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G = C810SD16order 128 = 27

1st semidirect product of C8 and SD16 acting via SD16/D4=C2

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

Aliases: D4.1D8, C810SD16, C42.224C23, C81C86C2, C82Q84C2, (C8×D4).5C2, C4.33(C2×D8), C4⋊C4.199D4, (C2×C8).343D4, (C2×D4).190D4, C4.86(C4○D8), C2.8(C87D4), (C4×C8).53C22, C4.58(C2×SD16), C4⋊Q8.47C22, C4.10D824C2, C4⋊C8.177C22, D4.D4.7C2, C2.7(D4.D4), C2.8(D4.5D4), (C4×D4).278C22, C4.112(C8.C22), C22.185(C4⋊D4), (C2×C4).9(C4○D4), (C2×C4).1259(C2×D4), SmallGroup(128,405)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C810SD16
C1C2C22C2×C4C42C4×D4C8×D4 — C810SD16
C1C22C42 — C810SD16
C1C22C42 — C810SD16
C1C22C22C42 — C810SD16

Generators and relations for C810SD16
 G = < a,b,c | a8=b8=c2=1, bab-1=a-1, ac=ca, cbc=b3 >

Subgroups: 192 in 86 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×4], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×4], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C2.D8 [×2], C4×D4, C4⋊Q8 [×2], C22×C8, C2×SD16 [×2], C4.10D8 [×2], C81C8, C8×D4, D4.D4 [×2], C82Q8, C810SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C2×SD16, C4○D8, C8.C22, D4.D4, C87D4, D4.5D4, C810SD16

Character table of C810SD16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111442222444161622224444448888
ρ111111111111111111111111111111    trivial
ρ21111-1-11111-11-11-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ31111-1-11111-11-1-1-11111-1-1-111-11111    linear of order 2
ρ41111111111111-11-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ511111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ61111-1-11111-11-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ71111-1-11111-11-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ81111111111111-1-11111111111-1-1-1-1    linear of order 2
ρ92222-2-2-22-222-220000000000000000    orthogonal lifted from D4
ρ102222002-22-20-20002222000-2-200000    orthogonal lifted from D4
ρ112222002-22-20-2000-2-2-2-20002200000    orthogonal lifted from D4
ρ12222222-22-22-2-2-20000000000000000    orthogonal lifted from D4
ρ1322-2-22-2020-2000002-2-22-222-22-20000    orthogonal lifted from D8
ρ1422-2-2-22020-2000002-2-222-2-2-2220000    orthogonal lifted from D8
ρ1522-2-22-2020-200000-222-22-2-22-220000    orthogonal lifted from D8
ρ1622-2-2-22020-200000-222-2-2222-2-20000    orthogonal lifted from D8
ρ17222200-2-2-2-2020000000-2i-2i2i002i0000    complex lifted from C4○D4
ρ1822-2-2000-202-2i02i002-2-22--2-2--22-2-20000    complex lifted from C4○D8
ρ192-22-200-2020000002-22-2000000-2-2--2--2    complex lifted from SD16
ρ202-22-200-202000000-22-22000000--2-2-2--2    complex lifted from SD16
ρ212-22-200-2020000002-22-2000000--2--2-2-2    complex lifted from SD16
ρ22222200-2-2-2-20200000002i2i-2i00-2i0000    complex lifted from C4○D4
ρ2322-2-2000-2022i0-2i00-222-2--2-2--2-22-20000    complex lifted from C4○D8
ρ2422-2-2000-2022i0-2i002-2-22-2--2-22-2--20000    complex lifted from C4○D8
ρ252-22-200-202000000-22-22000000-2--2--2-2    complex lifted from SD16
ρ2622-2-2000-202-2i02i00-222-2-2--2-2-22--20000    complex lifted from C4○D8
ρ274-44-40040-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-4-44000000000002222-22-220000000000    symplectic lifted from D4.5D4, Schur index 2
ρ294-4-4400000000000-22-2222220000000000    symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of C810SD16
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 16 35 59 49 30 43 18)(2 15 36 58 50 29 44 17)(3 14 37 57 51 28 45 24)(4 13 38 64 52 27 46 23)(5 12 39 63 53 26 47 22)(6 11 40 62 54 25 48 21)(7 10 33 61 55 32 41 20)(8 9 34 60 56 31 42 19)
(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)

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

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

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

Matrix representation of C810SD16 in GL4(𝔽17) generated by

15000
12800
0010
0001
,
16200
16100
00010
001210
,
1000
11600
0010
00116
G:=sub<GL(4,GF(17))| [15,12,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[16,16,0,0,2,1,0,0,0,0,0,12,0,0,10,10],[1,1,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C810SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_{10}{\rm SD}_{16}
% in TeX

G:=Group("C8:10SD16");
// GroupNames label

G:=SmallGroup(128,405);
// by ID

G=gap.SmallGroup(128,405);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,288,422,1123,136,2804,718,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^3>;
// generators/relations

Export

Character table of C810SD16 in TeX

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