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G = C8.7D8  order 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, C16⋊5C4⋊5C2, C8.44(C2×D4), (C2×C4).49D8, C4.12(C2×D8), (C2×Q32)⋊10C2, (C2×C8).139D4, C4⋊Q16⋊17C2, C4.6(C4⋊1D4), (C2×SD32).2C2, C2.17(C8⋊4D4), (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

Derived series
C1 — C2×C8 — C8.7D8
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C16⋊5C4 — C8.7D8
Lower central
C1 — C2 — C4 — C2×C8 — C8.7D8
Upper central
C1 — C22 — C42 — C4×C8 — C8.7D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C4×C8, C2×C16, SD32, Q32, C4.4D4, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C2×Q16, C2×Q16, C16⋊5C4, C4⋊Q16, C8.12D4, C2×SD32, C2×Q32, C8.7D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4⋊1D4, C2×D8, C8⋊4D4, Q32⋊C2, 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-√2√2√2-√2-√2√2-√2√2    orthogonal lifted from D8
ρ162-22-202-20000000002-2√2√2-√2-√2√2√2-√2-√2    orthogonal lifted from D8
ρ1722220-2-22-2000000000√2√2√2√2-√2-√2-√2-√2    orthogonal lifted from D8
ρ1822220-2-22-2000000000-√2-√2-√2-√2√2√2√2√2    orthogonal lifted from D8
ρ192-22-202-20000000002-2-√2-√2√2√2-√2-√2√2√2    orthogonal lifted from D8
ρ202-22-202-2000000000-22√2-√2-√2√2√2-√2√2-√2    orthogonal lifted from D8
ρ2122220-2-2-22000000000√2-√2√2-√2-√2√2√2-√2    orthogonal lifted from D8
ρ2222220-2-2-22000000000-√2√2-√2√2√2-√2-√2√2    orthogonal lifted from D8
ρ234-4-4400000000-2√22√2-2√22√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2444-4-400000000-2√2-2√22√22√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ254-4-44000000002√2-2√22√2-2√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2644-4-4000000002√22√2-2√2-2√20000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of C8.7D8
►On 64 points
Generators in S64
(1 58 45 19 9 50 37 27)(2 51 46 28 10 59 38 20)(3 60 47 21 11 52 39 29)(4 53 48 30 12 61 40 22)(5 62 33 23 13 54 41 31)(6 55 34 32 14 63 42 24)(7 64 35 25 15 56 43 17)(8 57 36 18 16 49 44 26)

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

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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