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## G = S3×C2×C20order 240 = 24·3·5

### Direct product of C2×C20 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C20
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C2×C20
 Lower central C3 — S3×C2×C20
 Upper central C1 — C2×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 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10AB 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 30A ··· 30L 60A ··· 60P order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 3 3 3 3 2 1 1 1 1 3 3 3 3 1 1 1 1 2 2 2 1 ··· 1 3 ··· 3 2 2 2 2 2 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C20 S3 D6 D6 C4×S3 C5×S3 S3×C10 S3×C10 S3×C20 kernel S3×C2×C20 S3×C20 C10×Dic3 C2×C60 S3×C2×C10 S3×C10 S3×C2×C4 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C20 C20 C2×C10 C10 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 4 16 4 4 4 32 1 2 1 4 4 8 4 16

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

 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 60 0 0 1 60
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 1 0
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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