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G = C16⋊3D4  order 128 = 27

3rd semidirect product of C16 and D4 acting via D4/C2=C22

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

Aliases: C16⋊3D4, C8.6D8, C42.156D4, C16⋊5C4⋊4C2, C8.43(C2×D4), (C2×C4).48D8, C4.11(C2×D8), (C2×D16)⋊10C2, C8⋊4D4⋊16C2, (C2×SD32)⋊3C2, (C2×C8).138D4, C4.5(C4⋊1D4), C8.12D4⋊13C2, C2.16(C8⋊4D4), (C2×C8).548C23, (C2×C16).29C22, (C4×C8).170C22, (C2×D8).19C22, C22.134(C2×D8), C2.22(C16⋊C22), (C2×Q16).20C22, (C2×C4).816(C2×D4), SmallGroup(128,982)

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

Derived series
C1 — C2×C8 — C16⋊3D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C16⋊5C4 — C16⋊3D4
Lower central
C1 — C2 — C4 — C2×C8 — C16⋊3D4
Upper central
C1 — C22 — C42 — C4×C8 — C16⋊3D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16⋊3D4

Generators and relations for C16⋊3D4
 G = < a,b,c | a16=b4=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >

Subgroups: 328 in 99 conjugacy classes, 36 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C16, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C4×C8, C2×C16, D16, SD32, C4.4D4, C4⋊1D4, C2×D8, C2×D8, C2×D8, C2×SD16, C2×Q16, C16⋊5C4, C8⋊4D4, C8.12D4, C2×D16, C2×SD32, C16⋊3D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4⋊1D4, C2×D8, C8⋊4D4, C16⋊C22, C16⋊3D4

Character table of C16⋊3D4

 class 12A2B2C2D2E2F4A4B4C4D4E8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111116161622441622224444444444
ρ111111111111111111111111111    trivial
ρ21111-11-111-1-111111-1-1-11-11-111-1    linear of order 2
ρ311111-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111-1-111111-1-11-11-11-1-11    linear of order 2
ρ51111-1-1-11111-111111111111111    linear of order 2
ρ611111-1111-1-1-11111-1-1-11-11-111-1    linear of order 2
ρ71111-1111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111-111-1-1-11111-1-11-11-11-1-11    linear of order 2
ρ92-22-20002-20002-2-2200020-20-220    orthogonal lifted from D4
ρ10222200022-2-20-2-2-2-22200000000    orthogonal lifted from D4
ρ11222200022220-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-22-20002-2000-222-200-2020200-2    orthogonal lifted from D4
ρ132-22-20002-20002-2-22000-20202-20    orthogonal lifted from D4
ρ142-22-20002-2000-222-20020-20-2002    orthogonal lifted from D4
ρ152222000-2-2-220000000√2√2√2√2-√2-√2-√2-√2    orthogonal lifted from D8
ρ162222000-2-22-20000000-√2√2-√2√2√2-√2-√2√2    orthogonal lifted from D8
ρ172222000-2-2-220000000-√2-√2-√2-√2√2√2√2√2    orthogonal lifted from D8
ρ182222000-2-22-20000000√2-√2√2-√2-√2√2√2-√2    orthogonal lifted from D8
ρ192-22-2000-220000000-22√2-√2-√2√2√2-√2√2-√2    orthogonal lifted from D8
ρ202-22-2000-2200000002-2-√2-√2√2√2-√2-√2√2√2    orthogonal lifted from D8
ρ212-22-2000-2200000002-2√2√2-√2-√2√2√2-√2-√2    orthogonal lifted from D8
ρ222-22-2000-220000000-22-√2√2√2-√2-√2√2-√2√2    orthogonal lifted from D8
ρ234-4-4400000000-2√22√2-2√22√20000000000    orthogonal lifted from C16⋊C22
ρ2444-4-400000000-2√2-2√22√22√20000000000    orthogonal lifted from C16⋊C22
ρ254-4-44000000002√2-2√22√2-2√20000000000    orthogonal lifted from C16⋊C22
ρ2644-4-4000000002√22√2-2√2-2√20000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of C16⋊3D4
►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 31 48 64)(2 24 33 57)(3 17 34 50)(4 26 35 59)(5 19 36 52)(6 28 37 61)(7 21 38 54)(8 30 39 63)(9 23 40 56)(10 32 41 49)(11 25 42 58)(12 18 43 51)(13 27 44 60)(14 20 45 53)(15 29 46 62)(16 22 47 55)

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

Matrix representation of C16⋊3D4 ►in GL6(𝔽17)

120000
16160000
001612516
0051615
0012115
001612121
,
120000
16160000
000010
000001
001000
000100
,
16150000
010000
0015121
0051615
0012115
0015516

G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,16,5,12,16,0,0,12,16,1,12,0,0,5,1,1,12,0,0,16,5,5,1],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,1,5,12,1,0,0,5,16,1,5,0,0,12,1,1,5,0,0,1,5,5,16] >;
 

C16⋊3D4 in GAP, Magma, Sage, TeX 

C_{16}\rtimes_3D_4
 
% in TeX
 
G:=Group("C16:3D4");
 
// GroupNames label
 
G:=SmallGroup(128,982);
 
// by ID
 
G=gap.SmallGroup(128,982);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,723,100,1123,360,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Character table of C16⋊3D4 in TeX

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