Copied to
clipboard

G = C163D4order 128 = 27

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

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

Aliases: C163D4, C8.6D8, C42.156D4, C165C44C2, C8.43(C2×D4), (C2×C4).48D8, C4.11(C2×D8), (C2×D16)⋊10C2, C84D416C2, (C2×SD32)⋊3C2, (C2×C8).138D4, C4.5(C41D4), C8.12D413C2, C2.16(C84D4), (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

C1C2×C8 — C163D4
C1C2C4C2×C4C2×C8C4×C8C165C4 — C163D4
C1C2C4C2×C8 — C163D4
C1C22C42C4×C8 — C163D4
C1C2C2C2C2C4C4C2×C8 — C163D4

Generators and relations for C163D4
 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 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], C16 [×4], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×8], SD16 [×2], Q16 [×2], C2×D4 [×5], C2×Q8, C4×C8, C2×C16 [×2], D16 [×4], SD32 [×4], C4.4D4, C41D4, C2×D8, C2×D8 [×2], C2×D8, C2×SD16, C2×Q16, C165C4, C84D4, C8.12D4, C2×D16 [×2], C2×SD32 [×2], C163D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C84D4, C16⋊C22 [×2], C163D4

Character table of C163D4

 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
ρ152222000-2-2-2200000002222-2-2-2-2    orthogonal lifted from D8
ρ162222000-2-22-20000000-22-222-2-22    orthogonal lifted from D8
ρ172222000-2-2-220000000-2-2-2-22222    orthogonal lifted from D8
ρ182222000-2-22-200000002-22-2-222-2    orthogonal lifted from D8
ρ192-22-2000-220000000-222-2-222-22-2    orthogonal lifted from D8
ρ202-22-2000-2200000002-2-2-222-2-222    orthogonal lifted from D8
ρ212-22-2000-2200000002-222-2-222-2-2    orthogonal lifted from D8
ρ222-22-2000-220000000-22-222-2-22-22    orthogonal lifted from D8
ρ234-4-4400000000-2222-22220000000000    orthogonal lifted from C16⋊C22
ρ2444-4-400000000-22-2222220000000000    orthogonal lifted from C16⋊C22
ρ254-4-440000000022-2222-220000000000    orthogonal lifted from C16⋊C22
ρ2644-4-4000000002222-22-220000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of C163D4
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 30 59 40)(2 23 60 33)(3 32 61 42)(4 25 62 35)(5 18 63 44)(6 27 64 37)(7 20 49 46)(8 29 50 39)(9 22 51 48)(10 31 52 41)(11 24 53 34)(12 17 54 43)(13 26 55 36)(14 19 56 45)(15 28 57 38)(16 21 58 47)
(1 50)(2 49)(3 64)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 26)(18 25)(19 24)(20 23)(21 22)(27 32)(28 31)(29 30)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(47 48)

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

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

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

Matrix representation of C163D4 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] >;

C163D4 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 C163D4 in TeX

׿
×
𝔽