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G = C83SD16order 128 = 27

3rd semidirect product of C8 and SD16 acting via SD16/C4=C22

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

Aliases: C83SD16, C42.242C23, C4⋊C4.64D4, C83Q81C2, C81C821C2, (C2×C8).94D4, (C2×D4).60D4, C86D4.2C2, C4.60(C2×SD16), C4⋊Q8.63C22, C4.10D828C2, C2.10(C8⋊D4), C4⋊C8.186C22, C4.42(C8⋊C22), (C4×C8).145C22, D42Q8.10C2, C4.6Q1617C2, D4.D4.9C2, (C4×D4).46C22, C2.9(D4.D4), C4.72(C8.C22), C2.18(D4.3D4), C22.203(C4⋊D4), (C2×C4).27(C4○D4), (C2×C4).1277(C2×D4), SmallGroup(128,423)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C83SD16
C1C2C22C2×C4C42C4×D4C86D4 — C83SD16
C1C22C42 — C83SD16
C1C22C42 — C83SD16
C1C22C22C42 — C83SD16

Generators and relations for C83SD16
 G = < a,b,c | a8=b8=c2=1, bab-1=a-1, cac=a5, cbc=b3 >

Subgroups: 184 in 80 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×4], C22, C22 [×3], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×3], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×3], C4.Q8 [×3], C4×D4, C4⋊Q8 [×2], C2×M4(2), C2×SD16, C4.10D8, C4.6Q16, C81C8, C86D4, D4.D4, D42Q8, C83Q8, C83SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8⋊C22, C8.C22 [×2], D4.D4, C8⋊D4, D4.3D4, C83SD16

Character table of C83SD16

 class 12A2B2C2D4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 1111822224816164444888888
ρ111111111111111111111111    trivial
ρ211111111111-11-1-1-1-11-11-1-1-1    linear of order 2
ρ31111-111111-1111111-1-1-1-1-1-1    linear of order 2
ρ41111-111111-1-11-1-1-1-1-11-1111    linear of order 2
ρ51111-111111-1-1-11111111-1-11    linear of order 2
ρ61111-111111-11-1-1-1-1-11-1111-1    linear of order 2
ρ711111111111-1-11111-1-1-111-1    linear of order 2
ρ8111111111111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ9222202-22-2-20002-2-22000000    orthogonal lifted from D4
ρ102222-2-22-22-22000000000000    orthogonal lifted from D4
ρ11222202-22-2-2000-222-2000000    orthogonal lifted from D4
ρ1222222-22-22-2-2000000000000    orthogonal lifted from D4
ρ1322220-2-2-2-2200000000002i-2i0    complex lifted from C4○D4
ρ1422220-2-2-2-220000000000-2i2i0    complex lifted from C4○D4
ρ152-2-220-202000000-220--2--2-200-2    complex lifted from SD16
ρ162-2-220-202000000-220-2-2--200--2    complex lifted from SD16
ρ172-2-220-2020000002-20--2-2-200--2    complex lifted from SD16
ρ182-2-220-2020000002-20-2--2--200-2    complex lifted from SD16
ρ194-44-40040-400000000000000    orthogonal lifted from C8⋊C22
ρ204-44-400-40400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ214-4-44040-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002-200-2-2000000    complex lifted from D4.3D4
ρ2344-4-4000000000-2-2002-2000000    complex lifted from D4.3D4

Smallest permutation representation of C83SD16
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 25 46 60 49 13 38 17)(2 32 47 59 50 12 39 24)(3 31 48 58 51 11 40 23)(4 30 41 57 52 10 33 22)(5 29 42 64 53 9 34 21)(6 28 43 63 54 16 35 20)(7 27 44 62 55 15 36 19)(8 26 45 61 56 14 37 18)
(2 6)(4 8)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)(25 60)(26 57)(27 62)(28 59)(29 64)(30 61)(31 58)(32 63)(33 45)(34 42)(35 47)(36 44)(37 41)(38 46)(39 43)(40 48)(50 54)(52 56)

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

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

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

Matrix representation of C83SD16 in GL8(𝔽17)

130200000
0016160000
00400000
011300000
0000010012
00007050
000001000
00007000
,
607100000
1401000000
7121430000
553140000
0000161143
000016161414
000098116
00009911
,
10000000
1616000000
00100000
1300160000
00001000
000001600
00000010
000000016

G:=sub<GL(8,GF(17))| [13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2,16,4,13,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,7,0,7,0,0,0,0,10,0,10,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0],[6,14,7,5,0,0,0,0,0,0,12,5,0,0,0,0,7,10,14,3,0,0,0,0,10,0,3,14,0,0,0,0,0,0,0,0,16,16,9,9,0,0,0,0,1,16,8,9,0,0,0,0,14,14,1,1,0,0,0,0,3,14,16,1],[1,16,0,13,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;

C83SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_3{\rm SD}_{16}
% in TeX

G:=Group("C8:3SD16");
// GroupNames label

G:=SmallGroup(128,423);
// by ID

G=gap.SmallGroup(128,423);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,64,422,387,1123,136,2804,718,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^3>;
// generators/relations

Export

Character table of C83SD16 in TeX

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