Copied to
clipboard

G = C2×C6.5S4order 288 = 25·32

Direct product of C2 and C6.5S4

direct product, non-abelian, soluble

Aliases: C2×C6.5S4, C6⋊CSU2(𝔽3), SL2(𝔽3).7D6, C6.31(C2×S4), (C2×C6).14S4, (C6×Q8).6S3, (C3×Q8).14D6, C22.4(C3⋊S4), C32(C2×CSU2(𝔽3)), (C2×SL2(𝔽3)).2S3, (C6×SL2(𝔽3)).3C2, (C3×SL2(𝔽3)).7C22, C2.5(C2×C3⋊S4), Q8.1(C2×C3⋊S3), (C2×Q8).2(C3⋊S3), SmallGroup(288,910)

Series: Derived Chief Lower central Upper central

C1C2Q8C3×SL2(𝔽3) — C2×C6.5S4
C1C2Q8C3×Q8C3×SL2(𝔽3)C6.5S4 — C2×C6.5S4
C3×SL2(𝔽3) — C2×C6.5S4
C1C22

Generators and relations for C2×C6.5S4
 G = < a,b,c,d,e,f | a2=b6=e3=1, c2=d2=f2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd-1=b3c, ece-1=b3cd, fcf-1=cd, ede-1=c, fdf-1=b3d, fef-1=e-1 >

Subgroups: 504 in 104 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×4], C22, C6, C6 [×2], C6 [×9], C8 [×2], C2×C4 [×2], Q8, Q8 [×4], C32, Dic3 [×8], C12 [×2], C2×C6, C2×C6 [×3], C2×C8, Q16 [×4], C2×Q8, C2×Q8, C3×C6 [×3], C3⋊C8 [×2], SL2(𝔽3) [×3], Dic6 [×3], C2×Dic3 [×4], C2×C12, C3×Q8, C3×Q8, C2×Q16, C3⋊Dic3 [×2], C62, C2×C3⋊C8, C3⋊Q16 [×4], CSU2(𝔽3) [×6], C2×SL2(𝔽3) [×3], C2×Dic6, C6×Q8, C3×SL2(𝔽3), C2×C3⋊Dic3, C2×C3⋊Q16, C2×CSU2(𝔽3) [×3], C6.5S4 [×2], C6×SL2(𝔽3), C2×C6.5S4
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], C3⋊S3, S4, C2×C3⋊S3, CSU2(𝔽3) [×2], C2×S4, C3⋊S4, C2×CSU2(𝔽3), C6.5S4 [×2], C2×C3⋊S4, C2×C6.5S4

Character table of C2×C6.5S4

 class 12A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K6L8A8B8C8D12A12B
 size 11112888663636222888888888181818181212
ρ1111111111111111111111111111111    trivial
ρ21111111111-1-1111111111111-1-1-1-111    linear of order 2
ρ311-1-111111-11-1-11-11-1-1-1-1-1-1111-1-11-11    linear of order 2
ρ411-1-111111-1-11-11-11-1-1-1-1-1-111-111-1-11    linear of order 2
ρ52222-1-12-12200-1-1-1-122-1-1-1-1-120000-1-1    orthogonal lifted from S3
ρ622-2-2-1-12-12-2001-11-1-2-21111-1200001-1    orthogonal lifted from D6
ρ722-2-2-1-1-122-2001-1121111-2-2-1-100001-1    orthogonal lifted from D6
ρ82222-12-1-12200-1-1-1-1-1-122-1-12-10000-1-1    orthogonal lifted from S3
ρ922-2-22-1-1-12-200-22-2-1111111-1-10000-22    orthogonal lifted from D6
ρ102222-1-1-122200-1-1-12-1-1-1-122-1-10000-1-1    orthogonal lifted from S3
ρ1122222-1-1-12200222-1-1-1-1-1-1-1-1-1000022    orthogonal lifted from S3
ρ1222-2-2-12-1-12-2001-11-111-2-2112-100001-1    orthogonal lifted from D6
ρ132-22-22-1-1-10000-2-2211-11-11-11122-2-200    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ142-2-222-1-1-100002-2-21-11-11-1111-22-2200    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ152-22-22-1-1-10000-2-2211-11-11-111-2-22200    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ162-2-222-1-1-100002-2-21-11-11-11112-22-200    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ1733333000-1-1-1-13330000000001111-1-1    orthogonal lifted from S4
ρ1833333000-1-111333000000000-1-1-1-1-1-1    orthogonal lifted from S4
ρ1933-3-33000-11-11-33-30000000001-1-111-1    orthogonal lifted from C2×S4
ρ2033-3-33000-111-1-33-3000000000-111-11-1    orthogonal lifted from C2×S4
ρ214-44-4-211-2000022-22-11-112-2-1-1000000    symplectic lifted from C6.5S4, Schur index 2
ρ224-4-44-21-210000-222-1-221-11-1-12000000    symplectic lifted from C6.5S4, Schur index 2
ρ234-4-44411100004-4-4-11-11-11-1-1-1000000    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ244-44-4-2-211000022-2-1-112-2-112-1000000    symplectic lifted from C6.5S4, Schur index 2
ρ254-4-44-211-20000-22221-11-1-22-1-1000000    symplectic lifted from C6.5S4, Schur index 2
ρ264-44-441110000-4-44-1-11-11-11-1-1000000    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ274-4-44-2-2110000-222-11-1-221-12-1000000    symplectic lifted from C6.5S4, Schur index 2
ρ284-44-4-21-21000022-2-12-2-11-11-12000000    symplectic lifted from C6.5S4, Schur index 2
ρ296666-3000-2-200-3-3-3000000000000011    orthogonal lifted from C3⋊S4
ρ3066-6-6-3000-22003-330000000000000-11    orthogonal lifted from C2×C3⋊S4

Smallest permutation representation of C2×C6.5S4
On 96 points
Generators in S96
(1 51)(2 52)(3 53)(4 54)(5 49)(6 50)(7 83)(8 84)(9 79)(10 80)(11 81)(12 82)(13 76)(14 77)(15 78)(16 73)(17 74)(18 75)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)(37 59)(38 60)(39 55)(40 56)(41 57)(42 58)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(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 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 29 4 26)(2 30 5 27)(3 25 6 28)(7 85 10 88)(8 86 11 89)(9 87 12 90)(13 91 16 94)(14 92 17 95)(15 93 18 96)(19 40 22 37)(20 41 23 38)(21 42 24 39)(31 54 34 51)(32 49 35 52)(33 50 36 53)(43 56 46 59)(44 57 47 60)(45 58 48 55)(61 80 64 83)(62 81 65 84)(63 82 66 79)(67 73 70 76)(68 74 71 77)(69 75 72 78)
(1 41 4 38)(2 42 5 39)(3 37 6 40)(7 14 10 17)(8 15 11 18)(9 16 12 13)(19 25 22 28)(20 26 23 29)(21 27 24 30)(31 44 34 47)(32 45 35 48)(33 46 36 43)(49 55 52 58)(50 56 53 59)(51 57 54 60)(61 71 64 68)(62 72 65 69)(63 67 66 70)(73 82 76 79)(74 83 77 80)(75 84 78 81)(85 95 88 92)(86 96 89 93)(87 91 90 94)
(7 85 95)(8 86 96)(9 87 91)(10 88 92)(11 89 93)(12 90 94)(19 25 37)(20 26 38)(21 27 39)(22 28 40)(23 29 41)(24 30 42)(31 57 47)(32 58 48)(33 59 43)(34 60 44)(35 55 45)(36 56 46)(61 71 83)(62 72 84)(63 67 79)(64 68 80)(65 69 81)(66 70 82)
(1 76 4 73)(2 75 5 78)(3 74 6 77)(7 59 10 56)(8 58 11 55)(9 57 12 60)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 64 28 61)(26 63 29 66)(27 62 30 65)(31 90 34 87)(32 89 35 86)(33 88 36 85)(37 80 40 83)(38 79 41 82)(39 84 42 81)(43 92 46 95)(44 91 47 94)(45 96 48 93)

G:=sub<Sym(96)| (1,51)(2,52)(3,53)(4,54)(5,49)(6,50)(7,83)(8,84)(9,79)(10,80)(11,81)(12,82)(13,76)(14,77)(15,78)(16,73)(17,74)(18,75)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32)(37,59)(38,60)(39,55)(40,56)(41,57)(42,58)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,29,4,26)(2,30,5,27)(3,25,6,28)(7,85,10,88)(8,86,11,89)(9,87,12,90)(13,91,16,94)(14,92,17,95)(15,93,18,96)(19,40,22,37)(20,41,23,38)(21,42,24,39)(31,54,34,51)(32,49,35,52)(33,50,36,53)(43,56,46,59)(44,57,47,60)(45,58,48,55)(61,80,64,83)(62,81,65,84)(63,82,66,79)(67,73,70,76)(68,74,71,77)(69,75,72,78), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,14,10,17)(8,15,11,18)(9,16,12,13)(19,25,22,28)(20,26,23,29)(21,27,24,30)(31,44,34,47)(32,45,35,48)(33,46,36,43)(49,55,52,58)(50,56,53,59)(51,57,54,60)(61,71,64,68)(62,72,65,69)(63,67,66,70)(73,82,76,79)(74,83,77,80)(75,84,78,81)(85,95,88,92)(86,96,89,93)(87,91,90,94), (7,85,95)(8,86,96)(9,87,91)(10,88,92)(11,89,93)(12,90,94)(19,25,37)(20,26,38)(21,27,39)(22,28,40)(23,29,41)(24,30,42)(31,57,47)(32,58,48)(33,59,43)(34,60,44)(35,55,45)(36,56,46)(61,71,83)(62,72,84)(63,67,79)(64,68,80)(65,69,81)(66,70,82), (1,76,4,73)(2,75,5,78)(3,74,6,77)(7,59,10,56)(8,58,11,55)(9,57,12,60)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,64,28,61)(26,63,29,66)(27,62,30,65)(31,90,34,87)(32,89,35,86)(33,88,36,85)(37,80,40,83)(38,79,41,82)(39,84,42,81)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;

G:=Group( (1,51)(2,52)(3,53)(4,54)(5,49)(6,50)(7,83)(8,84)(9,79)(10,80)(11,81)(12,82)(13,76)(14,77)(15,78)(16,73)(17,74)(18,75)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32)(37,59)(38,60)(39,55)(40,56)(41,57)(42,58)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,29,4,26)(2,30,5,27)(3,25,6,28)(7,85,10,88)(8,86,11,89)(9,87,12,90)(13,91,16,94)(14,92,17,95)(15,93,18,96)(19,40,22,37)(20,41,23,38)(21,42,24,39)(31,54,34,51)(32,49,35,52)(33,50,36,53)(43,56,46,59)(44,57,47,60)(45,58,48,55)(61,80,64,83)(62,81,65,84)(63,82,66,79)(67,73,70,76)(68,74,71,77)(69,75,72,78), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,14,10,17)(8,15,11,18)(9,16,12,13)(19,25,22,28)(20,26,23,29)(21,27,24,30)(31,44,34,47)(32,45,35,48)(33,46,36,43)(49,55,52,58)(50,56,53,59)(51,57,54,60)(61,71,64,68)(62,72,65,69)(63,67,66,70)(73,82,76,79)(74,83,77,80)(75,84,78,81)(85,95,88,92)(86,96,89,93)(87,91,90,94), (7,85,95)(8,86,96)(9,87,91)(10,88,92)(11,89,93)(12,90,94)(19,25,37)(20,26,38)(21,27,39)(22,28,40)(23,29,41)(24,30,42)(31,57,47)(32,58,48)(33,59,43)(34,60,44)(35,55,45)(36,56,46)(61,71,83)(62,72,84)(63,67,79)(64,68,80)(65,69,81)(66,70,82), (1,76,4,73)(2,75,5,78)(3,74,6,77)(7,59,10,56)(8,58,11,55)(9,57,12,60)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,64,28,61)(26,63,29,66)(27,62,30,65)(31,90,34,87)(32,89,35,86)(33,88,36,85)(37,80,40,83)(38,79,41,82)(39,84,42,81)(43,92,46,95)(44,91,47,94)(45,96,48,93) );

G=PermutationGroup([(1,51),(2,52),(3,53),(4,54),(5,49),(6,50),(7,83),(8,84),(9,79),(10,80),(11,81),(12,82),(13,76),(14,77),(15,78),(16,73),(17,74),(18,75),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,33),(26,34),(27,35),(28,36),(29,31),(30,32),(37,59),(38,60),(39,55),(40,56),(41,57),(42,58),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(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),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,29,4,26),(2,30,5,27),(3,25,6,28),(7,85,10,88),(8,86,11,89),(9,87,12,90),(13,91,16,94),(14,92,17,95),(15,93,18,96),(19,40,22,37),(20,41,23,38),(21,42,24,39),(31,54,34,51),(32,49,35,52),(33,50,36,53),(43,56,46,59),(44,57,47,60),(45,58,48,55),(61,80,64,83),(62,81,65,84),(63,82,66,79),(67,73,70,76),(68,74,71,77),(69,75,72,78)], [(1,41,4,38),(2,42,5,39),(3,37,6,40),(7,14,10,17),(8,15,11,18),(9,16,12,13),(19,25,22,28),(20,26,23,29),(21,27,24,30),(31,44,34,47),(32,45,35,48),(33,46,36,43),(49,55,52,58),(50,56,53,59),(51,57,54,60),(61,71,64,68),(62,72,65,69),(63,67,66,70),(73,82,76,79),(74,83,77,80),(75,84,78,81),(85,95,88,92),(86,96,89,93),(87,91,90,94)], [(7,85,95),(8,86,96),(9,87,91),(10,88,92),(11,89,93),(12,90,94),(19,25,37),(20,26,38),(21,27,39),(22,28,40),(23,29,41),(24,30,42),(31,57,47),(32,58,48),(33,59,43),(34,60,44),(35,55,45),(36,56,46),(61,71,83),(62,72,84),(63,67,79),(64,68,80),(65,69,81),(66,70,82)], [(1,76,4,73),(2,75,5,78),(3,74,6,77),(7,59,10,56),(8,58,11,55),(9,57,12,60),(13,54,16,51),(14,53,17,50),(15,52,18,49),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,64,28,61),(26,63,29,66),(27,62,30,65),(31,90,34,87),(32,89,35,86),(33,88,36,85),(37,80,40,83),(38,79,41,82),(39,84,42,81),(43,92,46,95),(44,91,47,94),(45,96,48,93)])

Matrix representation of C2×C6.5S4 in GL4(𝔽73) generated by

72000
07200
0010
0001
,
07200
17200
00720
00072
,
1000
0100
002123
00352
,
1000
0100
007122
00232
,
72100
72000
00252
004970
,
0100
1000
001022
001263
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,72,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,21,3,0,0,23,52],[1,0,0,0,0,1,0,0,0,0,71,23,0,0,22,2],[72,72,0,0,1,0,0,0,0,0,2,49,0,0,52,70],[0,1,0,0,1,0,0,0,0,0,10,12,0,0,22,63] >;

C2×C6.5S4 in GAP, Magma, Sage, TeX

C_2\times C_6._5S_4
% in TeX

G:=Group("C2xC6.5S4");
// GroupNames label

G:=SmallGroup(288,910);
// by ID

G=gap.SmallGroup(288,910);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,170,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic

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

Export

Character table of C2×C6.5S4 in TeX

׿
×
𝔽