Copied to
clipboard

## G = S3×C23.D5order 480 = 25·3·5

### Direct product of S3 and C23.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×C23.D5
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×S3×Dic5 — S3×C23.D5
 Lower central C15 — C30 — S3×C23.D5
 Upper central C1 — C22 — C23

Generators and relations for S3×C23.D5
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f5=1, g2=d, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg-1=ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg-1=f-1 >

Subgroups: 1036 in 264 conjugacy classes, 84 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, S3, C6, C6, C6, C2×C4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C5×S3, C5×S3, C30, C30, C30, C2×C22⋊C4, C2×Dic5, C2×Dic5, C22×C10, C22×C10, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C23.D5, C23.D5, C22×Dic5, C23×C10, S3×C22⋊C4, S3×Dic5, C6×Dic5, C2×Dic15, S3×C2×C10, S3×C2×C10, S3×C2×C10, C22×C30, C2×C23.D5, D6⋊Dic5, C3×C23.D5, C30.38D4, C2×S3×Dic5, S3×C22×C10, S3×C23.D5
Quotients:

Smallest permutation representation of S3×C23.D5
On 120 points
Generators in S120
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)(66 71)(67 72)(68 73)(69 74)(70 75)(81 86)(82 87)(83 88)(84 89)(85 90)(96 101)(97 102)(98 103)(99 104)(100 105)(111 116)(112 117)(113 118)(114 119)(115 120)
(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(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 93 31 63)(2 92 32 62)(3 91 33 61)(4 95 34 65)(5 94 35 64)(6 98 36 68)(7 97 37 67)(8 96 38 66)(9 100 39 70)(10 99 40 69)(11 103 41 73)(12 102 42 72)(13 101 43 71)(14 105 44 75)(15 104 45 74)(16 108 46 78)(17 107 47 77)(18 106 48 76)(19 110 49 80)(20 109 50 79)(21 113 51 83)(22 112 52 82)(23 111 53 81)(24 115 54 85)(25 114 55 84)(26 118 56 88)(27 117 57 87)(28 116 58 86)(29 120 59 90)(30 119 60 89)

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

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

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

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 Dic5 D10 D10 C4×S3 C5⋊D4 S3×D4 S3×D5 S3×Dic5 C2×S3×D5 S3×C5⋊D4 kernel S3×C23.D5 D6⋊Dic5 C3×C23.D5 C30.38D4 C2×S3×Dic5 S3×C22×C10 S3×C2×C10 C23.D5 S3×C10 S3×C23 C2×Dic5 C22×C10 C22×S3 C22×S3 C22×C6 C2×C10 D6 C10 C23 C22 C22 C2 # reps 1 2 1 1 2 1 8 1 4 2 2 1 8 4 2 4 16 2 2 4 2 8

Matrix representation of S3×C23.D5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 60 0 0 1 0
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 60 60
,
 1 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 34 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 0 52 0 0 34 0 0 0 0 0 50 0 0 0 0 50
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[34,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,34,0,0,52,0,0,0,0,0,50,0,0,0,0,50] >;

S3×C23.D5 in GAP, Magma, Sage, TeX

S_3\times C_2^3.D_5
% in TeX

G:=Group("S3xC2^3.D5");
// GroupNames label

G:=SmallGroup(480,630);
// by ID

G=gap.SmallGroup(480,630);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=e^2=f^5=1,g^2=d,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*c*g^-1=c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g^-1=f^-1>;
// generators/relations

׿
×
𝔽