Copied to
clipboard

## G = C2×C6.6S4order 288 = 25·32

### Direct product of C2 and C6.6S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — C2×C6.6S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.6S4 — C2×C6.6S4
 Lower central C3×SL2(𝔽3) — C2×C6.6S4
 Upper central C1 — C22

Generators and relations for C2×C6.6S4
G = < a,b,c,d,e,f | a2=b6=e3=f2=1, c2=d2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=b3c, ece-1=b3cd, fcf=cd, ede-1=c, fdf=b3d, fef=e-1 >

Subgroups: 1016 in 144 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, Q8, Q8, C23, C32, C12, D6, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3⋊C8, SL2(𝔽3), D12, C2×C12, C3×Q8, C3×Q8, C22×S3, C2×SD16, C2×C3⋊S3, C62, C2×C3⋊C8, Q82S3, GL2(𝔽3), C2×SL2(𝔽3), C2×D12, C6×Q8, C3×SL2(𝔽3), C22×C3⋊S3, C2×Q82S3, C2×GL2(𝔽3), C6.6S4, C6×SL2(𝔽3), C2×C6.6S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, GL2(𝔽3), C2×S4, C3⋊S4, C2×GL2(𝔽3), C6.6S4, C2×C3⋊S4, C2×C6.6S4

Character table of C2×C6.6S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 8C 8D 12A 12B size 1 1 1 1 36 36 2 8 8 8 6 6 2 2 2 8 8 8 8 8 8 8 8 8 18 18 18 18 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 linear of order 2 ρ3 1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 2 2 2 2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 2 -1 -1 2 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 -2 0 0 -1 -1 -1 2 2 -2 1 -1 1 -1 1 1 -2 -2 1 1 2 -1 0 0 0 0 1 -1 orthogonal lifted from D6 ρ7 2 2 -2 -2 0 0 -1 2 -1 -1 2 -2 1 -1 1 2 1 1 1 1 -2 -2 -1 -1 0 0 0 0 1 -1 orthogonal lifted from D6 ρ8 2 2 -2 -2 0 0 2 -1 -1 -1 2 -2 -2 2 -2 -1 1 1 1 1 1 1 -1 -1 0 0 0 0 -2 2 orthogonal lifted from D6 ρ9 2 2 2 2 0 0 -1 -1 2 -1 2 2 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 2 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 0 0 -1 -1 2 -1 2 -2 1 -1 1 -1 -2 -2 1 1 1 1 -1 2 0 0 0 0 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 2 -1 -1 -1 -1 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 0 0 2 -1 -1 -1 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 2 2 orthogonal lifted from S3 ρ13 2 -2 -2 2 0 0 2 -1 -1 -1 0 0 -2 -2 2 1 1 -1 1 -1 1 -1 1 1 -√-2 -√-2 √-2 √-2 0 0 complex lifted from GL2(𝔽3) ρ14 2 -2 2 -2 0 0 2 -1 -1 -1 0 0 2 -2 -2 1 -1 1 -1 1 -1 1 1 1 √-2 -√-2 √-2 -√-2 0 0 complex lifted from GL2(𝔽3) ρ15 2 -2 2 -2 0 0 2 -1 -1 -1 0 0 2 -2 -2 1 -1 1 -1 1 -1 1 1 1 -√-2 √-2 -√-2 √-2 0 0 complex lifted from GL2(𝔽3) ρ16 2 -2 -2 2 0 0 2 -1 -1 -1 0 0 -2 -2 2 1 1 -1 1 -1 1 -1 1 1 √-2 √-2 -√-2 -√-2 0 0 complex lifted from GL2(𝔽3) ρ17 3 3 -3 -3 -1 1 3 0 0 0 -1 1 -3 3 -3 0 0 0 0 0 0 0 0 0 1 -1 -1 1 1 -1 orthogonal lifted from C2×S4 ρ18 3 3 3 3 -1 -1 3 0 0 0 -1 -1 3 3 3 0 0 0 0 0 0 0 0 0 1 1 1 1 -1 -1 orthogonal lifted from S4 ρ19 3 3 3 3 1 1 3 0 0 0 -1 -1 3 3 3 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S4 ρ20 3 3 -3 -3 1 -1 3 0 0 0 -1 1 -3 3 -3 0 0 0 0 0 0 0 0 0 -1 1 1 -1 1 -1 orthogonal lifted from C2×S4 ρ21 4 -4 -4 4 0 0 -2 -2 1 1 0 0 2 2 -2 2 -1 1 -1 1 2 -2 -1 -1 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ22 4 -4 4 -4 0 0 -2 1 1 -2 0 0 -2 2 2 -1 1 -1 -2 2 1 -1 2 -1 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ23 4 -4 -4 4 0 0 -2 1 -2 1 0 0 2 2 -2 -1 2 -2 -1 1 -1 1 -1 2 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ24 4 -4 4 -4 0 0 4 1 1 1 0 0 4 -4 -4 -1 1 -1 1 -1 1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ25 4 -4 -4 4 0 0 4 1 1 1 0 0 -4 -4 4 -1 -1 1 -1 1 -1 1 -1 -1 0 0 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ26 4 -4 4 -4 0 0 -2 -2 1 1 0 0 -2 2 2 2 1 -1 1 -1 -2 2 -1 -1 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ27 4 -4 4 -4 0 0 -2 1 -2 1 0 0 -2 2 2 -1 -2 2 1 -1 1 -1 -1 2 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ28 4 -4 -4 4 0 0 -2 1 1 -2 0 0 2 2 -2 -1 -1 1 2 -2 -1 1 2 -1 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ29 6 6 -6 -6 0 0 -3 0 0 0 -2 2 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 orthogonal lifted from C2×C3⋊S4 ρ30 6 6 6 6 0 0 -3 0 0 0 -2 -2 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 orthogonal lifted from C3⋊S4

Smallest permutation representation of C2×C6.6S4
On 48 points
Generators in S48
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(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)
(1 26 4 29)(2 27 5 30)(3 28 6 25)(7 43 10 46)(8 44 11 47)(9 45 12 48)(13 32 16 35)(14 33 17 36)(15 34 18 31)(19 40 22 37)(20 41 23 38)(21 42 24 39)
(1 41 4 38)(2 42 5 39)(3 37 6 40)(7 17 10 14)(8 18 11 15)(9 13 12 16)(19 28 22 25)(20 29 23 26)(21 30 24 27)(31 44 34 47)(32 45 35 48)(33 46 36 43)
(1 5 3)(2 6 4)(7 47 35)(8 48 36)(9 43 31)(10 44 32)(11 45 33)(12 46 34)(13 17 15)(14 18 16)(19 26 39)(20 27 40)(21 28 41)(22 29 42)(23 30 37)(24 25 38)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(19 32)(20 31)(21 36)(22 35)(23 34)(24 33)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)

G:=sub<Sym(48)| (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,26,4,29)(2,27,5,30)(3,28,6,25)(7,43,10,46)(8,44,11,47)(9,45,12,48)(13,32,16,35)(14,33,17,36)(15,34,18,31)(19,40,22,37)(20,41,23,38)(21,42,24,39), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,17,10,14)(8,18,11,15)(9,13,12,16)(19,28,22,25)(20,29,23,26)(21,30,24,27)(31,44,34,47)(32,45,35,48)(33,46,36,43), (1,5,3)(2,6,4)(7,47,35)(8,48,36)(9,43,31)(10,44,32)(11,45,33)(12,46,34)(13,17,15)(14,18,16)(19,26,39)(20,27,40)(21,28,41)(22,29,42)(23,30,37)(24,25,38), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(19,32)(20,31)(21,36)(22,35)(23,34)(24,33)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)>;

G:=Group( (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,26,4,29)(2,27,5,30)(3,28,6,25)(7,43,10,46)(8,44,11,47)(9,45,12,48)(13,32,16,35)(14,33,17,36)(15,34,18,31)(19,40,22,37)(20,41,23,38)(21,42,24,39), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,17,10,14)(8,18,11,15)(9,13,12,16)(19,28,22,25)(20,29,23,26)(21,30,24,27)(31,44,34,47)(32,45,35,48)(33,46,36,43), (1,5,3)(2,6,4)(7,47,35)(8,48,36)(9,43,31)(10,44,32)(11,45,33)(12,46,34)(13,17,15)(14,18,16)(19,26,39)(20,27,40)(21,28,41)(22,29,42)(23,30,37)(24,25,38), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(19,32)(20,31)(21,36)(22,35)(23,34)(24,33)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46) );

G=PermutationGroup([[(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35)], [(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)], [(1,26,4,29),(2,27,5,30),(3,28,6,25),(7,43,10,46),(8,44,11,47),(9,45,12,48),(13,32,16,35),(14,33,17,36),(15,34,18,31),(19,40,22,37),(20,41,23,38),(21,42,24,39)], [(1,41,4,38),(2,42,5,39),(3,37,6,40),(7,17,10,14),(8,18,11,15),(9,13,12,16),(19,28,22,25),(20,29,23,26),(21,30,24,27),(31,44,34,47),(32,45,35,48),(33,46,36,43)], [(1,5,3),(2,6,4),(7,47,35),(8,48,36),(9,43,31),(10,44,32),(11,45,33),(12,46,34),(13,17,15),(14,18,16),(19,26,39),(20,27,40),(21,28,41),(22,29,42),(23,30,37),(24,25,38)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(19,32),(20,31),(21,36),(22,35),(23,34),(24,33),(25,45),(26,44),(27,43),(28,48),(29,47),(30,46)]])

Matrix representation of C2×C6.6S4 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 69 65 0 0 29 3 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 53 40 0 0 21 20
,
 1 0 0 0 0 1 0 0 0 0 53 52 0 0 33 20
,
 69 65 0 0 29 3 0 0 0 0 72 1 0 0 72 0
,
 72 0 0 0 10 1 0 0 0 0 72 1 0 0 0 1
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[69,29,0,0,65,3,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,53,21,0,0,40,20],[1,0,0,0,0,1,0,0,0,0,53,33,0,0,52,20],[69,29,0,0,65,3,0,0,0,0,72,72,0,0,1,0],[72,10,0,0,0,1,0,0,0,0,72,0,0,0,1,1] >;

C2×C6.6S4 in GAP, Magma, Sage, TeX

C_2\times C_6._6S_4
% in TeX

G:=Group("C2xC6.6S4");
// GroupNames label

G:=SmallGroup(288,911);
// by ID

G=gap.SmallGroup(288,911);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,170,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=e^3=f^2=1,c^2=d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,d*c*d^-1=b^3*c,e*c*e^-1=b^3*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=b^3*d,f*e*f=e^-1>;
// generators/relations

Export

׿
×
𝔽