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G = C8.7D8order 128 = 27

7th non-split extension by C8 of D8 acting via D8/C4=C22

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

Aliases: C8.7D8, C16.2D4, C42.157D4, C165C45C2, C8.44(C2×D4), (C2×C4).49D8, C4.12(C2×D8), (C2×Q32)⋊10C2, (C2×C8).139D4, C4⋊Q1617C2, C4.6(C41D4), (C2×SD32).2C2, C2.17(C84D4), (C4×C8).171C22, (C2×C8).549C23, (C2×C16).30C22, C8.12D4.6C2, (C2×D8).20C22, C22.135(C2×D8), C2.22(Q32⋊C2), (C2×Q16).21C22, (C2×C4).817(C2×D4), SmallGroup(128,983)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.7D8
C1C2C4C2×C4C2×C8C4×C8C165C4 — C8.7D8
C1C2C4C2×C8 — C8.7D8
C1C22C42C4×C8 — C8.7D8
C1C2C2C2C2C4C4C2×C8 — C8.7D8

Generators and relations for C8.7D8
 G = < a,b,c | a8=1, b8=c2=a4, bab-1=a5, cac-1=a3, cbc-1=a4b7 >

Subgroups: 232 in 89 conjugacy classes, 36 normal (16 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, C16 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×2], SD16 [×2], Q16 [×8], C2×D4, C2×Q8 [×3], C4×C8, C2×C16 [×2], SD32 [×4], Q32 [×4], C4.4D4, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C2×Q16 [×2], C2×Q16, C165C4, C4⋊Q16, C8.12D4, C2×SD32 [×2], C2×Q32 [×2], C8.7D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C84D4, Q32⋊C2 [×2], C8.7D8

Character table of C8.7D8

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111116224416161622224444444444
ρ111111111111111111111111111    trivial
ρ21111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111-1-1-11-11111-1-1-11-11-111-1    linear of order 2
ρ41111111-1-1-1-111111-1-11-11-11-1-11    linear of order 2
ρ51111-11111-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-11111-1-1-111111111111111    linear of order 2
ρ71111-111-1-111-11111-1-11-11-11-1-11    linear of order 2
ρ81111-111-1-11-111111-1-1-11-11-111-1    linear of order 2
ρ92222022-2-2000-2-2-2-22200000000    orthogonal lifted from D4
ρ102-22-20-22000002-2-2200020-20-220    orthogonal lifted from D4
ρ11222202222000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-22-20-2200000-222-20020-20-2002    orthogonal lifted from D4
ρ132-22-20-22000002-2-22000-20202-20    orthogonal lifted from D4
ρ142-22-20-2200000-222-200-2020200-2    orthogonal lifted from D4
ρ152-22-202-2000000000-22-222-2-22-22    orthogonal lifted from D8
ρ162-22-202-20000000002-222-2-222-2-2    orthogonal lifted from D8
ρ1722220-2-22-20000000002222-2-2-2-2    orthogonal lifted from D8
ρ1822220-2-22-2000000000-2-2-2-22222    orthogonal lifted from D8
ρ192-22-202-20000000002-2-2-222-2-222    orthogonal lifted from D8
ρ202-22-202-2000000000-222-2-222-22-2    orthogonal lifted from D8
ρ2122220-2-2-220000000002-22-2-222-2    orthogonal lifted from D8
ρ2222220-2-2-22000000000-22-222-2-22    orthogonal lifted from D8
ρ234-4-4400000000-2222-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2444-4-400000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ254-4-440000000022-2222-220000000000    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 C8.7D8
On 64 points
Generators in S64
(1 24 48 62 9 32 40 54)(2 17 33 55 10 25 41 63)(3 26 34 64 11 18 42 56)(4 19 35 57 12 27 43 49)(5 28 36 50 13 20 44 58)(6 21 37 59 14 29 45 51)(7 30 38 52 15 22 46 60)(8 23 39 61 16 31 47 53)
(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 9 62)(2 53 10 61)(3 52 11 60)(4 51 12 59)(5 50 13 58)(6 49 14 57)(7 64 15 56)(8 63 16 55)(17 39 25 47)(18 38 26 46)(19 37 27 45)(20 36 28 44)(21 35 29 43)(22 34 30 42)(23 33 31 41)(24 48 32 40)

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

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

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

Matrix representation of C8.7D8 in GL6(𝔽17)

0160000
100000
000033
0000143
00141400
0031400
,
0160000
100000
0011846
00911114
004669
0011486
,
010000
100000
00001414
0000143
003300
0031400

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,3,14,0,0,0,0,3,3,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,11,9,4,11,0,0,8,11,6,4,0,0,4,11,6,8,0,0,6,4,9,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,14,14,0,0,0,0,14,3,0,0] >;

C8.7D8 in GAP, Magma, Sage, TeX

C_8._7D_8
% in TeX

G:=Group("C8.7D8");
// GroupNames label

G:=SmallGroup(128,983);
// by ID

G=gap.SmallGroup(128,983);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,723,100,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^3,c*b*c^-1=a^4*b^7>;
// generators/relations

Export

Character table of C8.7D8 in TeX

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