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G = Q16⋊5D4  order 128 = 27

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

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

Aliases: Q16⋊5D4, C42.445C23, C4.1322+ (1+4), (D4×Q8)⋊5C2, C8⋊6D4⋊9C2, C8⋊5D4⋊9C2, C2.62(D42), C8.10(C2×D4), C8⋊D4⋊33C2, C4⋊C4.362D4, (C4×Q16)⋊32C2, Q8.25(C2×D4), Q8⋊D4⋊19C2, C4⋊SD16⋊20C2, C4⋊2Q16⋊37C2, (C2×D4).312D4, C4⋊C8.97C22, C4⋊3(C8.C22), Q8⋊6D4.4C2, C22⋊C4.45D4, C4.92(C22×D4), D4.7D4⋊41C2, 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⋊1D4.77C22, C4⋊D4.64C22, 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

Derived series
C1 — C2×C4 — Q16⋊5D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C22×Q8 — C2×C8.C22 — Q16⋊5D4
Lower central
C1 — C2 — C2×C4 — Q16⋊5D4
Upper central
C1 — C22 — C4×D4 — Q16⋊5D4
Jennings
C1 — C2 — C2 — C2×C4 — Q16⋊5D4

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], C4⋊1D4, C4⋊1D4, 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], C8⋊6D4, C4×Q16, Q8⋊D4 [×2], D4.7D4 [×2], C4⋊SD16, C4⋊2Q16, C8⋊D4 [×2], C8⋊5D4, D4×Q8, Q8⋊6D4, C2×C8.C22 [×2], Q16⋊5D4

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, Q16⋊5D4

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

Character table of Q16⋊5D4

 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√-22√-200    complex lifted from D4○SD16
ρ2944-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16

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
 

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