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## G = S3×C4○D20order 480 = 25·3·5

### Direct product of S3 and C4○D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×C4○D20
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C4×S3×D5 — S3×C4○D20
 Lower central C15 — C30 — S3×C4○D20
 Upper central C1 — C4 — C2×C4

Generators and relations for S3×C4○D20
G = < a,b,c,d,e | a3=b2=c4=e2=1, d10=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d9 >

Subgroups: 1612 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×2], S3 [×3], C6, C6 [×3], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C2×C6 [×2], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×4], Dic6 [×3], C4×S3 [×4], C4×S3 [×6], D12 [×3], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3, C22×S3 [×2], C5×S3 [×2], C5×S3, C3×D5 [×2], D15 [×2], C30, C30, C2×C4○D4, Dic10, Dic10 [×3], C4×D5 [×2], C4×D5 [×6], D20, D20 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×C10, S3×C2×C4, S3×C2×C4 [×2], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20, C4○D20 [×7], C2×C5⋊D4 [×2], C22×C20, S3×C4○D4, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], S3×C20 [×4], C10×Dic3, Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, C2×S3×D5 [×2], S3×C2×C10, C2×C4○D20, D205S3, S3×Dic10, D60⋊C2, D6.D10 [×2], C4×S3×D5 [×2], S3×D20, Dic5.D6 [×2], S3×C5⋊D4 [×2], C3×C4○D20, S3×C2×C20, D6011C2, S3×C4○D20
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C4○D20 [×2], C23×D5, S3×C4○D4, C2×S3×D5 [×3], C2×C4○D20, C22×S3×D5, S3×C4○D20

Smallest permutation representation of S3×C4○D20
On 120 points
Generators in S120
(1 31 43)(2 32 44)(3 33 45)(4 34 46)(5 35 47)(6 36 48)(7 37 49)(8 38 50)(9 39 51)(10 40 52)(11 21 53)(12 22 54)(13 23 55)(14 24 56)(15 25 57)(16 26 58)(17 27 59)(18 28 60)(19 29 41)(20 30 42)(61 114 100)(62 115 81)(63 116 82)(64 117 83)(65 118 84)(66 119 85)(67 120 86)(68 101 87)(69 102 88)(70 103 89)(71 104 90)(72 105 91)(73 106 92)(74 107 93)(75 108 94)(76 109 95)(77 110 96)(78 111 97)(79 112 98)(80 113 99)
(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)
(1 105 11 115)(2 106 12 116)(3 107 13 117)(4 108 14 118)(5 109 15 119)(6 110 16 120)(7 111 17 101)(8 112 18 102)(9 113 19 103)(10 114 20 104)(21 81 31 91)(22 82 32 92)(23 83 33 93)(24 84 34 94)(25 85 35 95)(26 86 36 96)(27 87 37 97)(28 88 38 98)(29 89 39 99)(30 90 40 100)(41 70 51 80)(42 71 52 61)(43 72 53 62)(44 73 54 63)(45 74 55 64)(46 75 56 65)(47 76 57 66)(48 77 58 67)(49 78 59 68)(50 79 60 69)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)

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

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

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

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 C4○D4 D10 D10 D10 D10 C4○D20 S3×D5 S3×C4○D4 C2×S3×D5 C2×S3×D5 S3×C4○D20 kernel S3×C4○D20 D20⋊5S3 S3×Dic10 D60⋊C2 D6.D10 C4×S3×D5 S3×D20 Dic5.D6 S3×C5⋊D4 C3×C4○D20 S3×C2×C20 D60⋊11C2 C4○D20 S3×C2×C4 Dic10 C4×D5 D20 C5⋊D4 C2×C20 C5×S3 C4×S3 C2×Dic3 C2×C12 C22×S3 S3 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 1 2 2 1 1 1 1 2 1 2 1 2 1 4 8 2 2 2 16 2 2 4 2 8

Matrix representation of S3×C4○D20 in GL4(𝔽61) generated by

 59 15 0 0 12 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 49 60 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 50 0 0 0 0 50
,
 60 0 0 0 0 60 0 0 0 0 54 29 0 0 32 59
,
 60 0 0 0 0 60 0 0 0 0 54 29 0 0 32 7
G:=sub<GL(4,GF(61))| [59,12,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[1,49,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,50,0,0,0,0,50],[60,0,0,0,0,60,0,0,0,0,54,32,0,0,29,59],[60,0,0,0,0,60,0,0,0,0,54,32,0,0,29,7] >;

S3×C4○D20 in GAP, Magma, Sage, TeX

S_3\times C_4\circ D_{20}
% in TeX

G:=Group("S3xC4oD20");
// GroupNames label

G:=SmallGroup(480,1091);
// by ID

G=gap.SmallGroup(480,1091);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^2=1,d^10=c^2,b*a*b=a^-1,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=c^2*d^9>;
// generators/relations

׿
×
𝔽