Copied to
clipboard

G = C162D4order 128 = 27

2nd semidirect product of C16 and D4 acting via D4/C2=C22

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

Aliases: C162D4, C23.18D8, (C2×C4).38D8, (C2×SD32)⋊1C2, C163C412C2, C8.104(C2×D4), (C2×C8).216D4, C2.D1618C2, C4.22(C4○D8), C8.49(C4○D4), C87D4.10C2, (C2×M5(2))⋊2C2, (C2×D8).8C22, C2.Q3218C2, C8.18D433C2, C4.95(C4⋊D4), C2.23(C87D4), (C2×C8).525C23, (C2×C16).25C22, (C22×C4).351D4, C22.111(C2×D8), (C2×Q16).9C22, C2.13(C16⋊C22), C2.D8.10C22, C2.13(Q32⋊C2), (C22×C8).301C22, (C2×C4).793(C2×D4), SmallGroup(128,952)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C162D4
C1C2C4C2×C4C2×C8C22×C8C2×M5(2) — C162D4
C1C2C4C2×C8 — C162D4
C1C22C22×C4C22×C8 — C162D4
C1C2C2C2C2C4C4C2×C8 — C162D4

Generators and relations for C162D4
 G = < a,b,c | a16=b4=c2=1, bab-1=a-1, cac=a7, cbc=b-1 >

Subgroups: 216 in 80 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, C16 [×2], C16, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], D8 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C2×Q8, D4⋊C4, Q8⋊C4, C2.D8 [×2], C2×C16 [×2], M5(2) [×2], SD32 [×2], C4⋊D4, C22⋊Q8, C22×C8, C2×D8, C2×Q16, C2.D16, C2.Q32, C163C4, C87D4, C8.18D4, C2×M5(2), C2×SD32, C162D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4, C16⋊C22, Q32⋊C2, C162D4

Character table of C162D4

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111141622416161622224444444444
ρ111111111111111111111111111    trivial
ρ211111-1111-1-1-111111111111111    linear of order 2
ρ3111111111-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1111-11-1-11111-1-11-11-11-1-11    linear of order 2
ρ61111-1-111-1-1111111-1-11-11-11-1-11    linear of order 2
ρ71111-1111-1-11-11111-1-1-11-11-111-1    linear of order 2
ρ81111-1-111-11-111111-1-1-11-11-111-1    linear of order 2
ρ92-22-2002-20000-222-20020-20-2002    orthogonal lifted from D4
ρ10222220222000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ112-22-2002-20000-222-200-2020200-2    orthogonal lifted from D4
ρ122222-2022-2000-2-2-2-22200000000    orthogonal lifted from D4
ρ13222220-2-2-2000000000-2-2-2-22222    orthogonal lifted from D8
ρ14222220-2-2-20000000002222-2-2-2-2    orthogonal lifted from D8
ρ152222-20-2-220000000002-22-2-222-2    orthogonal lifted from D8
ρ162222-20-2-22000000000-22-222-2-22    orthogonal lifted from D8
ρ172-22-2002-200002-2-22000-2i02i02i-2i0    complex lifted from C4○D4
ρ182-22-2002-200002-2-220002i0-2i0-2i2i0    complex lifted from C4○D4
ρ192-22-200-22000000002i-2i-2--22-2-2--2-22    complex lifted from C4○D8
ρ202-22-200-2200000000-2i2i-2-22--2-2-2--22    complex lifted from C4○D8
ρ212-22-200-2200000000-2i2i2--2-2-22--2-2-2    complex lifted from C4○D8
ρ222-22-200-22000000002i-2i2-2-2--22-2--2-2    complex lifted from C4○D8
ρ234-4-4400000000-2222-22220000000000    orthogonal lifted from C16⋊C22
ρ244-4-440000000022-2222-220000000000    orthogonal lifted from C16⋊C22
ρ2544-4-400000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2644-4-4000000002222-22-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of C162D4
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 54 36 31)(2 53 37 30)(3 52 38 29)(4 51 39 28)(5 50 40 27)(6 49 41 26)(7 64 42 25)(8 63 43 24)(9 62 44 23)(10 61 45 22)(11 60 46 21)(12 59 47 20)(13 58 48 19)(14 57 33 18)(15 56 34 17)(16 55 35 32)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 52)(18 59)(19 50)(20 57)(21 64)(22 55)(23 62)(24 53)(25 60)(26 51)(27 58)(28 49)(29 56)(30 63)(31 54)(32 61)(33 47)(34 38)(35 45)(37 43)(39 41)(40 48)(42 46)

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

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

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

Matrix representation of C162D4 in GL6(𝔽17)

1390000
040000
0011000
007100
008463
0012179
,
120000
16160000
0000160
0063115
001000
0015714
,
100000
16160000
001000
0001600
0000160
00014161

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,9,4,0,0,0,0,0,0,1,7,8,12,0,0,10,1,4,1,0,0,0,0,6,7,0,0,0,0,3,9],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,6,1,1,0,0,0,3,0,5,0,0,16,1,0,7,0,0,0,15,0,14],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,14,0,0,0,0,16,16,0,0,0,0,0,1] >;

C162D4 in GAP, Magma, Sage, TeX

C_{16}\rtimes_2D_4
% in TeX

G:=Group("C16:2D4");
// GroupNames label

G:=SmallGroup(128,952);
// by ID

G=gap.SmallGroup(128,952);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,723,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations

Export

Character table of C162D4 in TeX

׿
×
𝔽