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G = C82D4order 64 = 26

2nd semidirect product of C8 and D4 acting via D4/C2=C22

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

Aliases: C82D4, C23.17D4, (C2×D8)⋊7C2, C4.Q84C2, C4⋊D44C2, C4.57(C2×D4), (C2×C4).31D4, C4⋊C4.9C22, D4⋊C418C2, C4.11(C4○D4), (C2×M4(2))⋊2C2, (C2×C4).97C23, (C2×C8).16C22, C22.93(C2×D4), C2.21(C4⋊D4), C2.13(C8⋊C22), (C2×D4).18C22, (C22×C4).49C22, SmallGroup(64,150)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C82D4
C1C2C22C2×C4C22×C4C2×M4(2) — C82D4
C1C2C2×C4 — C82D4
C1C22C22×C4 — C82D4
C1C2C2C2×C4 — C82D4

Generators and relations for C82D4
 G = < a,b,c | a8=b4=c2=1, bab-1=a3, cac=a-1, cbc=b-1 >

Subgroups: 141 in 65 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], C22×C4, C2×D4 [×2], C2×D4 [×2], D4⋊C4 [×2], C4.Q8, C4⋊D4 [×2], C2×M4(2), C2×D8, C82D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, C4⋊D4, C8⋊C22 [×2], C82D4

Character table of C82D4

 class 12A2B2C2D2E2F4A4B4C4D4E8A8B8C8D
 size 1111488224884444
ρ11111111111111111    trivial
ρ2111111-1111-11-1-1-1-1    linear of order 2
ρ31111-1-1111-1-1111-1-1    linear of order 2
ρ41111-1-1-111-111-1-111    linear of order 2
ρ511111-111111-1-1-1-1-1    linear of order 2
ρ611111-1-1111-1-11111    linear of order 2
ρ71111-11111-1-1-1-1-111    linear of order 2
ρ81111-11-111-11-111-1-1    linear of order 2
ρ92222-200-2-22000000    orthogonal lifted from D4
ρ102-22-2000-2200000-22    orthogonal lifted from D4
ρ112222200-2-2-2000000    orthogonal lifted from D4
ρ122-22-2000-22000002-2    orthogonal lifted from D4
ρ132-22-20002-2000-2i2i00    complex lifted from C4○D4
ρ142-22-20002-20002i-2i00    complex lifted from C4○D4
ρ154-4-44000000000000    orthogonal lifted from C8⋊C22
ρ1644-4-4000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C82D4
On 32 points
Generators in S32
(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)
(1 13 27 19)(2 16 28 22)(3 11 29 17)(4 14 30 20)(5 9 31 23)(6 12 32 18)(7 15 25 21)(8 10 26 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 28)(26 27)(29 32)(30 31)

G:=sub<Sym(32)| (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), (1,13,27,19)(2,16,28,22)(3,11,29,17)(4,14,30,20)(5,9,31,23)(6,12,32,18)(7,15,25,21)(8,10,26,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,28)(26,27)(29,32)(30,31)>;

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), (1,13,27,19)(2,16,28,22)(3,11,29,17)(4,14,30,20)(5,9,31,23)(6,12,32,18)(7,15,25,21)(8,10,26,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,28)(26,27)(29,32)(30,31) );

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)], [(1,13,27,19),(2,16,28,22),(3,11,29,17),(4,14,30,20),(5,9,31,23),(6,12,32,18),(7,15,25,21),(8,10,26,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21),(25,28),(26,27),(29,32),(30,31)])

C82D4 is a maximal subgroup of
C23.SD16  M5(2).C22  C24.110D4  M4(2)⋊14D4  (C2×C8)⋊12D4  (C2×C8)⋊13D4  M4(2)⋊16D4  C42.260D4  C42.261D4  C24.125D4  C24.127D4  C24.130D4  C4.2+ 1+4  C4.142+ 1+4  C4.192+ 1+4  C42.299D4  C42.301D4  C42.304D4
 C23.D4p: C23.D8  D8⋊D4  C243D4  C403D4  C563D4 ...
 (C2p×D8)⋊C2: C42.255D4  C42.388C23  C42.391C23  C42.471C23  C42.474C23  C42.495C23  C42.496C23  C2412D4 ...
 C4⋊C4.D2p: C42.385C23  C4.2- 1+4  C42.26C23  C42.30C23  D89D4  Q1610D4  SD161D4  SD162D4 ...
C82D4 is a maximal quotient of
C24.76D4  C232D8
 C8⋊D4p: C82D8  C247D4  C82D20  C567D4 ...
 C4⋊C4.D2p: C82SD16  C84SD16  C82Q16  C42.252C23  C42.253C23  C24.84D4  C4⋊C4.106D4  C3⋊C8⋊D4 ...
 (C2×C8).D2p: C24.67D4  C2.(C82D4)  C8⋊(C4⋊C4)  (C2×D8)⋊10C4  (C2×C4)⋊3D8  C24.88D4  (C2×C4).21Q16  C243D4 ...

Matrix representation of C82D4 in GL6(𝔽17)

040000
400000
00111100
003000
001114314
003333
,
1300000
040000
0060116
00140011
0063143
00143314
,
040000
1300000
00111100
003600
001114314
00331414

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,11,3,11,3,0,0,11,0,14,3,0,0,0,0,3,3,0,0,0,0,14,3],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,6,14,6,14,0,0,0,0,3,3,0,0,11,0,14,3,0,0,6,11,3,14],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,11,3,11,3,0,0,11,6,14,3,0,0,0,0,3,14,0,0,0,0,14,14] >;

C82D4 in GAP, Magma, Sage, TeX

C_8\rtimes_2D_4
% in TeX

G:=Group("C8:2D4");
// GroupNames label

G:=SmallGroup(64,150);
// by ID

G=gap.SmallGroup(64,150);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,332,963,117]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^2=1,b*a*b^-1=a^3,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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Character table of C82D4 in TeX

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