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G = S3×C40⋊C2order 480 = 25·3·5

Direct product of S3 and C40⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — S3×C40⋊C2
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — S3×D20 — S3×C40⋊C2
 Lower central C15 — C30 — C60 — S3×C40⋊C2
 Upper central C1 — C2 — C4 — C8

Generators and relations for S3×C40⋊C2
G = < a,b,c,d | a3=b2=c40=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c19 >

Subgroups: 972 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×2], S3, C6, C6, C8, C8, C2×C4 [×2], D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20, C20, D10 [×4], C2×C10, C3⋊C8, C24, Dic6 [×2], C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C3×D5, D15, C30, C2×SD16, C40, C40, Dic10, Dic10 [×2], D20, D20 [×2], C2×Dic5, C2×C20, C22×D5, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C40⋊C2, C40⋊C2 [×3], C2×C40, C2×Dic10, C2×D20, S3×SD16, C5×C3⋊C8, C120, S3×Dic5, C3⋊D20, C15⋊Q8, C3×Dic10, C3×D20, S3×C20, Dic30, D60, C2×S3×D5, C2×C40⋊C2, C6.D20, C15⋊SD16, C3×C40⋊C2, S3×C40, C24⋊D5, S3×Dic10, S3×D20, S3×C40⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], C22×S3, C2×SD16, D20 [×2], C22×D5, S3×D4, S3×D5, C40⋊C2 [×2], C2×D20, S3×SD16, C2×S3×D5, C2×C40⋊C2, S3×D20, S3×C40⋊C2

Smallest permutation representation of S3×C40⋊C2
On 120 points
Generators in S120
(1 92 75)(2 93 76)(3 94 77)(4 95 78)(5 96 79)(6 97 80)(7 98 41)(8 99 42)(9 100 43)(10 101 44)(11 102 45)(12 103 46)(13 104 47)(14 105 48)(15 106 49)(16 107 50)(17 108 51)(18 109 52)(19 110 53)(20 111 54)(21 112 55)(22 113 56)(23 114 57)(24 115 58)(25 116 59)(26 117 60)(27 118 61)(28 119 62)(29 120 63)(30 81 64)(31 82 65)(32 83 66)(33 84 67)(34 85 68)(35 86 69)(36 87 70)(37 88 71)(38 89 72)(39 90 73)(40 91 74)
(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 69)(42 48)(43 67)(44 46)(45 65)(47 63)(49 61)(50 80)(51 59)(52 78)(53 57)(54 76)(56 74)(58 72)(60 70)(62 68)(64 66)(71 79)(73 77)(81 83)(82 102)(84 100)(85 119)(86 98)(87 117)(88 96)(89 115)(90 94)(91 113)(93 111)(95 109)(97 107)(99 105)(101 103)(104 120)(106 118)(108 116)(110 114)

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

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

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

69 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 24A 24B 30A 30B 40A ··· 40H 40I ··· 40P 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 8 8 8 8 10 10 10 10 10 10 12 12 15 15 20 20 20 20 20 20 20 20 24 24 30 30 40 ··· 40 40 ··· 40 60 60 60 60 120 ··· 120 size 1 1 3 3 20 60 2 2 6 20 60 2 2 2 40 2 2 6 6 2 2 6 6 6 6 4 40 4 4 2 2 2 2 6 6 6 6 4 4 4 4 2 ··· 2 6 ··· 6 4 4 4 4 4 ··· 4

69 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 SD16 D10 D10 D10 D20 D20 C40⋊C2 S3×D4 S3×D5 S3×SD16 C2×S3×D5 S3×D20 S3×C40⋊C2 kernel S3×C40⋊C2 C6.D20 C15⋊SD16 C3×C40⋊C2 S3×C40 C24⋊D5 S3×Dic10 S3×D20 C40⋊C2 C5×Dic3 S3×C10 S3×C8 C40 Dic10 D20 C5×S3 C3⋊C8 C24 C4×S3 Dic3 D6 S3 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 16 1 2 2 2 4 8

Matrix representation of S3×C40⋊C2 in GL4(𝔽241) generated by

 240 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 240 240 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 200 31 0 0 210 94
,
 1 0 0 0 0 1 0 0 0 0 240 0 0 0 190 1
G:=sub<GL(4,GF(241))| [240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,200,210,0,0,31,94],[1,0,0,0,0,1,0,0,0,0,240,190,0,0,0,1] >;

S3×C40⋊C2 in GAP, Magma, Sage, TeX

S_3\times C_{40}\rtimes C_2
% in TeX

G:=Group("S3xC40:C2");
// GroupNames label

G:=SmallGroup(480,327);
// by ID

G=gap.SmallGroup(480,327);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,58,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^40=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^19>;
// generators/relations

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