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G = D4.SD16  order 128 = 27

The non-split extension by D4 of SD16 acting via SD16/D4=C2

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

Aliases: D4.4SD16, C42.196C23, D4⋊C8⋊19C2, C4⋊C4.26D4, (C2×D4).49D4, D4⋊2Q8⋊32C2, D4⋊6D4.1C2, C4.34(C4○D8), (C4×C8).18C22, C4.SD16⋊1C2, C4.29(C2×SD16), C4⋊Q8.17C22, C4.10D8⋊19C2, C4⋊C8.164C22, C4.35(C8⋊C22), (C4×D4).28C22, C4.34(C8.C22), C2.16(D4.8D4), C2.16(D4.7D4), C22.162C22≀C2, C2.12(C22⋊SD16), (C2×C4).953(C2×D4), SmallGroup(128,367)

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⋊6D4 — 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=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=a2b, dcd-1=a2c3 >

Subgroups: 280 in 117 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, D4⋊C8, C4.10D8, D4⋊2Q8, C4.SD16, D4⋊6D4, D4.SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C4○D8, C8⋊C22, C8.C22, C22⋊SD16, D4.7D4, D4.8D4, D4.SD16

Character table of D4.SD16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111144822224448881644448888
ρ111111111111111111111111111    trivial
ρ21111-1-1-111111-1-1-111-1-1-1-1-11111    linear of order 2
ρ31111-1-111111111-1-1-11-1-1-1-1-11-11    linear of order 2
ρ4111111-111111-1-11-1-1-11111-11-11    linear of order 2
ρ511111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-1-111111-1-1-11111111-1-1-1-1    linear of order 2
ρ71111-1-111111111-1-1-1-111111-11-1    linear of order 2
ρ8111111-111111-1-11-1-11-1-1-1-11-11-1    linear of order 2
ρ922220022-22-2-2-2-2000000000000    orthogonal lifted from D4
ρ102222000-2-2-2-220002-2000000000    orthogonal lifted from D4
ρ11222200-22-22-2-222000000000000    orthogonal lifted from D4
ρ122222220-22-22-200-200000000000    orthogonal lifted from D4
ρ132222-2-20-22-22-200200000000000    orthogonal lifted from D4
ρ142222000-2-2-2-22000-22000000000    orthogonal lifted from D4
ρ1522-2-200020-200-2i2i0000√2√2-√2-√2-√-20√-20    complex lifted from C4○D8
ρ1622-2-200020-200-2i2i0000-√2-√2√2√2√-20-√-20    complex lifted from C4○D8
ρ172-22-2-220020-20000000√-2-√-2-√-2√-20√-20-√-2    complex lifted from SD16
ρ182-22-22-20020-20000000-√-2√-2√-2-√-20√-20-√-2    complex lifted from SD16
ρ192-22-2-220020-20000000-√-2√-2√-2-√-20-√-20√-2    complex lifted from SD16
ρ2022-2-200020-2002i-2i0000-√2-√2√2√2-√-20√-20    complex lifted from C4○D8
ρ212-22-22-20020-20000000√-2-√-2-√-2√-20-√-20√-2    complex lifted from SD16
ρ2222-2-200020-2002i-2i0000√2√2-√2-√2√-20-√-20    complex lifted from C4○D8
ρ234-44-40000-404000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-4000-4040000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-44000000000000002i-2i2i-2i0000    complex lifted from D4.8D4
ρ264-4-4400000000000000-2i2i-2i2i0000    complex lifted from D4.8D4

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

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

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

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

1000
0100
00130
0004
,
1000
0100
00013
0040
,
2700
0800
0020
0009
,
131500
16400
0004
0040
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,13,0],[2,0,0,0,7,8,0,0,0,0,2,0,0,0,0,9],[13,16,0,0,15,4,0,0,0,0,0,4,0,0,4,0] >;
 

D4.SD16 in GAP, Magma, Sage, TeX 

D_4.{\rm SD}_{16}
 
% in TeX
 
G:=Group("D4.SD16");
 
// GroupNames label
 
G:=SmallGroup(128,367);
 
// by ID
 
G=gap.SmallGroup(128,367);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
 
// generators/relations
 

Export

Character table of D4.SD16 in TeX

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