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G = C8⋊5SD16  order 128 = 27

5th semidirect product of C8 and SD16 acting via SD16/C4=C22

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

Aliases: C8⋊5SD16, C42.661C23, C8⋊C8⋊9C2, C8⋊3Q8⋊18C2, C8⋊2Q8⋊25C2, (C2×C8).150D4, C8⋊5D4.3C2, C4.5(C2×SD16), C2.6(C8⋊3D4), C2.9(C8⋊5D4), C4.1(C8⋊C22), C4⋊Q8.85C22, (C4×C8).151C22, C4.SD16⋊24C2, C4.4D8.12C2, C2.6(C8.2D4), C4.1(C8.C22), C4⋊1D4.46C22, C22.62(C4⋊1D4), (C2×C4).718(C2×D4), SmallGroup(128,446)

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

Derived series
C1 — C42 — C8⋊5SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊C8 — C8⋊5SD16
Lower central
C1 — C22 — C42 — C8⋊5SD16
Upper central
C1 — C22 — C42 — C8⋊5SD16
Jennings
C1 — C22 — C22 — C42 — C8⋊5SD16

Generators and relations for C8⋊5SD16
 G = < a,b,c | a8=b8=c2=1, bab-1=a5, cac=a3, cbc=b3 >

Subgroups: 256 in 99 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C4⋊1D4, C4⋊Q8, C2×SD16, C8⋊C8, C4.4D8, C4.SD16, C8⋊5D4, C8⋊3Q8, C8⋊2Q8, C8⋊5SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4⋊1D4, C2×SD16, C8⋊C22, C8.C22, C8⋊5D4, C8⋊3D4, C8.2D4, C8⋊5SD16

Character table of C8⋊5SD16

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 111116222222161616444444444444
ρ111111111111111111111111111    trivial
ρ21111-1111111-111-1-111-1-11-1-11-1-1    linear of order 2
ρ311111111111-11-1-1-1-1-11-1-11-1-111    linear of order 2
ρ41111-111111111-111-1-1-11-1-11-1-1-1    linear of order 2
ρ51111-11111111-11-1-1-1-11-1-11-1-111    linear of order 2
ρ611111111111-1-1111-1-1-11-1-11-1-1-1    linear of order 2
ρ71111-1111111-1-1-1111111111111    linear of order 2
ρ8111111111111-1-1-1-111-1-11-1-11-1-1    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
ρ152-22-200200-20000-√-2√-2-√-2√-22-√-2-√-20√-2√-20-2    complex lifted from SD16
ρ162-22-200-20020000√-2√-2-√-2√-20-√-2√-22-√-2-√-2-20    complex lifted from SD16
ρ172-22-200200-20000-√-2√-2√-2-√-2-2-√-2√-20√-2-√-202    complex lifted from SD16
ρ182-22-200-20020000-√-2-√-2-√-2√-20√-2√-2-2√-2-√-220    complex lifted from SD16
ρ192-22-200200-20000√-2-√-2√-2-√-22√-2√-20-√-2-√-20-2    complex lifted from SD16
ρ202-22-200-20020000-√-2-√-2√-2-√-20√-2-√-22√-2√-2-20    complex lifted from SD16
ρ212-22-200200-20000√-2-√-2-√-2√-2-2√-2-√-20-√-2√-202    complex lifted from SD16
ρ222-22-200-20020000√-2√-2√-2-√-20-√-2-√-2-2-√-2√-220    complex lifted from SD16
ρ2344-4-4000040-4000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-40000-404000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-44040-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-440-404000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8⋊5SD16
►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 63 17 46 11 25 53 35)(2 60 18 43 12 30 54 40)(3 57 19 48 13 27 55 37)(4 62 20 45 14 32 56 34)(5 59 21 42 15 29 49 39)(6 64 22 47 16 26 50 36)(7 61 23 44 9 31 51 33)(8 58 24 41 10 28 52 38)

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

Matrix representation of C8⋊5SD16 ►in GL6(𝔽17)

100000
010000
00413512
004455
00125134
0012121313
,
5120000
550000
000010
000001
001000
000100
,
1600000
010000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,12,12,0,0,13,4,5,12,0,0,5,5,13,13,0,0,12,5,4,13],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

C8⋊5SD16 in GAP, Magma, Sage, TeX 

C_8\rtimes_5{\rm SD}_{16}
 
% in TeX
 
G:=Group("C8:5SD16");
 
// GroupNames label
 
G:=SmallGroup(128,446);
 
// by ID
 
G=gap.SmallGroup(128,446);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,100,1123,136,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^3>;
 
// generators/relations
 

Export

Character table of C8⋊5SD16 in TeX

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