Copied to
clipboard

G = C3×SD16  order 48 = 24·3

Direct product of C3 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×SD16, C8⋊2C6, D4.C6, C24⋊6C2, Q8⋊2C6, C6.15D4, C12.18C22, C4.2(C2×C6), (C3×Q8)⋊4C2, C2.4(C3×D4), (C3×D4).2C2, SmallGroup(48,26)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C4 — C3×SD16
Chief series
C1 — C2 — C4 — C12 — C3×Q8 — C3×SD16
Lower central
C1 — C2 — C4 — C3×SD16
Upper central
C1 — C6 — C12 — C3×SD16

Generators and relations for C3×SD16
 G = < a,b,c | a3=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

C1
C2
4C2
C3
C4
2C4
2C22
C6
4C6
C8
D4
Q8
C12
2C12
2C2×C6
SD16
C3×Q8
C24
C3×D4
C3×SD16

Character table of C3×SD16

 class 12A2B3A3B4A4B6A6B6C6D8A8B12A12B12C12D24A24B24C24D
 size 114112411442222442222
ρ1111111111111111111111    trivial
ρ211-1111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ311-1111-111-1-11111-1-11111    linear of order 2
ρ4111111-11111-1-111-1-1-1-1-1-1    linear of order 2
ρ5111ζ32ζ31-1ζ32ζ3ζ32ζ3-1-1ζ32ζ3ζ6ζ65ζ65ζ6ζ65ζ6    linear of order 6
ρ6111ζ32ζ311ζ32ζ3ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ7111ζ3ζ3211ζ3ζ32ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ811-1ζ3ζ3211ζ3ζ32ζ65ζ6-1-1ζ3ζ32ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ911-1ζ32ζ311ζ32ζ3ζ6ζ65-1-1ζ32ζ3ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ1011-1ζ32ζ31-1ζ32ζ3ζ6ζ6511ζ32ζ3ζ6ζ65ζ3ζ32ζ3ζ32    linear of order 6
ρ11111ζ3ζ321-1ζ3ζ32ζ3ζ32-1-1ζ3ζ32ζ65ζ6ζ6ζ65ζ6ζ65    linear of order 6
ρ1211-1ζ3ζ321-1ζ3ζ32ζ65ζ611ζ3ζ32ζ65ζ6ζ32ζ3ζ32ζ3    linear of order 6
ρ1322022-20220000-2-2000000    orthogonal lifted from D4
ρ14220-1-√-3-1+√-3-20-1-√-3-1+√-300001+√-31-√-3000000    complex lifted from C3×D4
ρ15220-1+√-3-1-√-3-20-1+√-3-1-√-300001-√-31+√-3000000    complex lifted from C3×D4
ρ162-202200-2-200-√-2√-20000-√-2-√-2√-2√-2    complex lifted from SD16
ρ172-202200-2-200√-2-√-20000√-2√-2-√-2-√-2    complex lifted from SD16
ρ182-20-1-√-3-1+√-3001+√-31-√-300-√-2√-20000ζ87ζ3+ζ85ζ3ζ87ζ32+ζ85ζ32ζ83ζ3+ζ8ζ3ζ83ζ32+ζ8ζ32    complex faithful
ρ192-20-1-√-3-1+√-3001+√-31-√-300√-2-√-20000ζ83ζ3+ζ8ζ3ζ83ζ32+ζ8ζ32ζ87ζ3+ζ85ζ3ζ87ζ32+ζ85ζ32    complex faithful
ρ202-20-1+√-3-1-√-3001-√-31+√-300-√-2√-20000ζ87ζ32+ζ85ζ32ζ87ζ3+ζ85ζ3ζ83ζ32+ζ8ζ32ζ83ζ3+ζ8ζ3    complex faithful
ρ212-20-1+√-3-1-√-3001-√-31+√-300√-2-√-20000ζ83ζ32+ζ8ζ32ζ83ζ3+ζ8ζ3ζ87ζ32+ζ85ζ32ζ87ζ3+ζ85ζ3    complex faithful

Permutation representations of C3×SD16
►On 24 points - transitive group 24T41
Generators in S24
(1 10 23)(2 11 24)(3 12 17)(4 13 18)(5 14 19)(6 15 20)(7 16 21)(8 9 22)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 21)(18 24)(20 22)
 
G:=sub<Sym(24)| (1,10,23)(2,11,24)(3,12,17)(4,13,18)(5,14,19)(6,15,20)(7,16,21)(8,9,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)>;
 
G:=Group( (1,10,23)(2,11,24)(3,12,17)(4,13,18)(5,14,19)(6,15,20)(7,16,21)(8,9,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22) );
 
G=PermutationGroup([[(1,10,23),(2,11,24),(3,12,17),(4,13,18),(5,14,19),(6,15,20),(7,16,21),(8,9,22)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,21),(18,24),(20,22)]])
 
G:=TransitiveGroup(24,41);
 

C3×SD16 is a maximal subgroup of   Q8⋊3D6  D4.D6  Q8.7D6  SD16.A4  C56⋊C6  D4.F7  Q8⋊2F7
C3×SD16 is a maximal quotient of   C56⋊C6  D4.F7  Q8⋊2F7

Matrix representation of C3×SD16 ►in GL2(𝔽19) generated by

110
011
,
07
116
,
1815
01
G:=sub<GL(2,GF(19))| [11,0,0,11],[0,11,7,6],[18,0,15,1] >;
 

C3×SD16 in GAP, Magma, Sage, TeX 

C_3\times {\rm SD}_{16}
 
% in TeX
 
G:=Group("C3xSD16");
 
// GroupNames label
 
G:=SmallGroup(48,26);
 
// by ID
 
G=gap.SmallGroup(48,26);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-2,-2,120,141,723,368,58]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C3×SD16 in TeX
Character table of C3×SD16 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁