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G = S3×C2×C20order 240 = 24·3·5

Direct product of C2×C20 and S3

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

Aliases: S3×C2×C20, C6013C22, C30.49C23, C309(C2×C4), C61(C2×C20), C123(C2×C10), (C2×C60)⋊13C2, (C2×C12)⋊5C10, C31(C22×C20), C1510(C22×C4), D6.4(C2×C10), (C2×C10).37D6, (C2×Dic3)⋊5C10, Dic33(C2×C10), C6.2(C22×C10), C22.9(S3×C10), (C10×Dic3)⋊11C2, (C22×S3).2C10, (C2×C30).48C22, C10.39(C22×S3), (S3×C10).15C22, (C5×Dic3)⋊10C22, C2.1(S3×C2×C10), (S3×C2×C10).4C2, (C2×C6).9(C2×C10), (C2×C20)(C5×Dic3), (C2×C20)(C10×Dic3), SmallGroup(240,166)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C2×C20
C1C3C6C30S3×C10S3×C2×C10 — S3×C2×C20
C3 — S3×C2×C20
C1C2×C20

Generators and relations for S3×C2×C20
 G = < a,b,c,d | a2=b20=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 184 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×5], C23, C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C15, C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×4], C30, C30 [×2], C2×C20, C2×C20 [×5], C22×C10, S3×C2×C4, C5×Dic3 [×2], C60 [×2], S3×C10 [×6], C2×C30, C22×C20, S3×C20 [×4], C10×Dic3, C2×C60, S3×C2×C10, S3×C2×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], D6 [×3], C22×C4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C2×C20 [×6], C22×C10, S3×C2×C4, S3×C10 [×3], C22×C20, S3×C20 [×2], S3×C2×C10, S3×C2×C20

Smallest permutation representation of S3×C2×C20
On 120 points
Generators in S120
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 116)(42 117)(43 118)(44 119)(45 120)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)
(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 59 93)(2 60 94)(3 41 95)(4 42 96)(5 43 97)(6 44 98)(7 45 99)(8 46 100)(9 47 81)(10 48 82)(11 49 83)(12 50 84)(13 51 85)(14 52 86)(15 53 87)(16 54 88)(17 55 89)(18 56 90)(19 57 91)(20 58 92)(21 70 109)(22 71 110)(23 72 111)(24 73 112)(25 74 113)(26 75 114)(27 76 115)(28 77 116)(29 78 117)(30 79 118)(31 80 119)(32 61 120)(33 62 101)(34 63 102)(35 64 103)(36 65 104)(37 66 105)(38 67 106)(39 68 107)(40 69 108)
(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)

G:=sub<Sym(120)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,116)(42,117)(43,118)(44,119)(45,120)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115), (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), (1,59,93)(2,60,94)(3,41,95)(4,42,96)(5,43,97)(6,44,98)(7,45,99)(8,46,100)(9,47,81)(10,48,82)(11,49,83)(12,50,84)(13,51,85)(14,52,86)(15,53,87)(16,54,88)(17,55,89)(18,56,90)(19,57,91)(20,58,92)(21,70,109)(22,71,110)(23,72,111)(24,73,112)(25,74,113)(26,75,114)(27,76,115)(28,77,116)(29,78,117)(30,79,118)(31,80,119)(32,61,120)(33,62,101)(34,63,102)(35,64,103)(36,65,104)(37,66,105)(38,67,106)(39,68,107)(40,69,108), (21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)>;

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

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

S3×C2×C20 is a maximal subgroup of
C60.94D4  (S3×C20)⋊5C4  D6⋊Dic5⋊C2  D6⋊Dic10  C60.45D4  C60.46D4  (S3×C20)⋊7C4  C1517(C4×D4)  C1522(C4×D4)  D10⋊C4⋊S3  D6⋊D20  C604D4  C606D4

120 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H5A5B5C5D6A6B6C10A···10L10M···10AB12A12B12C12D15A15B15C15D20A···20P20Q···20AF30A···30L60A···60P
order12222222344444444555566610···1010···10121212121515151520···2020···2030···3060···60
size1111333321111333311112221···13···3222222221···13···32···22···2

120 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C4C5C10C10C10C10C20S3D6D6C4×S3C5×S3S3×C10S3×C10S3×C20
kernelS3×C2×C20S3×C20C10×Dic3C2×C60S3×C2×C10S3×C10S3×C2×C4C4×S3C2×Dic3C2×C12C22×S3D6C2×C20C20C2×C10C10C2×C4C4C22C2
# reps14111841644432121448416

Matrix representation of S3×C2×C20 in GL4(𝔽61) generated by

60000
06000
00600
00060
,
9000
05000
0010
0001
,
1000
0100
00060
00160
,
60000
06000
0001
0010
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[9,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,1,0] >;

S3×C2×C20 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{20}
% in TeX

G:=Group("S3xC2xC20");
// GroupNames label

G:=SmallGroup(240,166);
// by ID

G=gap.SmallGroup(240,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,194,5765]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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