-2000000    orthogonal lifted from D4
ρ102222-2-2-2-222200-2000000000000    orthogonal lifted from D4
ρ112222-2-222-2-2000000-2-222000000    orthogonal lifted from D4
ρ122222-2-2-2-222-2002000000000000    orthogonal lifted from D4
ρ132-2-22-222-22-200000000000200-20    symplectic lifted from Q8, Schur index 2
ρ142-2-222-22-2-22000000000000200-2    symplectic lifted from Q8, Schur index 2
ρ152-2-22-222-22-200000000000-20020    symplectic lifted from Q8, Schur index 2
ρ162-2-222-22-2-22000000000000-2002    symplectic lifted from Q8, Schur index 2
ρ172-2-222-2-222-20-2i002i00000000000    complex lifted from C4○D4
ρ182-2-22-22-22-2200-2i002i0000000000    complex lifted from C4○D4
ρ192-2-22-22-22-22002i00-2i0000000000    complex lifted from C4○D4
ρ20222222-2-2-2-200000000002i00-2i00    complex lifted from C4○D4
ρ212-2-222-2-222-202i00-2i00000000000    complex lifted from C4○D4
ρ22222222-2-2-2-20000000000-2i002i00    complex lifted from C4○D4
ρ2344-4-4000000000000-2√-22√-200000000    complex lifted from D4.3D4
ρ244-44-4000000000000002√-2-2√-2000000    complex lifted from D4.3D4
ρ254-44-400000000000000-2√-22√-2000000    complex lifted from D4.3D4
ρ2644-4-40000000000002√-2-2√-200000000    complex lifted from D4.3D4

Smallest permutation representation of M4(2).2Q8
►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)

(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)

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

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

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

0150000
900000
0000160
000001
000100
001000
,
1600000
0160000
001000
000100
0000160
0000016
,
080000
200000
0000512
000055
0051200
005500
,
1300000
040000
000001
0000160
0001600
001000

G:=sub<GL(6,GF(17))| [0,9,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,2,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,5,5,0,0,0,0,12,5,0,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0] >;
 

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

M_4(2)._2Q_8
 
% in TeX
 
G:=Group("M4(2).2Q8");
 
// GroupNames label
 
G:=SmallGroup(128,822);
 
// by ID
 
G=gap.SmallGroup(128,822);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,64,422,387,58,718,172,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2*b*c^2,b*a*b=a^5,c*a*c^-1=a^5*b,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*b*c^3>;
 
// generators/relations
 

Export

Character table of M4(2).2Q8 in TeX

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