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G = C8.2D8order 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), C82Q88C2, C8⋊C815C2, (C2×C8).38D4, C85D4.4C2, C4⋊Q1615C2, C2.9(C84D4), C4.4D8.7C2, C4⋊Q8.93C22, (C4×C8).155C22, C4.6(C8.C22), C2.11(C8.2D4), C41D4.52C22, C22.70(C41D4), (C2×C4).726(C2×D4), SmallGroup(128,454)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8.2D8
C1C2C22C2×C4C42C4×C8C8⋊C8 — C8.2D8
C1C22C42 — C8.2D8
C1C22C42 — C8.2D8
C1C22C22C42 — 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 [×2], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], Q8 [×6], C23, C42, C4⋊C4 [×6], C2×C8 [×6], SD16 [×4], Q16 [×4], C2×D4 [×3], C2×Q8 [×3], C4×C8, C4×C8 [×2], D4⋊C4 [×4], C2.D8 [×4], C41D4, C4⋊Q8, C4⋊Q8 [×2], C2×SD16 [×2], C2×Q16 [×2], C8⋊C8, C4.4D8 [×2], C85D4, C4⋊Q16, C82Q8 [×2], C8.2D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C8.C22 [×4], C84D4, C8.2D4 [×2], 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
ρ152-22-200-2002000022-220-222-2-2-20    orthogonal lifted from D8
ρ162-22-200200-200002-2-22-22-20-2202    orthogonal lifted from D8
ρ172-22-200-20020000222-20-2-2-2-2220    orthogonal lifted from D8
ρ182-22-200200-20000-22-222-2-20220-2    orthogonal lifted from D8
ρ192-22-200200-200002-22-22220-2-20-2    orthogonal lifted from D8
ρ202-22-200-20020000-2-2-22022-22-220    orthogonal lifted from D8
ρ212-22-200200-20000-222-2-2-2202-202    orthogonal lifted from D8
ρ222-22-200-20020000-2-22-202-2222-20    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 25 18 46 39 63 54 9)(2 30 19 43 40 60 55 14)(3 27 20 48 33 57 56 11)(4 32 21 45 34 62 49 16)(5 29 22 42 35 59 50 13)(6 26 23 47 36 64 51 10)(7 31 24 44 37 61 52 15)(8 28 17 41 38 58 53 12)
(1 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 17 13 21)(10 24 14 20)(11 23 15 19)(12 22 16 18)(33 64 37 60)(34 63 38 59)(35 62 39 58)(36 61 40 57)(41 50 45 54)(42 49 46 53)(43 56 47 52)(44 55 48 51)

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

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

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

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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