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## G = C2×C6.5S4order 288 = 25·32

### Direct product of C2 and C6.5S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — C2×C6.5S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.5S4 — C2×C6.5S4
 Lower central C3×SL2(𝔽3) — C2×C6.5S4
 Upper central C1 — C22

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 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 8C 8D 12A 12B size 1 1 1 1 2 8 8 8 6 6 36 36 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 -1 -1 2 -1 2 2 0 0 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 2 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 -2 -1 -1 2 -1 2 -2 0 0 1 -1 1 -1 -2 -2 1 1 1 1 -1 2 0 0 0 0 1 -1 orthogonal lifted from D6 ρ7 2 2 -2 -2 -1 -1 -1 2 2 -2 0 0 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 -1 2 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 2 2 -1 -1 2 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ9 2 2 -2 -2 2 -1 -1 -1 2 -2 0 0 -2 2 -2 -1 1 1 1 1 1 1 -1 -1 0 0 0 0 -2 2 orthogonal lifted from D6 ρ10 2 2 2 2 -1 -1 -1 2 2 2 0 0 -1 -1 -1 2 -1 -1 -1 -1 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 2 -1 -1 -1 2 2 0 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 2 2 orthogonal lifted from S3 ρ12 2 2 -2 -2 -1 2 -1 -1 2 -2 0 0 1 -1 1 -1 1 1 -2 -2 1 1 2 -1 0 0 0 0 1 -1 orthogonal lifted from D6 ρ13 2 -2 2 -2 2 -1 -1 -1 0 0 0 0 -2 -2 2 1 1 -1 1 -1 1 -1 1 1 √2 √2 -√2 -√2 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ14 2 -2 -2 2 2 -1 -1 -1 0 0 0 0 2 -2 -2 1 -1 1 -1 1 -1 1 1 1 -√2 √2 -√2 √2 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ15 2 -2 2 -2 2 -1 -1 -1 0 0 0 0 -2 -2 2 1 1 -1 1 -1 1 -1 1 1 -√2 -√2 √2 √2 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ16 2 -2 -2 2 2 -1 -1 -1 0 0 0 0 2 -2 -2 1 -1 1 -1 1 -1 1 1 1 √2 -√2 √2 -√2 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ17 3 3 3 3 3 0 0 0 -1 -1 -1 -1 3 3 3 0 0 0 0 0 0 0 0 0 1 1 1 1 -1 -1 orthogonal lifted from S4 ρ18 3 3 3 3 3 0 0 0 -1 -1 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 3 0 0 0 -1 1 -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 ρ20 3 3 -3 -3 3 0 0 0 -1 1 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 -2 1 1 -2 0 0 0 0 2 2 -2 2 -1 1 -1 1 2 -2 -1 -1 0 0 0 0 0 0 symplectic lifted from C6.5S4, Schur index 2 ρ22 4 -4 -4 4 -2 1 -2 1 0 0 0 0 -2 2 2 -1 -2 2 1 -1 1 -1 -1 2 0 0 0 0 0 0 symplectic lifted from C6.5S4, Schur index 2 ρ23 4 -4 -4 4 4 1 1 1 0 0 0 0 4 -4 -4 -1 1 -1 1 -1 1 -1 -1 -1 0 0 0 0 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ24 4 -4 4 -4 -2 -2 1 1 0 0 0 0 2 2 -2 -1 -1 1 2 -2 -1 1 2 -1 0 0 0 0 0 0 symplectic lifted from C6.5S4, Schur index 2 ρ25 4 -4 -4 4 -2 1 1 -2 0 0 0 0 -2 2 2 2 1 -1 1 -1 -2 2 -1 -1 0 0 0 0 0 0 symplectic lifted from C6.5S4, Schur index 2 ρ26 4 -4 4 -4 4 1 1 1 0 0 0 0 -4 -4 4 -1 -1 1 -1 1 -1 1 -1 -1 0 0 0 0 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ27 4 -4 -4 4 -2 -2 1 1 0 0 0 0 -2 2 2 -1 1 -1 -2 2 1 -1 2 -1 0 0 0 0 0 0 symplectic lifted from C6.5S4, Schur index 2 ρ28 4 -4 4 -4 -2 1 -2 1 0 0 0 0 2 2 -2 -1 2 -2 -1 1 -1 1 -1 2 0 0 0 0 0 0 symplectic lifted from C6.5S4, Schur index 2 ρ29 6 6 6 6 -3 0 0 0 -2 -2 0 0 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 orthogonal lifted from C3⋊S4 ρ30 6 6 -6 -6 -3 0 0 0 -2 2 0 0 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 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

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 1 72 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 21 23 0 0 3 52
,
 1 0 0 0 0 1 0 0 0 0 71 22 0 0 23 2
,
 72 1 0 0 72 0 0 0 0 0 2 52 0 0 49 70
,
 0 1 0 0 1 0 0 0 0 0 10 22 0 0 12 63
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

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