Copied to
clipboard

## G = S3×Dic18order 432 = 24·33

### Direct product of S3 and Dic18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — S3×Dic18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — S3×Dic9 — S3×Dic18
 Lower central C3×C9 — C3×C18 — S3×Dic18
 Upper central C1 — C2 — C4

Generators and relations for S3×Dic18
G = < a,b,c,d | a3=b2=c36=1, d2=c18, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 656 in 126 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×C9, Dic9, Dic9, C36, C36, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×Dic6, S3×Q8, S3×C9, C3×C18, Dic18, Dic18, C2×Dic9, C2×C36, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C324Q8, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, S3×C18, C2×Dic18, S3×Dic6, C9⋊Dic6, S3×Dic9, C3×Dic18, S3×C36, C12.D9, S3×Dic18
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, S32, C2×Dic6, S3×Q8, Dic18, C22×D9, C2×S32, S3×D9, C2×Dic18, S3×Dic6, C2×S3×D9, S3×Dic18

Smallest permutation representation of S3×Dic18
On 144 points
Generators in S144
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)(73 85 97)(74 86 98)(75 87 99)(76 88 100)(77 89 101)(78 90 102)(79 91 103)(80 92 104)(81 93 105)(82 94 106)(83 95 107)(84 96 108)(109 133 121)(110 134 122)(111 135 123)(112 136 124)(113 137 125)(114 138 126)(115 139 127)(116 140 128)(117 141 129)(118 142 130)(119 143 131)(120 144 132)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 41 19 59)(2 40 20 58)(3 39 21 57)(4 38 22 56)(5 37 23 55)(6 72 24 54)(7 71 25 53)(8 70 26 52)(9 69 27 51)(10 68 28 50)(11 67 29 49)(12 66 30 48)(13 65 31 47)(14 64 32 46)(15 63 33 45)(16 62 34 44)(17 61 35 43)(18 60 36 42)(73 140 91 122)(74 139 92 121)(75 138 93 120)(76 137 94 119)(77 136 95 118)(78 135 96 117)(79 134 97 116)(80 133 98 115)(81 132 99 114)(82 131 100 113)(83 130 101 112)(84 129 102 111)(85 128 103 110)(86 127 104 109)(87 126 105 144)(88 125 106 143)(89 124 107 142)(90 123 108 141)

G:=sub<Sym(144)| (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,85,97)(74,86,98)(75,87,99)(76,88,100)(77,89,101)(78,90,102)(79,91,103)(80,92,104)(81,93,105)(82,94,106)(83,95,107)(84,96,108)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,41,19,59)(2,40,20,58)(3,39,21,57)(4,38,22,56)(5,37,23,55)(6,72,24,54)(7,71,25,53)(8,70,26,52)(9,69,27,51)(10,68,28,50)(11,67,29,49)(12,66,30,48)(13,65,31,47)(14,64,32,46)(15,63,33,45)(16,62,34,44)(17,61,35,43)(18,60,36,42)(73,140,91,122)(74,139,92,121)(75,138,93,120)(76,137,94,119)(77,136,95,118)(78,135,96,117)(79,134,97,116)(80,133,98,115)(81,132,99,114)(82,131,100,113)(83,130,101,112)(84,129,102,111)(85,128,103,110)(86,127,104,109)(87,126,105,144)(88,125,106,143)(89,124,107,142)(90,123,108,141)>;

G:=Group( (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,85,97)(74,86,98)(75,87,99)(76,88,100)(77,89,101)(78,90,102)(79,91,103)(80,92,104)(81,93,105)(82,94,106)(83,95,107)(84,96,108)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,41,19,59)(2,40,20,58)(3,39,21,57)(4,38,22,56)(5,37,23,55)(6,72,24,54)(7,71,25,53)(8,70,26,52)(9,69,27,51)(10,68,28,50)(11,67,29,49)(12,66,30,48)(13,65,31,47)(14,64,32,46)(15,63,33,45)(16,62,34,44)(17,61,35,43)(18,60,36,42)(73,140,91,122)(74,139,92,121)(75,138,93,120)(76,137,94,119)(77,136,95,118)(78,135,96,117)(79,134,97,116)(80,133,98,115)(81,132,99,114)(82,131,100,113)(83,130,101,112)(84,129,102,111)(85,128,103,110)(86,127,104,109)(87,126,105,144)(88,125,106,143)(89,124,107,142)(90,123,108,141) );

G=PermutationGroup([[(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,49,61),(38,50,62),(39,51,63),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69),(46,58,70),(47,59,71),(48,60,72),(73,85,97),(74,86,98),(75,87,99),(76,88,100),(77,89,101),(78,90,102),(79,91,103),(80,92,104),(81,93,105),(82,94,106),(83,95,107),(84,96,108),(109,133,121),(110,134,122),(111,135,123),(112,136,124),(113,137,125),(114,138,126),(115,139,127),(116,140,128),(117,141,129),(118,142,130),(119,143,131),(120,144,132)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,136),(38,137),(39,138),(40,139),(41,140),(42,141),(43,142),(44,143),(45,144),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135)], [(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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,41,19,59),(2,40,20,58),(3,39,21,57),(4,38,22,56),(5,37,23,55),(6,72,24,54),(7,71,25,53),(8,70,26,52),(9,69,27,51),(10,68,28,50),(11,67,29,49),(12,66,30,48),(13,65,31,47),(14,64,32,46),(15,63,33,45),(16,62,34,44),(17,61,35,43),(18,60,36,42),(73,140,91,122),(74,139,92,121),(75,138,93,120),(76,137,94,119),(77,136,95,118),(78,135,96,117),(79,134,97,116),(80,133,98,115),(81,132,99,114),(82,131,100,113),(83,130,101,112),(84,129,102,111),(85,128,103,110),(86,127,104,109),(87,126,105,144),(88,125,106,143),(89,124,107,142),(90,123,108,141)]])

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 12I 18A 18B 18C 18D 18E 18F 18G ··· 18L 36A ··· 36F 36G ··· 36L 36M ··· 36R order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 3 3 2 2 4 2 6 18 18 54 54 2 2 4 6 6 2 2 2 4 4 4 2 2 4 4 4 6 6 36 36 2 2 2 4 4 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 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 4 type + + + + + + + + - + + + + + + - + + + - + - + + - + - image C1 C2 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 D6 D6 D9 Dic6 D18 D18 D18 Dic18 S32 S3×Q8 C2×S32 S3×D9 S3×Dic6 C2×S3×D9 S3×Dic18 kernel S3×Dic18 C9⋊Dic6 S3×Dic9 C3×Dic18 S3×C36 C12.D9 Dic18 S3×C12 S3×C9 Dic9 C36 C3×Dic3 C3×C12 S3×C6 C4×S3 C3×S3 Dic3 C12 D6 S3 C12 C9 C6 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 2 2 1 1 1 1 3 4 3 3 3 12 1 1 1 3 2 3 6

Matrix representation of S3×Dic18 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 36 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 15 34 0 0 0 0 26 22 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 7 0 0 0 0 0 0 16
,
 27 4 0 0 0 0 21 10 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 12 0 0 0 0 34 0

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,26,0,0,0,0,34,22,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,7,0,0,0,0,0,0,16],[27,21,0,0,0,0,4,10,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,34,0,0,0,0,12,0] >;

S3×Dic18 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{18}
% in TeX

G:=Group("S3xDic18");
// GroupNames label

G:=SmallGroup(432,284);
// by ID

G=gap.SmallGroup(432,284);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^36=1,d^2=c^18,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽