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G = C8.D8order 128 = 27

1st non-split extension by C8 of D8 acting via D8/C4=C22

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

Aliases: C8.1D8, C42.240C23, C4⋊C4.62D4, C82C816C2, C4.63(C2×D8), (C2×C8).92D4, (C2×D4).59D4, C86D4.1C2, C4⋊Q1614C2, C4⋊C8.30C22, D4⋊Q8.7C2, C4⋊Q8.61C22, C4.10D815C2, C2.11(C4⋊D8), (C4×C8).143C22, C2.6(C8.D4), (C4×D4).45C22, C4.122(C8⋊C22), C4.42(C8.C22), C2.12(D4.5D4), C22.201(C4⋊D4), (C2×C4).25(C4○D4), (C2×C4).1275(C2×D4), SmallGroup(128,421)

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=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 192 in 83 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×3], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×4], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], Q16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8, D4⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C2.D8 [×2], C4×D4, C4⋊Q8 [×2], C2×M4(2), C2×Q16 [×2], C4.10D8 [×2], C82C8, C86D4, D4⋊Q8 [×2], C4⋊Q16, C8.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C8.C22 [×2], C4⋊D8, C8.D4, D4.5D4, C8.D8

Character table of C8.D8

 class 12A2B2C2D4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 1111822224816164444888888
ρ111111111111111111111111    trivial
ρ211111111111-11-1-1-1-11-11-1-1-1    linear of order 2
ρ31111-111111-1111111-1-1-1-1-1-1    linear of order 2
ρ41111-111111-1-11-1-1-1-1-11-1111    linear of order 2
ρ51111-111111-1-1-11111111-1-11    linear of order 2
ρ61111-111111-11-1-1-1-1-11-1111-1    linear of order 2
ρ711111111111-1-11111-1-1-111-1    linear of order 2
ρ8111111111111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ9222202-22-2-20002-2-22000000    orthogonal lifted from D4
ρ102222-2-22-22-22000000000000    orthogonal lifted from D4
ρ11222202-22-2-2000-222-2000000    orthogonal lifted from D4
ρ1222222-22-22-2-2000000000000    orthogonal lifted from D4
ρ132-2-220-202000000-220-2-22002    orthogonal lifted from D8
ρ142-2-220-2020000002-202-2-2002    orthogonal lifted from D8
ρ152-2-220-202000000-22022-200-2    orthogonal lifted from D8
ρ162-2-220-2020000002-20-22200-2    orthogonal lifted from D8
ρ1722220-2-2-2-2200000000002i-2i0    complex lifted from C4○D4
ρ1822220-2-2-2-220000000000-2i2i0    complex lifted from C4○D4
ρ194-4-44040-4000000000000000    orthogonal lifted from C8⋊C22
ρ204-44-40040-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ214-44-400-40400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002200-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ2344-4-4000000000-220022000000    symplectic lifted from D4.5D4, Schur index 2

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

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

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

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

Matrix representation of C8.D8 in GL6(𝔽17)

100000
010000
000100
0013000
000004
000010
,
060000
1460000
000010
0000016
0016000
000100
,
060000
300000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,16,0,0],[0,3,0,0,0,0,6,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C8.D8 in GAP, Magma, Sage, TeX

C_8.D_8
% in TeX

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

G:=SmallGroup(128,421);
// by ID

G=gap.SmallGroup(128,421);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,387,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C8.D8 in TeX

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