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G = Q16.5D4order 128 = 27

2nd non-split extension by Q16 of D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: Q16.5D4, C42.141D4, C4⋊C168C2, (C2×Q32)⋊4C2, (C2×C4).35D8, C8.75(C2×D4), (C4×Q16)⋊19C2, (C2×C8).175D4, C2.D16.2C2, C2.9(C4○D16), C8.93(C4○D4), (C2×C16).6C22, (C2×SD32).4C2, (C2×D8).4C22, C2.Q3211C2, C2.23(C4⋊D8), C4.54(C4⋊D4), C4.19(C8⋊C22), (C4×C8).104C22, (C2×C8).519C23, C8.12D4.2C2, C22.105(C2×D8), (C2×Q16).5C22, C2.11(Q32⋊C2), C2.D8.161C22, (C2×C4).787(C2×D4), SmallGroup(128,943)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — Q16.5D4
C1C2C4C8C2×C8C2×Q16C4×Q16 — Q16.5D4
C1C2C4C2×C8 — Q16.5D4
C1C22C42C4×C8 — Q16.5D4
C1C2C2C2C2C4C4C2×C8 — Q16.5D4

Generators and relations for Q16.5D4
 G = < a,b,c,d | a8=c4=1, b2=d2=a4, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a4c-1 >

Subgroups: 204 in 79 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C8 [×2], C8, C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×5], C23, C16 [×2], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×2], SD16 [×2], Q16 [×2], Q16 [×3], C2×D4, C2×Q8 [×2], C4×C8, Q8⋊C4, C2.D8, C2×C16 [×2], SD32 [×2], Q32 [×2], C4×Q8, C4.4D4, C2×D8, C2×SD16, C2×Q16 [×2], C2.D16, C2.Q32, C4⋊C16, C4×Q16, C8.12D4, C2×SD32, C2×Q32, Q16.5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C4○D16, Q32⋊C2, Q16.5D4

Character table of Q16.5D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111162222488881622224444444444
ρ111111111111111111111111111111    trivial
ρ21111-11-1-11-11-11-111111-1-1111-1-11-1-1    linear of order 2
ρ3111111-1-11-1-11-11-11111-1-1111-1-11-1-1    linear of order 2
ρ41111-111111-1-1-1-1-111111111111111    linear of order 2
ρ51111111111-1-1-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-11-1-11-1-11-1111111-1-1-1-1-111-111    linear of order 2
ρ7111111-1-11-11-11-1-11111-1-1-1-1-111-111    linear of order 2
ρ81111-1111111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ92-2-220200-2020-200-222-20000000000    orthogonal lifted from D4
ρ10222202-2-22-200000-2-2-2-22200000000    orthogonal lifted from D4
ρ112-2-220200-20-20200-222-20000000000    orthogonal lifted from D4
ρ12222202222200000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1322220-2-2-2-2200000000000-22-2-2222-2    orthogonal lifted from D8
ρ1422220-222-2-2000000000002-22-22-22-2    orthogonal lifted from D8
ρ1522220-222-2-200000000000-22-22-22-22    orthogonal lifted from D8
ρ1622220-2-2-2-22000000000002-222-2-2-22    orthogonal lifted from D8
ρ172-2-220200-200-2i02i02-2-220000000000    complex lifted from C4○D4
ρ182-2-220200-2002i0-2i02-2-220000000000    complex lifted from C4○D4
ρ1922-2-2002i-2i0000000-22-22--2-2ζ16716ζ16516316716ζ1615169ζ16131611165163ζ165163ζ16716    complex lifted from C4○D16
ρ2022-2-200-2i2i0000000-22-22-2--216716165163ζ16716ζ1615169ζ16131611ζ165163ζ165163ζ16716    complex lifted from C4○D16
ρ2122-2-200-2i2i0000000-22-22-2--2ζ16716ζ16516316716ζ16716ζ165163165163ζ16131611ζ1615169    complex lifted from C4○D16
ρ2222-2-2002i-2i0000000-22-22--2-216716165163ζ16716ζ16716ζ165163ζ165163ζ16131611ζ1615169    complex lifted from C4○D16
ρ2322-2-2002i-2i00000002-22-2-2--2ζ16516316716165163ζ165163ζ1615169ζ16716ζ16716ζ16131611    complex lifted from C4○D16
ρ2422-2-2002i-2i00000002-22-2-2--2165163ζ16716ζ165163ζ16131611ζ1671616716ζ1615169ζ165163    complex lifted from C4○D16
ρ2522-2-200-2i2i00000002-22-2--2-2ζ16516316716165163ζ16131611ζ16716ζ16716ζ1615169ζ165163    complex lifted from C4○D16
ρ2622-2-200-2i2i00000002-22-2--2-2165163ζ16716ζ165163ζ165163ζ161516916716ζ16716ζ16131611    complex lifted from C4○D16
ρ274-4-440-400400000000000000000000    orthogonal lifted from C8⋊C22
ρ284-44-400000000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ294-44-4000000000002222-22-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q16.5D4
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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 51 45 55)(42 50 46 54)(43 49 47 53)(44 56 48 52)
(1 47 15 35)(2 48 16 36)(3 41 9 37)(4 42 10 38)(5 43 11 39)(6 44 12 40)(7 45 13 33)(8 46 14 34)(17 61 29 49)(18 62 30 50)(19 63 31 51)(20 64 32 52)(21 57 25 53)(22 58 26 54)(23 59 27 55)(24 60 28 56)
(1 39 5 35)(2 38 6 34)(3 37 7 33)(4 36 8 40)(9 41 13 45)(10 48 14 44)(11 47 15 43)(12 46 16 42)(17 54 21 50)(18 53 22 49)(19 52 23 56)(20 51 24 55)(25 62 29 58)(26 61 30 57)(27 60 31 64)(28 59 32 63)

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

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

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

Matrix representation of Q16.5D4 in GL4(𝔽17) generated by

14300
141400
0010
0001
,
1700
71600
0010
0001
,
4000
0400
00115
00116
,
4000
01300
00115
00016
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[1,7,0,0,7,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,15,16] >;

Q16.5D4 in GAP, Magma, Sage, TeX

Q_{16}._5D_4
% in TeX

G:=Group("Q16.5D4");
// GroupNames label

G:=SmallGroup(128,943);
// by ID

G=gap.SmallGroup(128,943);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,352,1684,438,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of Q16.5D4 in TeX

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