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G = C85SD16order 128 = 27

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

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

Aliases: C85SD16, C42.661C23, C8⋊C89C2, C83Q818C2, C82Q825C2, (C2×C8).150D4, C85D4.3C2, C4.5(C2×SD16), C2.6(C83D4), C2.9(C85D4), C4.1(C8⋊C22), C4⋊Q8.85C22, (C4×C8).151C22, C4.SD1624C2, C4.4D8.12C2, C2.6(C8.2D4), C4.1(C8.C22), C41D4.46C22, C22.62(C41D4), (C2×C4).718(C2×D4), SmallGroup(128,446)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C85SD16
C1C2C22C2×C4C42C4×C8C8⋊C8 — C85SD16
C1C22C42 — C85SD16
C1C22C42 — C85SD16
C1C22C22C42 — C85SD16

Generators and relations for C85SD16
 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 [×3], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×3], D4 [×6], Q8 [×6], C23, C42, C4⋊C4 [×6], C2×C8 [×6], SD16 [×8], C2×D4 [×3], C2×Q8 [×3], C4×C8 [×3], D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C41D4, C4⋊Q8 [×3], C2×SD16 [×4], C8⋊C8, C4.4D8, C4.SD16, C85D4 [×2], C83Q8, C82Q8, C85SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×4], C2×D4 [×3], C41D4, C2×SD16 [×2], C8⋊C22 [×2], C8.C22 [×2], C85D4, C83D4, C8.2D4, C85SD16

Character table of C85SD16

 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 C85SD16
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 54 46 39 25 18 9)(2 60 55 43 40 30 19 14)(3 57 56 48 33 27 20 11)(4 62 49 45 34 32 21 16)(5 59 50 42 35 29 22 13)(6 64 51 47 36 26 23 10)(7 61 52 44 37 31 24 15)(8 58 53 41 38 28 17 12)
(2 4)(3 7)(6 8)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)(17 51)(18 54)(19 49)(20 52)(21 55)(22 50)(23 53)(24 56)(33 37)(34 40)(36 38)(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), (1,63,54,46,39,25,18,9)(2,60,55,43,40,30,19,14)(3,57,56,48,33,27,20,11)(4,62,49,45,34,32,21,16)(5,59,50,42,35,29,22,13)(6,64,51,47,36,26,23,10)(7,61,52,44,37,31,24,15)(8,58,53,41,38,28,17,12), (2,4)(3,7)(6,8)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30)(17,51)(18,54)(19,49)(20,52)(21,55)(22,50)(23,53)(24,56)(33,37)(34,40)(36,38)(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), (1,63,54,46,39,25,18,9)(2,60,55,43,40,30,19,14)(3,57,56,48,33,27,20,11)(4,62,49,45,34,32,21,16)(5,59,50,42,35,29,22,13)(6,64,51,47,36,26,23,10)(7,61,52,44,37,31,24,15)(8,58,53,41,38,28,17,12), (2,4)(3,7)(6,8)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30)(17,51)(18,54)(19,49)(20,52)(21,55)(22,50)(23,53)(24,56)(33,37)(34,40)(36,38)(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)], [(1,63,54,46,39,25,18,9),(2,60,55,43,40,30,19,14),(3,57,56,48,33,27,20,11),(4,62,49,45,34,32,21,16),(5,59,50,42,35,29,22,13),(6,64,51,47,36,26,23,10),(7,61,52,44,37,31,24,15),(8,58,53,41,38,28,17,12)], [(2,4),(3,7),(6,8),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30),(17,51),(18,54),(19,49),(20,52),(21,55),(22,50),(23,53),(24,56),(33,37),(34,40),(36,38),(41,64),(42,59),(43,62),(44,57),(45,60),(46,63),(47,58),(48,61)])

Matrix representation of C85SD16 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] >;

C85SD16 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 C85SD16 in TeX

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