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G = Q165D4order 128 = 27

4th semidirect product of Q16 and D4 acting via D4/C4=C2

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

Aliases: Q165D4, C42.445C23, C4.1322+ 1+4, C2.62D42, (D4×Q8)⋊5C2, C86D49C2, C85D49C2, C8.10(C2×D4), C8⋊D433C2, C4⋊C4.362D4, (C4×Q16)⋊32C2, Q8.25(C2×D4), C4⋊SD1620C2, Q8⋊D419C2, C42Q1637C2, (C2×D4).312D4, C4⋊C8.97C22, C43(C8.C22), Q86D4.4C2, C22⋊C4.45D4, C4.92(C22×D4), D4.7D441C2, C4⋊C4.217C23, (C2×C4).476C24, (C2×C8).286C23, (C4×C8).191C22, C23.318(C2×D4), C4⋊Q8.136C22, C2.62(D4○SD16), (C2×D4).214C23, (C4×D4).150C22, C4⋊D4.64C22, C41D4.77C22, C22⋊C8.75C22, (C4×Q8).142C22, (C2×Q8).198C23, C2.D8.223C22, C22⋊Q8.63C22, D4⋊C4.68C22, (C22×C4).326C23, (C2×Q16).161C22, (C2×SD16).93C22, C22.736(C22×D4), Q8⋊C4.177C22, (C22×Q8).333C22, (C2×M4(2)).107C22, (C2×C4).159(C2×D4), (C2×C8.C22)⋊33C2, C2.73(C2×C8.C22), (C2×C4○D4).191C22, SmallGroup(128,2010)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q165D4
C1C2C22C2×C4C2×Q8C22×Q8C2×C8.C22 — Q165D4
C1C2C2×C4 — Q165D4
C1C22C4×D4 — Q165D4
C1C2C2C2×C4 — Q165D4

Generators and relations for Q165D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 488 in 241 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×12], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×15], Q8 [×4], Q8 [×10], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×10], Q16 [×4], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×8], C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C41D4, C41D4, C4⋊Q8, C4⋊Q8, C2×M4(2) [×2], C2×SD16 [×6], C2×Q16, C2×Q16 [×2], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4 [×2], C86D4, C4×Q16, Q8⋊D4 [×2], D4.7D4 [×2], C4⋊SD16, C42Q16, C8⋊D4 [×2], C85D4, D4×Q8, Q86D4, C2×C8.C22 [×2], Q165D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8.C22 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C8.C22, D4○SD16, Q165D4

Character table of Q165D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-111-1-111-11-111-111-11-1-111-1-11-1    linear of order 2
ρ31111-1-1-1-111111-11-11-1-1111-1-1-1-1-111    linear of order 2
ρ411111-1-11-111-1111-1-1-1-1-11-11-1-1111-1    linear of order 2
ρ5111111-1-11111-11-1111111-11-1-1-1-1-1-1    linear of order 2
ρ61111-11-11-111-1-1-1-11-111-111-1-1-111-11    linear of order 2
ρ71111-1-1111111-1-1-1-11-1-111-1-11111-1-1    linear of order 2
ρ811111-11-1-111-1-11-1-1-1-1-1-111111-1-1-11    linear of order 2
ρ9111111-1-11111-11-111-1-1-1-1-1-1111111    linear of order 2
ρ101111-11-11-111-1-1-1-11-1-1-11-11111-1-11-1    linear of order 2
ρ111111-1-1111111-1-1-1-1111-1-1-11-1-1-1-111    linear of order 2
ρ1211111-11-1-111-1-11-1-1-1111-11-1-1-1111-1    linear of order 2
ρ1311111111111111111-1-1-1-11-1-1-1-1-1-1-1    linear of order 2
ρ141111-111-1-111-11-111-1-1-11-1-11-1-111-11    linear of order 2
ρ151111-1-1-1-111111-11-1111-1-1111111-1-1    linear of order 2
ρ1611111-1-11-111-1111-1-1111-1-1-111-1-1-11    linear of order 2
ρ1722222-200-2-2-2-20-2022000000000000    orthogonal lifted from D4
ρ182-22-2000002-20-20200-2200002-20000    orthogonal lifted from D4
ρ192222-2-2002-2-220202-2000000000000    orthogonal lifted from D4
ρ202-22-2000002-2020-200-220000-220000    orthogonal lifted from D4
ρ212-22-2000002-2020-2002-200002-20000    orthogonal lifted from D4
ρ222222-2200-2-2-2-2020-22000000000000    orthogonal lifted from D4
ρ232-22-2000002-20-202002-20000-220000    orthogonal lifted from D4
ρ24222222002-2-220-20-2-2000000000000    orthogonal lifted from D4
ρ254-44-400000-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-440000400-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-440000-400400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of Q165D4
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 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 36 13 40)(10 35 14 39)(11 34 15 38)(12 33 16 37)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 34 19 60)(2 35 20 61)(3 36 21 62)(4 37 22 63)(5 38 23 64)(6 39 24 57)(7 40 17 58)(8 33 18 59)(9 49 41 30)(10 50 42 31)(11 51 43 32)(12 52 44 25)(13 53 45 26)(14 54 46 27)(15 55 47 28)(16 56 48 29)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 57)(10 62)(11 59)(12 64)(13 61)(14 58)(15 63)(16 60)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)

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

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

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

Matrix representation of Q165D4 in GL6(𝔽17)

1600000
0160000
0000125
000055
00121200
0012500
,
1600000
0160000
0012500
005500
000055
0000512
,
440000
0130000
000001
000010
0001600
0016000
,
100000
15160000
0000016
0000160
0001600
0016000

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

Q165D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_5D_4
% in TeX

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

G:=SmallGroup(128,2010);
// by ID

G=gap.SmallGroup(128,2010);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,346,248,2804,1411,375,172]);
// Polycyclic

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

Export

Character table of Q165D4 in TeX

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