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G = C8.2D8  order 128 = 27

2nd non-split extension by C8 of D8 acting via D8/C4=C22

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

Aliases: C8.2D8, C42.669C23, C4.6(C2×D8), C8⋊2Q8⋊8C2, C8⋊C8⋊15C2, (C2×C8).38D4, C8⋊5D4.4C2, C4⋊Q16⋊15C2, C2.9(C8⋊4D4), C4.4D8.7C2, C4⋊Q8.93C22, (C4×C8).155C22, C4.6(C8.C22), C2.11(C8.2D4), C4⋊1D4.52C22, C22.70(C4⋊1D4), (C2×C4).726(C2×D4), SmallGroup(128,454)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — C8.2D8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊C8 — C8.2D8
Lower central
C1 — C22 — C42 — C8.2D8
Upper central
C1 — C22 — C42 — C8.2D8
Jennings
C1 — C22 — C22 — C42 — C8.2D8

Generators and relations for C8.2D8
 G = < a,b,c | a8=b8=1, c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b-1 >

Subgroups: 256 in 99 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C4×C8, C4×C8, D4⋊C4, C2.D8, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×Q16, C8⋊C8, C4.4D8, C8⋊5D4, C4⋊Q16, C8⋊2Q8, C8.2D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4⋊1D4, C2×D8, C8.C22, C8⋊4D4, C8.2D4, C8.2D8

Character table of C8.2D8

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 111116222222161616444444444444
ρ111111111111111111111111111    trivial
ρ211111111111-11-1-1-1-1-11-1-11-1-111    linear of order 2
ρ31111-11111111-11-1-1-1-11-1-11-1-111    linear of order 2
ρ41111-1111111-1-1-1111111111111    linear of order 2
ρ51111-111111111-1-1-111-1-11-1-11-1-1    linear of order 2
ρ61111-1111111-11111-1-1-11-1-11-1-1-1    linear of order 2
ρ7111111111111-1-111-1-1-11-1-11-1-1-1    linear of order 2
ρ811111111111-1-11-1-111-1-11-1-11-1-1    linear of order 2
ρ922220-22-2-22-20000000200-200-22    orthogonal lifted from D4
ρ10222202-22-2-2-20002-2000-2002000    orthogonal lifted from D4
ρ1122220-2-2-22-2200000-2-200200200    orthogonal lifted from D4
ρ1222220-2-2-22-22000002200-200-200    orthogonal lifted from D4
ρ1322220-22-2-22-20000000-2002002-2    orthogonal lifted from D4
ρ14222202-22-2-2-2000-22000200-2000    orthogonal lifted from D4
ρ152-22-200-20020000√2√2-√2√20-√2√22-√2-√2-20    orthogonal lifted from D8
ρ162-22-200200-20000√2-√2-√2√2-2√2-√20-√2√202    orthogonal lifted from D8
ρ172-22-200-20020000√2√2√2-√20-√2-√2-2-√2√220    orthogonal lifted from D8
ρ182-22-200200-20000-√2√2-√2√22-√2-√20√2√20-2    orthogonal lifted from D8
ρ192-22-200200-20000√2-√2√2-√22√2√20-√2-√20-2    orthogonal lifted from D8
ρ202-22-200-20020000-√2-√2-√2√20√2√2-2√2-√220    orthogonal lifted from D8
ρ212-22-200200-20000-√2√2√2-√2-2-√2√20√2-√202    orthogonal lifted from D8
ρ222-22-200-20020000-√2-√2√2-√20√2-√22√2√2-20    orthogonal lifted from D8
ρ2344-4-4000040-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-4-44040-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-440-404000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-40000-404000000000000000    symplectic lifted from C8.C22, Schur index 2

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

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

Matrix representation of C8.2D8 ►in GL6(𝔽17)

100000
010000
0014307
001414100
00100314
0001033
,
0110000
3110000
000010
000001
0001600
001000
,
6110000
3110000
0001033
00100314
003370
00314010

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,14,10,0,0,0,3,14,0,10,0,0,0,10,3,3,0,0,7,0,14,3],[0,3,0,0,0,0,11,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0],[6,3,0,0,0,0,11,11,0,0,0,0,0,0,0,10,3,3,0,0,10,0,3,14,0,0,3,3,7,0,0,0,3,14,0,10] >;
 

C8.2D8 in GAP, Magma, Sage, TeX 

C_8._2D_8
 
% in TeX
 
G:=Group("C8.2D8");
 
// GroupNames label
 
G:=SmallGroup(128,454);
 
// by ID
 
G=gap.SmallGroup(128,454);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,64,422,387,436,1123,136,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^8=1,c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Character table of C8.2D8 in TeX

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