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G = C8⋊D8order 128 = 27

1st semidirect product of C8 and D8 acting via D8/C4=C22

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

Aliases: C81D8, C42.236C23, C85D41C2, C86D41C2, C4⋊C4.58D4, C81C819C2, (C2×C8).88D4, C4.61(C2×D8), D4⋊Q87C2, (C2×D4).57D4, C4⋊D8.7C2, C4.D89C2, C2.8(C8⋊D4), C2.9(C4⋊D8), C4⋊C8.28C22, C4⋊Q8.59C22, C4.10D814C2, (C4×C8).139C22, (C4×D4).43C22, C4.120(C8⋊C22), C41D4.32C22, C4.70(C8.C22), C2.16(D4.3D4), C22.197(C4⋊D4), (C2×C4).21(C4○D4), (C2×C4).1271(C2×D4), SmallGroup(128,417)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8⋊D8
C1C2C22C2×C4C42C4×D4C86D4 — C8⋊D8
C1C22C42 — C8⋊D8
C1C22C42 — C8⋊D8
C1C22C22C42 — C8⋊D8

Generators and relations for C8⋊D8
 G = < a,b,c | a8=b8=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >

Subgroups: 248 in 91 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×3], C22, C22 [×6], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×4], D4 [×7], Q8 [×2], C23 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], D8 [×2], SD16 [×4], C22×C4, C2×D4, C2×D4 [×3], C2×Q8, C4×C8, C22⋊C8, D4⋊C4 [×2], C4⋊C8 [×3], C2.D8, C4×D4, C41D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16 [×2], C4.D8, C4.10D8, C81C8, C86D4, C4⋊D8, D4⋊Q8, C85D4, C8⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22 [×2], C8.C22, C4⋊D8, C8⋊D4, D4.3D4, C8⋊D8

Character table of C8⋊D8

 class 12A2B2C2D2E4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J
 size 1111816222248164444888888
ρ111111111111111111111111    trivial
ρ21111-1111111-111111-1-1-1-1-1-1    linear of order 2
ρ311111-11111111-1-1-1-11-11-1-1-1    linear of order 2
ρ41111-1-111111-11-1-1-1-1-11-1111    linear of order 2
ρ511111-1111111-11111-1-1-1-111    linear of order 2
ρ61111-1-111111-1-111111111-1-1    linear of order 2
ρ7111111111111-1-1-1-1-1-11-11-1-1    linear of order 2
ρ81111-1111111-1-1-1-1-1-11-11-111    linear of order 2
ρ92222002-22-2-2002-2-22000000    orthogonal lifted from D4
ρ102222002-22-2-200-222-2000000    orthogonal lifted from D4
ρ11222220-22-22-2-200000000000    orthogonal lifted from D4
ρ122222-20-22-22-2200000000000    orthogonal lifted from D4
ρ132-2-2200-202000002-2022-2-200    orthogonal lifted from D8
ρ142-2-2200-20200000-220-222-200    orthogonal lifted from D8
ρ152-2-2200-202000002-20-2-22200    orthogonal lifted from D8
ρ162-2-2200-20200000-2202-2-2200    orthogonal lifted from D8
ρ17222200-2-2-2-220000000000-2i2i    complex lifted from C4○D4
ρ18222200-2-2-2-2200000000002i-2i    complex lifted from C4○D4
ρ194-4-440040-400000000000000    orthogonal lifted from C8⋊C22
ρ204-44-400040-40000000000000    orthogonal lifted from C8⋊C22
ρ214-44-4000-4040000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002-200-2-2000000    complex lifted from D4.3D4
ρ2344-4-4000000000-2-2002-2000000    complex lifted from D4.3D4

Smallest permutation representation of C8⋊D8
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 12 35 29 55 59 21 43)(2 11 36 28 56 58 22 42)(3 10 37 27 49 57 23 41)(4 9 38 26 50 64 24 48)(5 16 39 25 51 63 17 47)(6 15 40 32 52 62 18 46)(7 14 33 31 53 61 19 45)(8 13 34 30 54 60 20 44)
(2 4)(3 7)(6 8)(9 42)(10 45)(11 48)(12 43)(13 46)(14 41)(15 44)(16 47)(17 39)(18 34)(19 37)(20 40)(21 35)(22 38)(23 33)(24 36)(25 63)(26 58)(27 61)(28 64)(29 59)(30 62)(31 57)(32 60)(49 53)(50 56)(52 54)

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

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

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

Matrix representation of C8⋊D8 in GL6(𝔽17)

1600000
0160000
002111515
005111615
0011266
001111215
,
060000
1460000
000010
0010016
001000
0001610
,
100000
1160000
001000
0011600
000010
00201616

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,2,5,11,1,0,0,11,11,2,11,0,0,15,16,6,12,0,0,15,15,6,15],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,1,0,0,1,0,0,0,16,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,2,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C8⋊D8 in GAP, Magma, Sage, TeX

C_8\rtimes D_8
% in TeX

G:=Group("C8:D8");
// GroupNames label

G:=SmallGroup(128,417);
// by ID

G=gap.SmallGroup(128,417);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C8⋊D8 in TeX

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