direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C22×C10, C15⋊4C24, C30⋊4C23, C3⋊(C23×C10), C6⋊(C22×C10), (C22×C30)⋊7C2, (C22×C6)⋊3C10, (C2×C30)⋊14C22, (C2×C6)⋊4(C2×C10), SmallGroup(240,206)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C22×C10 |
Generators and relations for S3×C22×C10
G = < a,b,c,d,e | a2=b2=c10=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 472 in 268 conjugacy classes, 166 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C23, C10, C10, D6, C2×C6, C15, C24, C2×C10, C2×C10, C22×S3, C22×C6, C5×S3, C30, C22×C10, C22×C10, S3×C23, S3×C10, C2×C30, C23×C10, S3×C2×C10, C22×C30, S3×C22×C10
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C24, C2×C10, C22×S3, C5×S3, C22×C10, S3×C23, S3×C10, C23×C10, S3×C2×C10, S3×C22×C10
►On 120 points
Generators in S120
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 61)(28 62)(29 63)(30 64)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 101)(18 102)(19 103)(20 104)(21 100)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)
(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 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)
(1 42 39)(2 43 40)(3 44 31)(4 45 32)(5 46 33)(6 47 34)(7 48 35)(8 49 36)(9 50 37)(10 41 38)(11 23 114)(12 24 115)(13 25 116)(14 26 117)(15 27 118)(16 28 119)(17 29 120)(18 30 111)(19 21 112)(20 22 113)(51 77 64)(52 78 65)(53 79 66)(54 80 67)(55 71 68)(56 72 69)(57 73 70)(58 74 61)(59 75 62)(60 76 63)(81 107 94)(82 108 95)(83 109 96)(84 110 97)(85 101 98)(86 102 99)(87 103 100)(88 104 91)(89 105 92)(90 106 93)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 67)(12 68)(13 69)(14 70)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 101)(39 102)(40 103)(41 98)(42 99)(43 100)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
G:=sub<Sym(120)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,61)(28,62)(29,63)(30,64)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,101)(18,102)(19,103)(20,104)(21,100)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,76)(42,77)(43,78)(44,79)(45,80)(46,71)(47,72)(48,73)(49,74)(50,75)(81,116)(82,117)(83,118)(84,119)(85,120)(86,111)(87,112)(88,113)(89,114)(90,115), (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)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,42,39)(2,43,40)(3,44,31)(4,45,32)(5,46,33)(6,47,34)(7,48,35)(8,49,36)(9,50,37)(10,41,38)(11,23,114)(12,24,115)(13,25,116)(14,26,117)(15,27,118)(16,28,119)(17,29,120)(18,30,111)(19,21,112)(20,22,113)(51,77,64)(52,78,65)(53,79,66)(54,80,67)(55,71,68)(56,72,69)(57,73,70)(58,74,61)(59,75,62)(60,76,63)(81,107,94)(82,108,95)(83,109,96)(84,110,97)(85,101,98)(86,102,99)(87,103,100)(88,104,91)(89,105,92)(90,106,93), (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,101)(39,102)(40,103)(41,98)(42,99)(43,100)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,80)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,61)(28,62)(29,63)(30,64)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,101)(18,102)(19,103)(20,104)(21,100)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,76)(42,77)(43,78)(44,79)(45,80)(46,71)(47,72)(48,73)(49,74)(50,75)(81,116)(82,117)(83,118)(84,119)(85,120)(86,111)(87,112)(88,113)(89,114)(90,115), (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)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,42,39)(2,43,40)(3,44,31)(4,45,32)(5,46,33)(6,47,34)(7,48,35)(8,49,36)(9,50,37)(10,41,38)(11,23,114)(12,24,115)(13,25,116)(14,26,117)(15,27,118)(16,28,119)(17,29,120)(18,30,111)(19,21,112)(20,22,113)(51,77,64)(52,78,65)(53,79,66)(54,80,67)(55,71,68)(56,72,69)(57,73,70)(58,74,61)(59,75,62)(60,76,63)(81,107,94)(82,108,95)(83,109,96)(84,110,97)(85,101,98)(86,102,99)(87,103,100)(88,104,91)(89,105,92)(90,106,93), (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,101)(39,102)(40,103)(41,98)(42,99)(43,100)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,81),(7,82),(8,83),(9,84),(10,85),(11,80),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,61),(28,62),(29,63),(30,64),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,101),(18,102),(19,103),(20,104),(21,100),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65),(41,76),(42,77),(43,78),(44,79),(45,80),(46,71),(47,72),(48,73),(49,74),(50,75),(81,116),(82,117),(83,118),(84,119),(85,120),(86,111),(87,112),(88,113),(89,114),(90,115)], [(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),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120)], [(1,42,39),(2,43,40),(3,44,31),(4,45,32),(5,46,33),(6,47,34),(7,48,35),(8,49,36),(9,50,37),(10,41,38),(11,23,114),(12,24,115),(13,25,116),(14,26,117),(15,27,118),(16,28,119),(17,29,120),(18,30,111),(19,21,112),(20,22,113),(51,77,64),(52,78,65),(53,79,66),(54,80,67),(55,71,68),(56,72,69),(57,73,70),(58,74,61),(59,75,62),(60,76,63),(81,107,94),(82,108,95),(83,109,96),(84,110,97),(85,101,98),(86,102,99),(87,103,100),(88,104,91),(89,105,92),(90,106,93)], [(1,86),(2,87),(3,88),(4,89),(5,90),(6,81),(7,82),(8,83),(9,84),(10,85),(11,67),(12,68),(13,69),(14,70),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,78),(22,79),(23,80),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,101),(39,102),(40,103),(41,98),(42,99),(43,100),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)]])
S3×C22×C10 is a maximal subgroup of
C15⋊C22≀C2 (C2×C10)⋊11D12
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10AB | 10AC | ··· | 10BH | 15A | 15B | 15C | 15D | 30A | ··· | 30AB |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | C5×S3 | S3×C10 |
| kernel | S3×C22×C10 | S3×C2×C10 | C22×C30 | S3×C23 | C22×S3 | C22×C6 | C22×C10 | C2×C10 | C23 | C22 |
| # reps | 1 | 14 | 1 | 4 | 56 | 4 | 1 | 7 | 4 | 28 |
Matrix representation of S3×C22×C10 ►in GL4(𝔽31) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 1 | 30 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 30 | 0 |
G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[30,0,0,0,0,30,0,0,0,0,15,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,30],[30,0,0,0,0,30,0,0,0,0,0,30,0,0,30,0] >;
S3×C22×C10 in GAP, Magma, Sage, TeX
S_3\times C_2^2\times C_{10} % in TeX
G:=Group("S3xC2^2xC10"); // GroupNames label
G:=SmallGroup(240,206);
// by ID
G=gap.SmallGroup(240,206);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-3,5765]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^10=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations