direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C9⋊C8, C36.41D6, C12.41D18, D6.2Dic9, Dic3.2Dic9, (S3×C9)⋊C8, C9⋊3(S3×C8), C12.63S32, (C4×S3).3D9, C4.26(S3×D9), (S3×C18).2C4, (S3×C36).3C2, C18.17(C4×S3), C6.1(C2×Dic9), C2.1(S3×Dic9), (S3×C12).10S3, C36.S3⋊9C2, (C3×C12).159D6, (C9×Dic3).2C4, (S3×C6).1Dic3, C6.22(S3×Dic3), (C3×C36).40C22, (C3×Dic3).4Dic3, C3⋊1(C2×C9⋊C8), (C3×C9⋊C8)⋊9C2, (C3×C9)⋊3(C2×C8), C3.3(S3×C3⋊C8), (C3×S3).(C3⋊C8), C32.2(C2×C3⋊C8), (C3×C18).5(C2×C4), (C3×C6).29(C2×Dic3), SmallGroup(432,66)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — S3×C9⋊C8 |
Generators and relations for S3×C9⋊C8
G = < a,b,c,d | a3=b2=c9=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 244 in 74 conjugacy classes, 37 normal (31 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C3⋊C8, S3×C9, C3×C18, C9⋊C8, C9⋊C8, C2×C36, C3×C3⋊C8, C32⋊4C8, S3×C12, C9×Dic3, C3×C36, S3×C18, C2×C9⋊C8, S3×C3⋊C8, C3×C9⋊C8, C36.S3, S3×C36, S3×C9⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, D9, C3⋊C8, C4×S3, C2×Dic3, Dic9, D18, S32, S3×C8, C2×C3⋊C8, C9⋊C8, C2×Dic9, S3×Dic3, S3×D9, C2×C9⋊C8, S3×C3⋊C8, S3×Dic9, S3×C9⋊C8
►On 144 points
Generators in S144
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 73)(8 74)(9 75)(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)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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 68 31 50 13 59 22 41)(2 67 32 49 14 58 23 40)(3 66 33 48 15 57 24 39)(4 65 34 47 16 56 25 38)(5 64 35 46 17 55 26 37)(6 72 36 54 18 63 27 45)(7 71 28 53 10 62 19 44)(8 70 29 52 11 61 20 43)(9 69 30 51 12 60 21 42)(73 143 100 125 82 134 91 116)(74 142 101 124 83 133 92 115)(75 141 102 123 84 132 93 114)(76 140 103 122 85 131 94 113)(77 139 104 121 86 130 95 112)(78 138 105 120 87 129 96 111)(79 137 106 119 88 128 97 110)(80 136 107 118 89 127 98 109)(81 144 108 126 90 135 99 117)
G:=sub<Sym(144)| (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,73)(8,74)(9,75)(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)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,68,31,50,13,59,22,41)(2,67,32,49,14,58,23,40)(3,66,33,48,15,57,24,39)(4,65,34,47,16,56,25,38)(5,64,35,46,17,55,26,37)(6,72,36,54,18,63,27,45)(7,71,28,53,10,62,19,44)(8,70,29,52,11,61,20,43)(9,69,30,51,12,60,21,42)(73,143,100,125,82,134,91,116)(74,142,101,124,83,133,92,115)(75,141,102,123,84,132,93,114)(76,140,103,122,85,131,94,113)(77,139,104,121,86,130,95,112)(78,138,105,120,87,129,96,111)(79,137,106,119,88,128,97,110)(80,136,107,118,89,127,98,109)(81,144,108,126,90,135,99,117)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,73)(8,74)(9,75)(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)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,68,31,50,13,59,22,41)(2,67,32,49,14,58,23,40)(3,66,33,48,15,57,24,39)(4,65,34,47,16,56,25,38)(5,64,35,46,17,55,26,37)(6,72,36,54,18,63,27,45)(7,71,28,53,10,62,19,44)(8,70,29,52,11,61,20,43)(9,69,30,51,12,60,21,42)(73,143,100,125,82,134,91,116)(74,142,101,124,83,133,92,115)(75,141,102,123,84,132,93,114)(76,140,103,122,85,131,94,113)(77,139,104,121,86,130,95,112)(78,138,105,120,87,129,96,111)(79,137,106,119,88,128,97,110)(80,136,107,118,89,127,98,109)(81,144,108,126,90,135,99,117) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,106,103),(101,107,104),(102,108,105),(109,112,115),(110,113,116),(111,114,117),(118,121,124),(119,122,125),(120,123,126),(127,130,133),(128,131,134),(129,132,135),(136,139,142),(137,140,143),(138,141,144)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,73),(8,74),(9,75),(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),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(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)], [(1,68,31,50,13,59,22,41),(2,67,32,49,14,58,23,40),(3,66,33,48,15,57,24,39),(4,65,34,47,16,56,25,38),(5,64,35,46,17,55,26,37),(6,72,36,54,18,63,27,45),(7,71,28,53,10,62,19,44),(8,70,29,52,11,61,20,43),(9,69,30,51,12,60,21,42),(73,143,100,125,82,134,91,116),(74,142,101,124,83,133,92,115),(75,141,102,123,84,132,93,114),(76,140,103,122,85,131,94,113),(77,139,104,121,86,130,95,112),(78,138,105,120,87,129,96,111),(79,137,106,119,88,128,97,110),(80,136,107,118,89,127,98,109),(81,144,108,126,90,135,99,117)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 36M | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 6 | 6 | 9 | 9 | 9 | 9 | 27 | 27 | 27 | 27 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
72 irreducible representations
| dim | 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 | C4 | C4 | C8 | S3 | S3 | D6 | Dic3 | D6 | Dic3 | D9 | C4×S3 | C3⋊C8 | Dic9 | D18 | Dic9 | S3×C8 | C9⋊C8 | S32 | S3×Dic3 | S3×D9 | S3×C3⋊C8 | S3×Dic9 | S3×C9⋊C8 |
| kernel | S3×C9⋊C8 | C3×C9⋊C8 | C36.S3 | S3×C36 | C9×Dic3 | S3×C18 | S3×C9 | C9⋊C8 | S3×C12 | C36 | C3×Dic3 | C3×C12 | S3×C6 | C4×S3 | C18 | C3×S3 | Dic3 | C12 | D6 | C9 | S3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 4 | 3 | 3 | 3 | 4 | 12 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of S3×C9⋊C8 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 37 |
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,37],[63,0,0,0,0,63,0,0,0,0,0,8,0,0,9,0] >;
S3×C9⋊C8 in GAP, Magma, Sage, TeX
S_3\times C_9\rtimes C_8
% in TeX
G:=Group("S3xC9:C8"); // GroupNames label
G:=SmallGroup(432,66);
// by ID
G=gap.SmallGroup(432,66);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^9=d^8=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^-1=c^-1>;
// generators/relations