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G = D4⋊4SD16  order 128 = 27

3rd semidirect product of D4 and SD16 acting via SD16/D4=C2

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

Aliases: D4⋊4SD16, C42.215C23, D4⋊C8⋊23C2, C4⋊C4.37D4, D4⋊Q8⋊5C2, C8⋊3Q8⋊14C2, (C2×D4).56D4, D4⋊6D4.3C2, C4.63(C4○D8), C4.37(C2×SD16), C4⋊Q8.35C22, D4.D4⋊34C2, C4.10D8⋊13C2, C4⋊C8.172C22, C4.41(C8⋊C22), (C4×C8).248C22, (C4×D4).42C22, C2.26(D4⋊D4), C2.17(C22⋊SD16), C22.181C22≀C2, C2.19(D4.10D4), (C2×C4).972(C2×D4), SmallGroup(128,386)

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

Derived series
C1 — C42 — D4⋊4SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4⋊6D4 — D4⋊4SD16
Lower central
C1 — C22 — C42 — D4⋊4SD16
Upper central
C1 — C22 — C42 — D4⋊4SD16
Jennings
C1 — C22 — C22 — C42 — D4⋊4SD16

Generators and relations for D4⋊4SD16
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd=a2b, dcd=c3 >

Subgroups: 288 in 120 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, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×SD16, C2×C4○D4, D4⋊C8, C4.10D8, D4.D4, D4⋊Q8, C8⋊3Q8, D4⋊6D4, D4⋊4SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C4○D8, C8⋊C22, D4⋊D4, C22⋊SD16, D4.10D4, D4⋊4SD16

Character table of D4⋊4SD16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111144822224448881644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-11111-1-111-11-1-1-1-1-11111    linear of order 2
ρ41111-1-1-11111-1-111-1111111-1-1-1-1    linear of order 2
ρ51111-1-111111111-1-1-1-11111-11-11    linear of order 2
ρ61111-1-111111111-1-1-11-1-1-1-11-11-1    linear of order 2
ρ7111111-11111-1-11-11-11-1-1-1-1-11-11    linear of order 2
ρ8111111-11111-1-11-11-1-111111-11-1    linear of order 2
ρ92222000-2-2-2-2002-202000000000    orthogonal lifted from D4
ρ102222000-2-2-2-200220-2000000000    orthogonal lifted from D4
ρ11222200-2-222-222-2000000000000    orthogonal lifted from D4
ρ122222-2-202-2-2200-2020000000000    orthogonal lifted from D4
ρ132222002-222-2-2-2-2000000000000    orthogonal lifted from D4
ρ1422222202-2-2200-20-20000000000    orthogonal lifted from D4
ρ152-2-22-220200-20000000-√-2√-2√-2-√-2√-20-√-20    complex lifted from SD16
ρ162-2-222-20200-20000000√-2-√-2-√-2√-2√-20-√-20    complex lifted from SD16
ρ1722-2-200002-202i-2i00000-√-2-√-2√-2√-20-√20√2    complex lifted from C4○D8
ρ1822-2-200002-20-2i2i00000-√-2-√-2√-2√-20√20-√2    complex lifted from C4○D8
ρ192-2-22-220200-20000000√-2-√-2-√-2√-2-√-20√-20    complex lifted from SD16
ρ2022-2-200002-202i-2i00000√-2√-2-√-2-√-20√20-√2    complex lifted from C4○D8
ρ212-2-222-20200-20000000-√-2√-2√-2-√-2-√-20√-20    complex lifted from SD16
ρ2222-2-200002-20-2i2i00000√-2√-2-√-2-√-20-√20√2    complex lifted from C4○D8
ρ234-4-44000-4004000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-40000-440000000000000000    orthogonal lifted from C8⋊C22
ρ254-44-400000000000000-22-220000    symplectic lifted from D4.10D4, Schur index 2
ρ264-44-4000000000000002-22-20000    symplectic lifted from D4.10D4, Schur index 2

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

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

Matrix representation of D4⋊4SD16 ►in GL4(𝔽17) generated by

13000
8400
0010
0001
,
2200
71500
00160
00016
,
131300
0400
00512
0055
,
1000
151600
0001
0010
G:=sub<GL(4,GF(17))| [13,8,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,7,0,0,2,15,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,13,4,0,0,0,0,5,5,0,0,12,5],[1,15,0,0,0,16,0,0,0,0,0,1,0,0,1,0] >;
 

D4⋊4SD16 in GAP, Magma, Sage, TeX 

D_4\rtimes_4{\rm SD}_{16}
 
% in TeX
 
G:=Group("D4:4SD16");
 
// GroupNames label
 
G:=SmallGroup(128,386);
 
// by ID
 
G=gap.SmallGroup(128,386);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,680,422,520,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d=a^2*b,d*c*d=c^3>;
 
// generators/relations
 

Export

Character table of D4⋊4SD16 in TeX

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