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G = C86SD16order 128 = 27

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

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

Aliases: C86SD16, C42.662C23, C83Q82C2, C8⋊C810C2, C85D418C2, (C2×C8).151D4, C4.6(C2×SD16), C4.4D825C2, C84D4.14C2, C2.7(C83D4), C4.2(C8⋊C22), C4⋊Q8.86C22, C2.10(C85D4), (C4×C8).152C22, C41D4.47C22, C22.63(C41D4), (C2×C4).719(C2×D4), SmallGroup(128,447)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C86SD16
C1C2C22C2×C4C42C4×C8C8⋊C8 — C86SD16
C1C22C42 — C86SD16
C1C22C42 — C86SD16
C1C22C22C42 — C86SD16

Generators and relations for C86SD16
 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 [×2], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×12], Q8 [×4], C23 [×2], C42, C4⋊C4 [×4], C2×C8 [×6], D8 [×4], SD16 [×8], C2×D4 [×6], C2×Q8 [×2], C4×C8, C4×C8 [×2], D4⋊C4 [×4], C4.Q8 [×2], C41D4 [×2], C4⋊Q8 [×2], C2×D8 [×2], C2×SD16 [×4], C8⋊C8, C4.4D8 [×2], C85D4 [×2], C84D4, C83Q8, C86SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×4], C2×D4 [×3], C41D4, C2×SD16 [×2], C8⋊C22 [×4], C85D4, C83D4 [×2], C86SD16

Character table of C86SD16

 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 C86SD16
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 49 24 48 35 29 63 15)(2 54 17 45 36 26 64 12)(3 51 18 42 37 31 57 9)(4 56 19 47 38 28 58 14)(5 53 20 44 39 25 59 11)(6 50 21 41 40 30 60 16)(7 55 22 46 33 27 61 13)(8 52 23 43 34 32 62 10)
(2 8)(3 7)(4 6)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 64)(24 63)(33 37)(34 36)(38 40)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)

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

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

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

Matrix representation of C86SD16 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] >;

C86SD16 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 C86SD16 in TeX

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