Copied to
clipboard

G = D4⋊SD16  order 128 = 27

1st semidirect product of D4 and SD16 acting via SD16/Q8=C2

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

Aliases: D4⋊5SD16, C42.183C23, D4⋊C8⋊15C2, Q8⋊C8⋊15C2, C4⋊C4.48D4, C8⋊5D4⋊12C2, D4⋊3Q8⋊1C2, C4⋊D8.1C2, C4⋊C8.1C22, (C2×Q8).42D4, C4⋊Q8.5C22, C4⋊SD16⋊31C2, (C2×D4).249D4, C4.51(C4○D8), C4.6Q16⋊6C2, C4.23(C2×SD16), C4.31(C8⋊C22), (C4×C8).240C22, (C4×D4).19C22, C4⋊1D4.7C22, (C4×Q8).19C22, C2.15(D4⋊4D4), C2.14(D4⋊D4), C2.10(Q8⋊D4), C4.55(C8.C22), C22.149C22≀C2, (C2×C4).940(C2×D4), SmallGroup(128,354)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — D4⋊SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4⋊3Q8 — D4⋊SD16
Lower central
C1 — C22 — C42 — D4⋊SD16
Upper central
C1 — C22 — C42 — D4⋊SD16
Jennings
C1 — C22 — C22 — C42 — D4⋊SD16

Generators and relations for D4⋊SD16
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=dad=a-1, ac=ca, cbc-1=dbd=ab, dcd=c3 >

Subgroups: 304 in 118 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, D4⋊C4, C4⋊C8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, D4⋊C8, Q8⋊C8, C4.6Q16, C4⋊D8, C4⋊SD16, C8⋊5D4, D4⋊3Q8, D4⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, Q8⋊D4, D4⋊D4, D4⋊4D4, D4⋊SD16

Character table of D4⋊SD16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111144162222444888844448888
ρ111111111111111111111111111    trivial
ρ21111-1-1111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ3111111111111-1-11-1-1-1-1-1-1-11-1-11    linear of order 2
ρ41111-1-111111111-11-1-1-1-1-1-1-111-1    linear of order 2
ρ5111111-111111-1-11-1-1-11111-111-1    linear of order 2
ρ61111-1-1-11111111-11-1-111111-1-11    linear of order 2
ρ7111111-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-111111-1-1-1-111-1-1-1-11111    linear of order 2
ρ92222000-2-2-2-2200002-200000000    orthogonal lifted from D4
ρ102222-2-20-2-222-200200000000000    orthogonal lifted from D4
ρ112222000-2-2-2-220000-2200000000    orthogonal lifted from D4
ρ12222200022-2-2-2-2-2020000000000    orthogonal lifted from D4
ρ132222220-2-222-200-200000000000    orthogonal lifted from D4
ρ14222200022-2-2-2220-20000000000    orthogonal lifted from D4
ρ1522-2-22-2000-220000000-√-2-√-2√-2√-20√-2-√-20    complex lifted from SD16
ρ1622-2-2-22000-220000000√-2√-2-√-2-√-20√-2-√-20    complex lifted from SD16
ρ172-2-220002-2000-2i2i0000-√-2√-2√-2-√-2-√200√2    complex lifted from C4○D8
ρ182-2-220002-20002i-2i0000√-2-√-2-√-2√-2-√200√2    complex lifted from C4○D8
ρ1922-2-22-2000-220000000√-2√-2-√-2-√-20-√-2√-20    complex lifted from SD16
ρ202-2-220002-2000-2i2i0000√-2-√-2-√-2√-2√200-√2    complex lifted from C4○D8
ρ2122-2-2-22000-220000000-√-2-√-2√-2√-20-√-2√-20    complex lifted from SD16
ρ222-2-220002-20002i-2i0000-√-2√-2√-2-√-2√200-√2    complex lifted from C4○D8
ρ234-44-400000000000000-22-220000    orthogonal lifted from D4⋊4D4
ρ244-44-4000000000000002-22-20000    orthogonal lifted from D4⋊4D4
ρ254-4-44000-4400000000000000000    orthogonal lifted from C8⋊C22
ρ2644-4-4000004-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of D4⋊SD16
►On 64 points
Generators in S64
(1 57 17 34)(2 58 18 35)(3 59 19 36)(4 60 20 37)(5 61 21 38)(6 62 22 39)(7 63 23 40)(8 64 24 33)(9 54 45 31)(10 55 46 32)(11 56 47 25)(12 49 48 26)(13 50 41 27)(14 51 42 28)(15 52 43 29)(16 53 44 30)

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

Matrix representation of D4⋊SD16 ►in GL4(𝔽17) generated by

1000
0100
0001
00160
,
16000
01600
00143
0033
,
5500
12500
001212
00512
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,14,3,0,0,3,3],[5,12,0,0,5,5,0,0,0,0,12,5,0,0,12,12],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
 

D4⋊SD16 in GAP, Magma, Sage, TeX 

D_4\rtimes {\rm SD}_{16}
 
% in TeX
 
G:=Group("D4:SD16");
 
// GroupNames label
 
G:=SmallGroup(128,354);
 
// by ID
 
G=gap.SmallGroup(128,354);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,232,422,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=d*b*d=a*b,d*c*d=c^3>;
 
// generators/relations
 

Export

Character table of D4⋊SD16 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁