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G = C84SD16order 128 = 27

4th semidirect product of C8 and SD16 acting via SD16/C4=C22

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

Aliases: C84SD16, C42.244C23, C4⋊C4.66D4, C82C817C2, (C2×C8).96D4, C82Q823C2, (C2×D4).61D4, C86D4.3C2, C2.8(C82D4), C4.61(C2×SD16), C4⋊Q8.65C22, C4.10D829C2, C4⋊C8.187C22, C4.71(C8⋊C22), (C4×C8).147C22, (C4×D4).47C22, D4.D4.10C2, C2.10(D4.D4), C2.14(D4.5D4), C4.116(C8.C22), C22.205(C4⋊D4), (C2×C4).29(C4○D4), (C2×C4).1279(C2×D4), SmallGroup(128,425)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C84SD16
C1C2C22C2×C4C42C4×D4C86D4 — C84SD16
C1C22C42 — C84SD16
C1C22C42 — C84SD16
C1C22C22C42 — C84SD16

Generators and relations for C84SD16
 G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a5, cbc=b3 >

Subgroups: 192 in 83 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×3], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×4], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], 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], C2×M4(2), C2×SD16 [×2], C4.10D8 [×2], C82C8, C86D4, D4.D4 [×2], C82Q8, C84SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8⋊C22 [×2], C8.C22, D4.D4, C82D4, D4.5D4, C84SD16

Character table of C84SD16

 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    orthogonal lifted from C8⋊C22
ρ214-4-44040-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2244-4-40000000002200-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ2344-4-4000000000-220022000000    symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of C84SD16
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 31 47 58 49 16 34 19)(2 26 48 61 50 11 35 22)(3 29 41 64 51 14 36 17)(4 32 42 59 52 9 37 20)(5 27 43 62 53 12 38 23)(6 30 44 57 54 15 39 18)(7 25 45 60 55 10 40 21)(8 28 46 63 56 13 33 24)
(2 6)(4 8)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)(25 60)(26 57)(27 62)(28 59)(29 64)(30 61)(31 58)(32 63)(33 42)(34 47)(35 44)(36 41)(37 46)(38 43)(39 48)(40 45)(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), (1,31,47,58,49,16,34,19)(2,26,48,61,50,11,35,22)(3,29,41,64,51,14,36,17)(4,32,42,59,52,9,37,20)(5,27,43,62,53,12,38,23)(6,30,44,57,54,15,39,18)(7,25,45,60,55,10,40,21)(8,28,46,63,56,13,33,24), (2,6)(4,8)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19)(25,60)(26,57)(27,62)(28,59)(29,64)(30,61)(31,58)(32,63)(33,42)(34,47)(35,44)(36,41)(37,46)(38,43)(39,48)(40,45)(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), (1,31,47,58,49,16,34,19)(2,26,48,61,50,11,35,22)(3,29,41,64,51,14,36,17)(4,32,42,59,52,9,37,20)(5,27,43,62,53,12,38,23)(6,30,44,57,54,15,39,18)(7,25,45,60,55,10,40,21)(8,28,46,63,56,13,33,24), (2,6)(4,8)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19)(25,60)(26,57)(27,62)(28,59)(29,64)(30,61)(31,58)(32,63)(33,42)(34,47)(35,44)(36,41)(37,46)(38,43)(39,48)(40,45)(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)], [(1,31,47,58,49,16,34,19),(2,26,48,61,50,11,35,22),(3,29,41,64,51,14,36,17),(4,32,42,59,52,9,37,20),(5,27,43,62,53,12,38,23),(6,30,44,57,54,15,39,18),(7,25,45,60,55,10,40,21),(8,28,46,63,56,13,33,24)], [(2,6),(4,8),(9,24),(10,21),(11,18),(12,23),(13,20),(14,17),(15,22),(16,19),(25,60),(26,57),(27,62),(28,59),(29,64),(30,61),(31,58),(32,63),(33,42),(34,47),(35,44),(36,41),(37,46),(38,43),(39,48),(40,45),(50,54),(52,56)])

Matrix representation of C84SD16 in GL8(𝔽17)

10200000
01020000
1601600000
0160160000
00008200
00004900
0000113119
000011356
,
4134130000
44440000
1521340000
151513130000
00000010
0000158816
00001000
0000912119
,
10000000
016000000
00100000
000160000
00001000
000091600
00000010
000000716

G:=sub<GL(8,GF(17))| [1,0,16,0,0,0,0,0,0,1,0,16,0,0,0,0,2,0,16,0,0,0,0,0,0,2,0,16,0,0,0,0,0,0,0,0,8,4,1,11,0,0,0,0,2,9,13,3,0,0,0,0,0,0,11,5,0,0,0,0,0,0,9,6],[4,4,15,15,0,0,0,0,13,4,2,15,0,0,0,0,4,4,13,13,0,0,0,0,13,4,4,13,0,0,0,0,0,0,0,0,0,15,1,9,0,0,0,0,0,8,0,12,0,0,0,0,1,8,0,11,0,0,0,0,0,16,0,9],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,16] >;

C84SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C8:4SD16");
// GroupNames label

G:=SmallGroup(128,425);
// by ID

G=gap.SmallGroup(128,425);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,288,422,387,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C84SD16 in TeX

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