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G = M4(2).Q8order 128 = 27

1st non-split extension by M4(2) of Q8 acting via Q8/C2=C22

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

Aliases: M4(2).1Q8, (C2×C8).58D4, C4⋊C4.113D4, C4.38(C4⋊Q8), C4.14(C22⋊Q8), C4.C42.4C2, C2.23(D4.5D4), C2.23(D4.4D4), C23.284(C4○D4), C22.C42.5C2, M4(2)⋊C4.8C2, (C22×C8).122C22, (C22×C4).740C23, C22.262(C4⋊D4), C42⋊C2.71C22, (C2×M4(2)).33C22, C42.6C22.6C2, C22.11(C42.C2), C4.118(C22.D4), C2.11(C23.81C23), (C2×C4).22(C2×Q8), (C2×C2.D8).19C2, (C2×C4).1382(C2×D4), (C2×C4).356(C4○D4), (C2×C4⋊C4).160C22, SmallGroup(128,821)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).Q8
C1C2C4C2×C4C22×C4C2×C4⋊C4M4(2)⋊C4 — M4(2).Q8
C1C2C22×C4 — M4(2).Q8
C1C22C22×C4 — M4(2).Q8
C1C2C2C22×C4 — M4(2).Q8

Generators and relations for M4(2).Q8
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6bc2, bab=a5, cac-1=ab, dad-1=a3, cbc-1=dbd-1=a4b, dcd-1=a2bc3 >

Subgroups: 168 in 88 conjugacy classes, 42 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×7], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×4], C2×C4⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C4.C42, C22.C42 [×2], C42.6C22, C2×C2.D8, M4(2)⋊C4 [×2], M4(2).Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C23.81C23, D4.4D4, D4.5D4, M4(2).Q8

Character table of M4(2).Q8

 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
ρ234-44-400000000000000-2222000000    orthogonal lifted from D4.4D4
ρ244-44-40000000000000022-22000000    orthogonal lifted from D4.4D4
ρ2544-4-400000000000022-2200000000    symplectic lifted from D4.5D4, Schur index 2
ρ2644-4-4000000000000-222200000000    symplectic lifted from D4.5D4, Schur index 2

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

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

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

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

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

100000
0160000
0031123
00104315
00155110
0071199
,
1600000
0160000
001000
000100
0010160
0034016
,
050000
700000
005566
00115116
004433
00112144
,
400000
0130000
0034015
0016020
001011016
0026414

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,3,10,15,7,0,0,11,4,5,11,0,0,2,3,1,9,0,0,3,15,10,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,3,0,0,0,1,0,4,0,0,0,0,16,0,0,0,0,0,0,16],[0,7,0,0,0,0,5,0,0,0,0,0,0,0,5,11,4,11,0,0,5,5,4,2,0,0,6,11,3,14,0,0,6,6,3,4],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,3,16,10,2,0,0,4,0,11,6,0,0,0,2,0,4,0,0,15,0,16,14] >;

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

M_4(2).Q_8
% in TeX

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

G:=SmallGroup(128,821);
// by ID

G=gap.SmallGroup(128,821);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,58,718,172,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*b*c^2,b*a*b=a^5,c*a*c^-1=a*b,d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations

Export

Character table of M4(2).Q8 in TeX

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