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G = C8⋊6SD16  order 128 = 27

6th semidirect product of C8 and SD16 acting via SD16/C4=C22

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

Aliases: C8⋊6SD16, C42.662C23, C8⋊3Q8⋊2C2, C8⋊C8⋊10C2, C8⋊5D4⋊18C2, (C2×C8).151D4, C4.6(C2×SD16), C4.4D8⋊25C2, C8⋊4D4.14C2, C2.7(C8⋊3D4), C4.2(C8⋊C22), C4⋊Q8.86C22, C2.10(C8⋊5D4), (C4×C8).152C22, C4⋊1D4.47C22, C22.63(C4⋊1D4), (C2×C4).719(C2×D4), SmallGroup(128,447)

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

Derived series
C1 — C42 — C8⋊6SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊C8 — C8⋊6SD16
Lower central
C1 — C22 — C42 — C8⋊6SD16
Upper central
C1 — C22 — C42 — C8⋊6SD16
Jennings
C1 — C22 — C22 — C42 — C8⋊6SD16

Generators and relations for C8⋊6SD16
 G = < a,b,c | a8=b8=c2=1, bab-1=a5, cac=a-1, cbc=b3 >

Subgroups: 320 in 110 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, D8, SD16, C2×D4, C2×Q8, C4×C8, C4×C8, D4⋊C4, C4.Q8, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C8⋊C8, C4.4D8, C8⋊5D4, C8⋊4D4, C8⋊3Q8, C8⋊6SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4⋊1D4, C2×SD16, C8⋊C22, C8⋊5D4, C8⋊3D4, C8⋊6SD16

Character table of C8⋊6SD16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L
 size 111116162222221616444444444444
ρ111111111111111111111111111    trivial
ρ21111-1-111111111-1-1-1-11-1-11-1-111    linear of order 2
ρ311111-11111111-1-1-111-1-11-1-11-1-1    linear of order 2
ρ41111-111111111-111-1-1-11-1-11-1-1-1    linear of order 2
ρ511111-1111111-1111-1-1-11-1-11-1-1-1    linear of order 2
ρ61111-11111111-11-1-111-1-11-1-11-1-1    linear of order 2
ρ7111111111111-1-1-1-1-1-11-1-11-1-111    linear of order 2
ρ81111-1-1111111-1-1111111111111    linear of order 2
ρ9222200-22-2-22-2000000200-200-22    orthogonal lifted from D4
ρ10222200-2-2-22-220000-2-200200200    orthogonal lifted from D4
ρ112222002-22-2-2-200-22000200-2000    orthogonal lifted from D4
ρ12222200-2-2-22-2200002200-200-200    orthogonal lifted from D4
ρ13222200-22-2-22-2000000-2002002-2    orthogonal lifted from D4
ρ142222002-22-2-2-2002-2000-2002000    orthogonal lifted from D4
ρ152-22-2000200-2000-√-2√-2-√-2√-22-√-2-√-20√-2√-20-2    complex lifted from SD16
ρ162-22-2000-2002000-√-2-√-2-√-2√-20√-2√-2-2√-2-√-220    complex lifted from SD16
ρ172-22-2000-2002000√-2√-2-√-2√-20-√-2√-22-√-2-√-2-20    complex lifted from SD16
ρ182-22-2000200-2000-√-2√-2√-2-√-2-2-√-2√-20√-2-√-202    complex lifted from SD16
ρ192-22-2000-2002000√-2√-2√-2-√-20-√-2-√-2-2-√-2√-220    complex lifted from SD16
ρ202-22-2000200-2000√-2-√-2√-2-√-22√-2√-20-√-2-√-20-2    complex lifted from SD16
ρ212-22-2000200-2000√-2-√-2-√-2√-2-2√-2-√-20-√-2√-202    complex lifted from SD16
ρ222-22-2000-2002000-√-2-√-2√-2-√-20√-2-√-22√-2√-2-20    complex lifted from SD16
ρ234-4-4400000-40400000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400-40400000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-440000040-400000000000000    orthogonal lifted from C8⋊C22
ρ2644-4-40040-400000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C8⋊6SD16
►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 20 48 35 29 52 15)(2 64 21 45 36 26 53 12)(3 61 22 42 37 31 54 9)(4 58 23 47 38 28 55 14)(5 63 24 44 39 25 56 11)(6 60 17 41 40 30 49 16)(7 57 18 46 33 27 50 13)(8 62 19 43 34 32 51 10)

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

Matrix representation of C8⋊6SD16 ►in GL6(𝔽17)

1600000
0160000
0000107
005007
00012512
00125512
,
5120000
550000
0010150
0000161
0000160
0001160
,
010000
100000
001000
0011600
000010
0010016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,12,0,0,0,0,12,5,0,0,10,0,5,5,0,0,7,7,12,12],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

C8⋊6SD16 in GAP, Magma, Sage, TeX 

C_8\rtimes_6{\rm SD}_{16}
 
% in TeX
 
G:=Group("C8:6SD16");
 
// GroupNames label
 
G:=SmallGroup(128,447);
 
// by ID
 
G=gap.SmallGroup(128,447);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,387,100,1123,136,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^3>;
 
// generators/relations
 

Export

Character table of C8⋊6SD16 in TeX

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