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G = C8⋊3SD16  order 128 = 27

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

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

Aliases: C8⋊3SD16, C42.242C23, C4⋊C4.64D4, C8⋊3Q8⋊1C2, C8⋊1C8⋊21C2, (C2×C8).94D4, (C2×D4).60D4, C8⋊6D4.2C2, C4.60(C2×SD16), C4⋊Q8.63C22, C4.10D8⋊28C2, C2.10(C8⋊D4), C4⋊C8.186C22, C4.42(C8⋊C22), (C4×C8).145C22, D4⋊2Q8.10C2, C4.6Q16⋊17C2, 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

Derived series
C1 — C42 — C8⋊3SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — C8⋊6D4 — C8⋊3SD16
Lower central
C1 — C22 — C42 — C8⋊3SD16
Upper central
C1 — C22 — C42 — C8⋊3SD16
Jennings
C1 — C22 — C22 — C42 — C8⋊3SD16

Generators and relations for C8⋊3SD16
 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, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C2×M4(2), C2×SD16, C4.10D8, C4.6Q16, C8⋊1C8, C8⋊6D4, D4.D4, D4⋊2Q8, C8⋊3Q8, C8⋊3SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C8⋊C22, C8.C22, D4.D4, C8⋊D4, D4.3D4, C8⋊3SD16

Character table of C8⋊3SD16

 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 C8⋊3SD16
►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 45 64 49 22 38 14)(2 32 46 63 50 21 39 13)(3 31 47 62 51 20 40 12)(4 30 48 61 52 19 33 11)(5 29 41 60 53 18 34 10)(6 28 42 59 54 17 35 9)(7 27 43 58 55 24 36 16)(8 26 44 57 56 23 37 15)

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

Matrix representation of C8⋊3SD16 ►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] >;
 

C8⋊3SD16 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 C8⋊3SD16 in TeX

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