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G = Q162D4order 128 = 27

1st semidirect product of Q16 and D4 acting via D4/C4=C2

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

Aliases: Q162D4, C43SD32, C42.137D4, C4⋊C1610C2, (C4×Q16)⋊2C2, (C2×C8).66D4, C8.71(C2×D4), C84D4.8C2, (C2×C4).148D8, C2.D1611C2, C2.7(C2×SD32), (C2×SD32)⋊13C2, C8.89(C4○D4), (C4×C8).62C22, (C2×D8).3C22, C4.50(C4⋊D4), C2.19(C4⋊D8), C4.15(C8⋊C22), (C2×C8).515C23, (C2×C16).39C22, C22.101(C2×D8), C2.10(C16⋊C22), C2.D8.157C22, (C2×Q16).109C22, (C2×C4).783(C2×D4), SmallGroup(128,939)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — Q162D4
C1C2C4C8C2×C8C2×Q16C4×Q16 — Q162D4
C1C2C4C2×C8 — Q162D4
C1C22C42C4×C8 — Q162D4
C1C2C2C2C2C4C4C2×C8 — Q162D4

Generators and relations for Q162D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Subgroups: 268 in 87 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×2], C8, C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×3], C23 [×2], C16 [×2], C42, C42, C4⋊C4 [×2], C2×C8 [×2], D8 [×6], Q16 [×2], Q16, C2×D4 [×4], C2×Q8, C4×C8, Q8⋊C4, C2.D8, C2×C16 [×2], SD32 [×4], C4×Q8, C41D4, C2×D8 [×2], C2×D8, C2×Q16, C2.D16 [×2], C4⋊C16, C4×Q16, C84D4, C2×SD32 [×2], Q162D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, SD32 [×2], C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C2×SD32, C16⋊C22, Q162D4

Character table of Q162D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111161622224888822224444444444
ρ111111111111111111111111111111    trivial
ρ21111-111-1-11-1-11-111111-1-1111-11-1-1-1    linear of order 2
ρ31111-1-111111-1-1-1-111111111111111    linear of order 2
ρ411111-11-1-11-11-11-11111-1-1111-11-1-1-1    linear of order 2
ρ511111111111-1-1-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-111-1-11-11-11-11111-1-1-1-1-11-1111    linear of order 2
ρ71111-1-1111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ811111-11-1-11-1-11-111111-1-1-1-1-11-1111    linear of order 2
ρ92222002-2-22-20000-2-2-2-22200000000    orthogonal lifted from D4
ρ102-2-2200200-20-2020-222-20000000000    orthogonal lifted from D4
ρ11222200222220000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-2200200-2020-20-222-20000000000    orthogonal lifted from D4
ρ13222200-2-2-2-2200000000002-222-2-2-22    orthogonal lifted from D8
ρ14222200-222-2-20000000000-22-222-2-22    orthogonal lifted from D8
ρ15222200-222-2-200000000002-22-2-222-2    orthogonal lifted from D8
ρ16222200-2-2-2-220000000000-22-2-2222-2    orthogonal lifted from D8
ρ172-2-2200200-2002i0-2i2-2-220000000000    complex lifted from C4○D4
ρ182-2-2200200-200-2i02i2-2-220000000000    complex lifted from C4○D4
ρ192-22-2000-220000002-22-2-22ζ1615169ζ165163ζ16716ζ165163ζ16131611ζ1615169ζ16716ζ16131611    complex lifted from SD32
ρ202-22-2000-220000002-22-2-22ζ16716ζ16131611ζ1615169ζ16131611ζ165163ζ16716ζ1615169ζ165163    complex lifted from SD32
ρ212-22-2000-22000000-22-222-2ζ16131611ζ1615169ζ165163ζ1615169ζ16716ζ16131611ζ165163ζ16716    complex lifted from SD32
ρ222-22-2000-22000000-22-222-2ζ165163ζ16716ζ16131611ζ16716ζ1615169ζ165163ζ16131611ζ1615169    complex lifted from SD32
ρ232-22-20002-20000002-22-22-2ζ16716ζ16131611ζ1615169ζ165163ζ165163ζ1615169ζ16716ζ16131611    complex lifted from SD32
ρ242-22-20002-20000002-22-22-2ζ1615169ζ165163ζ16716ζ16131611ζ16131611ζ16716ζ1615169ζ165163    complex lifted from SD32
ρ252-22-20002-2000000-22-22-22ζ16131611ζ1615169ζ165163ζ16716ζ16716ζ165163ζ16131611ζ1615169    complex lifted from SD32
ρ262-22-20002-2000000-22-22-22ζ165163ζ16716ζ16131611ζ1615169ζ1615169ζ16131611ζ165163ζ16716    complex lifted from SD32
ρ274-4-4400-40040000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000-22-2222220000000000    orthogonal lifted from C16⋊C22
ρ2944-4-4000000000002222-22-220000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of Q162D4
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 43 11 35)(2 44 12 36)(3 45 13 37)(4 46 14 38)(5 47 15 39)(6 48 16 40)(7 41 9 33)(8 42 10 34)(17 57 25 49)(18 58 26 50)(19 59 27 51)(20 60 28 52)(21 61 29 53)(22 62 30 54)(23 63 31 55)(24 64 32 56)
(1 35)(2 34)(3 33)(4 40)(5 39)(6 38)(7 37)(8 36)(9 45)(10 44)(11 43)(12 42)(13 41)(14 48)(15 47)(16 46)(17 50)(18 49)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 58)(26 57)(27 64)(28 63)(29 62)(30 61)(31 60)(32 59)

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

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

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

Matrix representation of Q162D4 in GL4(𝔽17) generated by

16000
01600
00314
0033
,
01600
16000
00101
0017
,
01300
13000
00160
00016
,
01300
4000
00160
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,3,3,0,0,14,3],[0,16,0,0,16,0,0,0,0,0,10,1,0,0,1,7],[0,13,0,0,13,0,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,13,0,0,0,0,0,16,0,0,0,0,1] >;

Q162D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_2D_4
% in TeX

G:=Group("Q16:2D4");
// GroupNames label

G:=SmallGroup(128,939);
// by ID

G=gap.SmallGroup(128,939);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,64,422,352,1684,438,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of Q162D4 in TeX

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