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G = SD16⋊C4order 64 = 26

1st semidirect product of SD16 and C4 acting via C4/C2=C2

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

Aliases: SD161C4, C42.9C22, C82(C2×C4), Q82(C2×C4), (C4×Q8)⋊2C2, C8⋊C41C2, D4.2(C2×C4), C2.D811C2, (C4×D4).5C2, C2.15(C4×D4), C4.4(C4○D4), (C2×C4).101D4, Q8⋊C416C2, D4⋊C4.6C2, C4⋊C4.53C22, C2.4(C8⋊C22), (C2×C8).49C22, C4.12(C22×C4), (C2×C4).76C23, (C2×SD16).1C2, C22.54(C2×D4), (C2×D4).52C22, C2.4(C8.C22), (C2×Q8).46C22, SmallGroup(64,121)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — SD16⋊C4
C1C2C22C2×C4C42C4×D4 — SD16⋊C4
C1C2C4 — SD16⋊C4
C1C22C42 — SD16⋊C4
C1C2C2C2×C4 — SD16⋊C4

Generators and relations for SD16⋊C4
 G = < a,b,c | a8=b2=c4=1, bab=a3, cac-1=a5, bc=cb >

Subgroups: 101 in 60 conjugacy classes, 35 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], Q8, C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, SD16⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, C4×D4, C8⋊C22, C8.C22, SD16⋊C4

Character table of SD16⋊C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 1111442222224444444444
ρ11111111111111111111111    trivial
ρ21111-1-111-1-1-1-1111-1-11-11-11    linear of order 2
ρ311111111-1-1-1-1-1-1-111-1-11-11    linear of order 2
ρ41111-1-1111111-1-1-1-1-1-11111    linear of order 2
ρ51111-1-11111111-1111-1-1-1-1-1    linear of order 2
ρ611111111-1-1-1-11-11-1-1-11-11-1    linear of order 2
ρ71111-1-111-1-1-1-1-11-11111-11-1    linear of order 2
ρ8111111111111-11-1-1-11-1-1-1-1    linear of order 2
ρ91-11-11-11-1-ii-ii-i-ii-11ii-1-i1    linear of order 4
ρ101-11-1-111-1i-ii-i-i-ii1-1i-i-1i1    linear of order 4
ρ111-11-1-111-1-ii-ii-iii-11-i-i1i-1    linear of order 4
ρ121-11-11-11-1i-ii-i-iii1-1-ii1-i-1    linear of order 4
ρ131-11-1-111-1i-ii-ii-i-i-11ii1-i-1    linear of order 4
ρ141-11-11-11-1-ii-iii-i-i1-1i-i1i-1    linear of order 4
ρ151-11-11-11-1i-ii-iii-i-11-i-i-1i1    linear of order 4
ρ161-11-1-111-1-ii-iiii-i1-1-ii-1-i1    linear of order 4
ρ17222200-2-2-222-20000000000    orthogonal lifted from D4
ρ18222200-2-22-2-220000000000    orthogonal lifted from D4
ρ192-22-200-22-2i-2i2i2i0000000000    complex lifted from C4○D4
ρ202-22-200-222i2i-2i-2i0000000000    complex lifted from C4○D4
ρ214-4-44000000000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of SD16⋊C4
On 32 points
Generators in S32
(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)
(1 32)(2 27)(3 30)(4 25)(5 28)(6 31)(7 26)(8 29)(9 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)
(1 11 32 19)(2 16 25 24)(3 13 26 21)(4 10 27 18)(5 15 28 23)(6 12 29 20)(7 9 30 17)(8 14 31 22)

G:=sub<Sym(32)| (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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18), (1,11,32,19)(2,16,25,24)(3,13,26,21)(4,10,27,18)(5,15,28,23)(6,12,29,20)(7,9,30,17)(8,14,31,22)>;

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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18), (1,11,32,19)(2,16,25,24)(3,13,26,21)(4,10,27,18)(5,15,28,23)(6,12,29,20)(7,9,30,17)(8,14,31,22) );

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)], [(1,32),(2,27),(3,30),(4,25),(5,28),(6,31),(7,26),(8,29),(9,21),(10,24),(11,19),(12,22),(13,17),(14,20),(15,23),(16,18)], [(1,11,32,19),(2,16,25,24),(3,13,26,21),(4,10,27,18),(5,15,28,23),(6,12,29,20),(7,9,30,17),(8,14,31,22)])

SD16⋊C4 is a maximal subgroup of
C42.352C23  C42.353C23  C42.354C23  C42.355C23  C42.357C23  C42.358C23  C42.360C23  C42.387C23  C42.390C23  SD166D4  SD168D4  C42.45C23  C42.46C23  C42.49C23  C42.50C23  C42.55C23  C42.57C23  C42.60C23  C42.62C23  C42.64C23  C42.508C23  C42.509C23  C42.511C23  C42.512C23  C42.513C23  C42.516C23  C42.517C23  C42.518C23  SD16⋊Q8  SD162Q8  SD163Q8  C42.75C23  C42.532C23  GL2(𝔽3)⋊C4
 C42.D2p: C42.383D4  C4×C8⋊C22  C4×C8.C22  C42.228D4  C42.229D4  C42.230D4  C42.232D4  C42.233D4 ...
 C2p.(C4×D4): C42.275C23  C42.276C23  C42.278C23  C42.280C23  D4.S3⋊C4  Q83(C4×S3)  C24⋊C2⋊C4  SD16⋊Dic3 ...
 C8pD4⋊C2: C42.386C23  C42.391C23  SD16⋊D4  SD167D4  C42.43C23  C42.44C23  C42.56C23  C42.72C23 ...
SD16⋊C4 is a maximal quotient of
SD16⋊C8  D4.M4(2)  Q82M4(2)  C8⋊M4(2)  C8⋊C42  C4.Q89C4  C4.Q810C4  C8⋊(C4⋊C4)
 C42.D2p: D4⋊C42  Q8⋊C42  C42.16D6  C42.51D6  C42.56D6  C42.16D10  C42.51D10  C42.56D10 ...
 C2p.(C4×D4): C2.(C4×D8)  Q8⋊(C4⋊C4)  (C2×SD16)⋊14C4  (C2×SD16)⋊15C4  D4⋊C4⋊C4  C2.(C4×Q16)  C2.(C8⋊D4)  C2.(C82D4) ...

Matrix representation of SD16⋊C4 in GL6(𝔽17)

080000
200000
0008011
00138311
0001109
0031149
,
1600000
010000
0016000
0016100
0000160
0000161
,
400000
040000
000010
000001
0016000
0001600

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

SD16⋊C4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes C_4
% in TeX

G:=Group("SD16:C4");
// GroupNames label

G:=SmallGroup(64,121);
// by ID

G=gap.SmallGroup(64,121);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^3,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations

Export

Character table of SD16⋊C4 in TeX

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