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G = C3×SD16order 48 = 24·3

Direct product of C3 and SD16

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

Aliases: C3×SD16, C82C6, D4.C6, C246C2, Q82C6, 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

C1C4 — C3×SD16
C1C2C4C12C3×Q8 — C3×SD16
C1C2C4 — C3×SD16
C1C6C12 — C3×SD16

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

4C2
2C4
2C22
4C6
2C12
2C2×C6

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--2-20000ζ87ζ385ζ3ζ87ζ3285ζ32ζ83ζ38ζ3ζ83ζ328ζ32    complex faithful
ρ192-20-1--3-1+-3001+-31--300-2--20000ζ83ζ38ζ3ζ83ζ328ζ32ζ87ζ385ζ3ζ87ζ3285ζ32    complex faithful
ρ202-20-1+-3-1--3001--31+-300--2-20000ζ87ζ3285ζ32ζ87ζ385ζ3ζ83ζ328ζ32ζ83ζ38ζ3    complex faithful
ρ212-20-1+-3-1--3001--31+-300-2--20000ζ83ζ328ζ32ζ83ζ38ζ3ζ87ζ3285ζ32ζ87ζ385ζ3    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);

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

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