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G = S3×C22×C10order 240 = 24·3·5

Direct product of C22×C10 and S3

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C22×C10, C154C24, C304C23, 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

C1C3 — S3×C22×C10
C1C3C15C5×S3S3×C10S3×C2×C10 — S3×C22×C10
C3 — S3×C22×C10
C1C22×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 [×7], C2 [×8], C3, C22 [×7], C22 [×28], C5, S3 [×8], C6 [×7], C23, C23 [×14], C10 [×7], C10 [×8], D6 [×28], C2×C6 [×7], C15, C24, C2×C10 [×7], C2×C10 [×28], C22×S3 [×14], C22×C6, C5×S3 [×8], C30 [×7], C22×C10, C22×C10 [×14], S3×C23, S3×C10 [×28], C2×C30 [×7], C23×C10, S3×C2×C10 [×14], C22×C30, S3×C22×C10
Quotients: C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C22×C10 [×15], S3×C23, S3×C10 [×7], C23×C10, S3×C2×C10 [×7], S3×C22×C10

Smallest permutation representation of 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 45 39)(2 46 40)(3 47 31)(4 48 32)(5 49 33)(6 50 34)(7 41 35)(8 42 36)(9 43 37)(10 44 38)(11 30 111)(12 21 112)(13 22 113)(14 23 114)(15 24 115)(16 25 116)(17 26 117)(18 27 118)(19 28 119)(20 29 120)(51 80 64)(52 71 65)(53 72 66)(54 73 67)(55 74 68)(56 75 69)(57 76 70)(58 77 61)(59 78 62)(60 79 63)(81 110 94)(82 101 95)(83 102 96)(84 103 97)(85 104 98)(86 105 99)(87 106 100)(88 107 91)(89 108 92)(90 109 93)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 61)(19 62)(20 63)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 107)(32 108)(33 109)(34 110)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 91)(48 92)(49 93)(50 94)(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), (1,45,39)(2,46,40)(3,47,31)(4,48,32)(5,49,33)(6,50,34)(7,41,35)(8,42,36)(9,43,37)(10,44,38)(11,30,111)(12,21,112)(13,22,113)(14,23,114)(15,24,115)(16,25,116)(17,26,117)(18,27,118)(19,28,119)(20,29,120)(51,80,64)(52,71,65)(53,72,66)(54,73,67)(55,74,68)(56,75,69)(57,76,70)(58,77,61)(59,78,62)(60,79,63)(81,110,94)(82,101,95)(83,102,96)(84,103,97)(85,104,98)(86,105,99)(87,106,100)(88,107,91)(89,108,92)(90,109,93), (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,61)(19,62)(20,63)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,107)(32,108)(33,109)(34,110)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,91)(48,92)(49,93)(50,94)(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), (1,45,39)(2,46,40)(3,47,31)(4,48,32)(5,49,33)(6,50,34)(7,41,35)(8,42,36)(9,43,37)(10,44,38)(11,30,111)(12,21,112)(13,22,113)(14,23,114)(15,24,115)(16,25,116)(17,26,117)(18,27,118)(19,28,119)(20,29,120)(51,80,64)(52,71,65)(53,72,66)(54,73,67)(55,74,68)(56,75,69)(57,76,70)(58,77,61)(59,78,62)(60,79,63)(81,110,94)(82,101,95)(83,102,96)(84,103,97)(85,104,98)(86,105,99)(87,106,100)(88,107,91)(89,108,92)(90,109,93), (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,61)(19,62)(20,63)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,107)(32,108)(33,109)(34,110)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,91)(48,92)(49,93)(50,94)(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)], [(1,45,39),(2,46,40),(3,47,31),(4,48,32),(5,49,33),(6,50,34),(7,41,35),(8,42,36),(9,43,37),(10,44,38),(11,30,111),(12,21,112),(13,22,113),(14,23,114),(15,24,115),(16,25,116),(17,26,117),(18,27,118),(19,28,119),(20,29,120),(51,80,64),(52,71,65),(53,72,66),(54,73,67),(55,74,68),(56,75,69),(57,76,70),(58,77,61),(59,78,62),(60,79,63),(81,110,94),(82,101,95),(83,102,96),(84,103,97),(85,104,98),(86,105,99),(87,106,100),(88,107,91),(89,108,92),(90,109,93)], [(1,86),(2,87),(3,88),(4,89),(5,90),(6,81),(7,82),(8,83),(9,84),(10,85),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,61),(19,62),(20,63),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,107),(32,108),(33,109),(34,110),(35,101),(36,102),(37,103),(38,104),(39,105),(40,106),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,91),(48,92),(49,93),(50,94),(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···2G2H···2O 3 5A5B5C5D6A···6G10A···10AB10AC···10BH15A15B15C15D30A···30AB
order12···22···2355556···610···1010···101515151530···30
size11···13···3211112···21···13···322222···2

120 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10S3D6C5×S3S3×C10
kernelS3×C22×C10S3×C2×C10C22×C30S3×C23C22×S3C22×C6C22×C10C2×C10C23C22
# reps1141456417428

Matrix representation of S3×C22×C10 in GL4(𝔽31) generated by

1000
0100
00300
00030
,
1000
03000
00300
00030
,
30000
03000
00150
00015
,
1000
0100
00030
00130
,
30000
03000
00030
00300
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

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