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G = S3×C23×C6order 288 = 25·32

Direct product of C23×C6 and S3

Aliases: S3×C23×C6, C322C25, C6210C23, C3⋊(C24×C6), C6⋊(C23×C6), (C3×C6)⋊2C24, (C23×C6)⋊13C6, (C22×C62)⋊7C2, (C2×C62)⋊17C22, (C2×C6)⋊6(C22×C6), (C22×C6)⋊10(C2×C6), SmallGroup(288,1043)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C23×C6
 Chief series C1 — C3 — C32 — C3×S3 — S3×C6 — S3×C2×C6 — S3×C22×C6 — S3×C23×C6
 Lower central C3 — S3×C23×C6
 Upper central C1 — C23×C6

Generators and relations for S3×C23×C6
G = < a,b,c,d,e,f | a2=b2=c2=d6=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2858 in 1563 conjugacy classes, 882 normal (10 characteristic)
C1, C2 [×15], C2 [×16], C3 [×2], C3, C22 [×35], C22 [×120], S3 [×16], C6 [×30], C6 [×31], C23 [×15], C23 [×140], C32, D6 [×120], C2×C6 [×70], C2×C6 [×155], C24, C24 [×30], C3×S3 [×16], C3×C6 [×15], C22×S3 [×140], C22×C6 [×30], C22×C6 [×155], C25, S3×C6 [×120], C62 [×35], S3×C23 [×30], C23×C6 [×2], C23×C6 [×31], S3×C2×C6 [×140], C2×C62 [×15], S3×C24, C24×C6, S3×C22×C6 [×30], C22×C62, S3×C23×C6
Quotients: C1, C2 [×31], C3, C22 [×155], S3, C6 [×31], C23 [×155], D6 [×15], C2×C6 [×155], C24 [×31], C3×S3, C22×S3 [×35], C22×C6 [×155], C25, S3×C6 [×15], S3×C23 [×15], C23×C6 [×31], S3×C2×C6 [×35], S3×C24, C24×C6, S3×C22×C6 [×15], S3×C23×C6

Smallest permutation representation of S3×C23×C6
On 96 points
Generators in S96
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)(49 58)(50 59)(51 60)(52 55)(53 56)(54 57)(61 70)(62 71)(63 72)(64 67)(65 68)(66 69)(73 82)(74 83)(75 84)(76 79)(77 80)(78 81)(85 94)(86 95)(87 96)(88 91)(89 92)(90 93)
(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 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(1 53)(2 54)(3 49)(4 50)(5 51)(6 52)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 3A 3B 3C 3D 3E 6A ··· 6AD 6AE ··· 6BW 6BX ··· 6DC order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 ··· 1 3 ··· 3 1 1 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 kernel S3×C23×C6 S3×C22×C6 C22×C62 S3×C24 S3×C23 C23×C6 C23×C6 C22×C6 C24 C23 # reps 1 30 1 2 60 2 1 15 2 30

Matrix representation of S3×C23×C6 in GL5(𝔽7)

 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 6 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6
,
 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6
,
 4 0 0 0 0 0 2 0 0 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 5
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 2
,
 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(7))| [1,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6],[6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6],[4,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,1,0] >;

S3×C23×C6 in GAP, Magma, Sage, TeX

S_3\times C_2^3\times C_6
% in TeX

G:=Group("S3xC2^3xC6");
// GroupNames label

G:=SmallGroup(288,1043);
// by ID

G=gap.SmallGroup(288,1043);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,9414]);
// Polycyclic

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

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