direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32×C3⋊C16, C33⋊5C16, C32⋊5C48, C3⋊(C3×C48), C6.(C3×C24), C24.9(C3×C6), (C3×C6).9C24, C24.28(C3×S3), (C3×C24).21C6, (C3×C24).28S3, C12.2(C3×C12), C8.2(S3×C32), (C32×C6).5C8, (C3×C12).22C12, (C32×C24).5C2, (C32×C12).10C4, C12.25(C3×Dic3), (C3×C12).32Dic3, C4.2(C32×Dic3), C2.(C32×C3⋊C8), C6.10(C3×C3⋊C8), (C3×C6).13(C3⋊C8), SmallGroup(432,229)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C32×C3⋊C16 |
Generators and relations for C32×C3⋊C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 136 in 92 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C8, C32, C32, C32, C12, C12, C12, C16, C3×C6, C3×C6, C3×C6, C24, C24, C24, C33, C3×C12, C3×C12, C3×C12, C3⋊C16, C48, C32×C6, C3×C24, C3×C24, C3×C24, C32×C12, C3×C3⋊C16, C3×C48, C32×C24, C32×C3⋊C16
Quotients: C1, C2, C3, C4, S3, C6, C8, C32, Dic3, C12, C16, C3×S3, C3×C6, C3⋊C8, C24, C3×Dic3, C3×C12, C3⋊C16, C48, S3×C32, C3×C3⋊C8, C3×C24, C32×Dic3, C3×C3⋊C16, C3×C48, C32×C3⋊C8, C32×C3⋊C16
►On 144 points
Generators in S144
(1 118 141)(2 119 142)(3 120 143)(4 121 144)(5 122 129)(6 123 130)(7 124 131)(8 125 132)(9 126 133)(10 127 134)(11 128 135)(12 113 136)(13 114 137)(14 115 138)(15 116 139)(16 117 140)(17 108 59)(18 109 60)(19 110 61)(20 111 62)(21 112 63)(22 97 64)(23 98 49)(24 99 50)(25 100 51)(26 101 52)(27 102 53)(28 103 54)(29 104 55)(30 105 56)(31 106 57)(32 107 58)(33 77 83)(34 78 84)(35 79 85)(36 80 86)(37 65 87)(38 66 88)(39 67 89)(40 68 90)(41 69 91)(42 70 92)(43 71 93)(44 72 94)(45 73 95)(46 74 96)(47 75 81)(48 76 82)
(1 84 27)(2 85 28)(3 86 29)(4 87 30)(5 88 31)(6 89 32)(7 90 17)(8 91 18)(9 92 19)(10 93 20)(11 94 21)(12 95 22)(13 96 23)(14 81 24)(15 82 25)(16 83 26)(33 101 117)(34 102 118)(35 103 119)(36 104 120)(37 105 121)(38 106 122)(39 107 123)(40 108 124)(41 109 125)(42 110 126)(43 111 127)(44 112 128)(45 97 113)(46 98 114)(47 99 115)(48 100 116)(49 137 74)(50 138 75)(51 139 76)(52 140 77)(53 141 78)(54 142 79)(55 143 80)(56 144 65)(57 129 66)(58 130 67)(59 131 68)(60 132 69)(61 133 70)(62 134 71)(63 135 72)(64 136 73)
(1 34 53)(2 54 35)(3 36 55)(4 56 37)(5 38 57)(6 58 39)(7 40 59)(8 60 41)(9 42 61)(10 62 43)(11 44 63)(12 64 45)(13 46 49)(14 50 47)(15 48 51)(16 52 33)(17 124 68)(18 69 125)(19 126 70)(20 71 127)(21 128 72)(22 73 113)(23 114 74)(24 75 115)(25 116 76)(26 77 117)(27 118 78)(28 79 119)(29 120 80)(30 65 121)(31 122 66)(32 67 123)(81 138 99)(82 100 139)(83 140 101)(84 102 141)(85 142 103)(86 104 143)(87 144 105)(88 106 129)(89 130 107)(90 108 131)(91 132 109)(92 110 133)(93 134 111)(94 112 135)(95 136 97)(96 98 137)
(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)
G:=sub<Sym(144)| (1,118,141)(2,119,142)(3,120,143)(4,121,144)(5,122,129)(6,123,130)(7,124,131)(8,125,132)(9,126,133)(10,127,134)(11,128,135)(12,113,136)(13,114,137)(14,115,138)(15,116,139)(16,117,140)(17,108,59)(18,109,60)(19,110,61)(20,111,62)(21,112,63)(22,97,64)(23,98,49)(24,99,50)(25,100,51)(26,101,52)(27,102,53)(28,103,54)(29,104,55)(30,105,56)(31,106,57)(32,107,58)(33,77,83)(34,78,84)(35,79,85)(36,80,86)(37,65,87)(38,66,88)(39,67,89)(40,68,90)(41,69,91)(42,70,92)(43,71,93)(44,72,94)(45,73,95)(46,74,96)(47,75,81)(48,76,82), (1,84,27)(2,85,28)(3,86,29)(4,87,30)(5,88,31)(6,89,32)(7,90,17)(8,91,18)(9,92,19)(10,93,20)(11,94,21)(12,95,22)(13,96,23)(14,81,24)(15,82,25)(16,83,26)(33,101,117)(34,102,118)(35,103,119)(36,104,120)(37,105,121)(38,106,122)(39,107,123)(40,108,124)(41,109,125)(42,110,126)(43,111,127)(44,112,128)(45,97,113)(46,98,114)(47,99,115)(48,100,116)(49,137,74)(50,138,75)(51,139,76)(52,140,77)(53,141,78)(54,142,79)(55,143,80)(56,144,65)(57,129,66)(58,130,67)(59,131,68)(60,132,69)(61,133,70)(62,134,71)(63,135,72)(64,136,73), (1,34,53)(2,54,35)(3,36,55)(4,56,37)(5,38,57)(6,58,39)(7,40,59)(8,60,41)(9,42,61)(10,62,43)(11,44,63)(12,64,45)(13,46,49)(14,50,47)(15,48,51)(16,52,33)(17,124,68)(18,69,125)(19,126,70)(20,71,127)(21,128,72)(22,73,113)(23,114,74)(24,75,115)(25,116,76)(26,77,117)(27,118,78)(28,79,119)(29,120,80)(30,65,121)(31,122,66)(32,67,123)(81,138,99)(82,100,139)(83,140,101)(84,102,141)(85,142,103)(86,104,143)(87,144,105)(88,106,129)(89,130,107)(90,108,131)(91,132,109)(92,110,133)(93,134,111)(94,112,135)(95,136,97)(96,98,137), (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)>;
G:=Group( (1,118,141)(2,119,142)(3,120,143)(4,121,144)(5,122,129)(6,123,130)(7,124,131)(8,125,132)(9,126,133)(10,127,134)(11,128,135)(12,113,136)(13,114,137)(14,115,138)(15,116,139)(16,117,140)(17,108,59)(18,109,60)(19,110,61)(20,111,62)(21,112,63)(22,97,64)(23,98,49)(24,99,50)(25,100,51)(26,101,52)(27,102,53)(28,103,54)(29,104,55)(30,105,56)(31,106,57)(32,107,58)(33,77,83)(34,78,84)(35,79,85)(36,80,86)(37,65,87)(38,66,88)(39,67,89)(40,68,90)(41,69,91)(42,70,92)(43,71,93)(44,72,94)(45,73,95)(46,74,96)(47,75,81)(48,76,82), (1,84,27)(2,85,28)(3,86,29)(4,87,30)(5,88,31)(6,89,32)(7,90,17)(8,91,18)(9,92,19)(10,93,20)(11,94,21)(12,95,22)(13,96,23)(14,81,24)(15,82,25)(16,83,26)(33,101,117)(34,102,118)(35,103,119)(36,104,120)(37,105,121)(38,106,122)(39,107,123)(40,108,124)(41,109,125)(42,110,126)(43,111,127)(44,112,128)(45,97,113)(46,98,114)(47,99,115)(48,100,116)(49,137,74)(50,138,75)(51,139,76)(52,140,77)(53,141,78)(54,142,79)(55,143,80)(56,144,65)(57,129,66)(58,130,67)(59,131,68)(60,132,69)(61,133,70)(62,134,71)(63,135,72)(64,136,73), (1,34,53)(2,54,35)(3,36,55)(4,56,37)(5,38,57)(6,58,39)(7,40,59)(8,60,41)(9,42,61)(10,62,43)(11,44,63)(12,64,45)(13,46,49)(14,50,47)(15,48,51)(16,52,33)(17,124,68)(18,69,125)(19,126,70)(20,71,127)(21,128,72)(22,73,113)(23,114,74)(24,75,115)(25,116,76)(26,77,117)(27,118,78)(28,79,119)(29,120,80)(30,65,121)(31,122,66)(32,67,123)(81,138,99)(82,100,139)(83,140,101)(84,102,141)(85,142,103)(86,104,143)(87,144,105)(88,106,129)(89,130,107)(90,108,131)(91,132,109)(92,110,133)(93,134,111)(94,112,135)(95,136,97)(96,98,137), (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) );
G=PermutationGroup([[(1,118,141),(2,119,142),(3,120,143),(4,121,144),(5,122,129),(6,123,130),(7,124,131),(8,125,132),(9,126,133),(10,127,134),(11,128,135),(12,113,136),(13,114,137),(14,115,138),(15,116,139),(16,117,140),(17,108,59),(18,109,60),(19,110,61),(20,111,62),(21,112,63),(22,97,64),(23,98,49),(24,99,50),(25,100,51),(26,101,52),(27,102,53),(28,103,54),(29,104,55),(30,105,56),(31,106,57),(32,107,58),(33,77,83),(34,78,84),(35,79,85),(36,80,86),(37,65,87),(38,66,88),(39,67,89),(40,68,90),(41,69,91),(42,70,92),(43,71,93),(44,72,94),(45,73,95),(46,74,96),(47,75,81),(48,76,82)], [(1,84,27),(2,85,28),(3,86,29),(4,87,30),(5,88,31),(6,89,32),(7,90,17),(8,91,18),(9,92,19),(10,93,20),(11,94,21),(12,95,22),(13,96,23),(14,81,24),(15,82,25),(16,83,26),(33,101,117),(34,102,118),(35,103,119),(36,104,120),(37,105,121),(38,106,122),(39,107,123),(40,108,124),(41,109,125),(42,110,126),(43,111,127),(44,112,128),(45,97,113),(46,98,114),(47,99,115),(48,100,116),(49,137,74),(50,138,75),(51,139,76),(52,140,77),(53,141,78),(54,142,79),(55,143,80),(56,144,65),(57,129,66),(58,130,67),(59,131,68),(60,132,69),(61,133,70),(62,134,71),(63,135,72),(64,136,73)], [(1,34,53),(2,54,35),(3,36,55),(4,56,37),(5,38,57),(6,58,39),(7,40,59),(8,60,41),(9,42,61),(10,62,43),(11,44,63),(12,64,45),(13,46,49),(14,50,47),(15,48,51),(16,52,33),(17,124,68),(18,69,125),(19,126,70),(20,71,127),(21,128,72),(22,73,113),(23,114,74),(24,75,115),(25,116,76),(26,77,117),(27,118,78),(28,79,119),(29,120,80),(30,65,121),(31,122,66),(32,67,123),(81,138,99),(82,100,139),(83,140,101),(84,102,141),(85,142,103),(86,104,143),(87,144,105),(88,106,129),(89,130,107),(90,108,131),(91,132,109),(92,110,133),(93,134,111),(94,112,135),(95,136,97),(96,98,137)], [(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)]])
216 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12AH | 16A | ··· | 16H | 24A | ··· | 24AF | 24AG | ··· | 24BP | 48A | ··· | 48BL |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3⋊C16 | C3×C3⋊C8 | C3×C3⋊C16 |
| kernel | C32×C3⋊C16 | C32×C24 | C3×C3⋊C16 | C32×C12 | C3×C24 | C32×C6 | C3×C12 | C33 | C3×C6 | C32 | C3×C24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 8 | 32 | 64 | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C32×C3⋊C16 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 61 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 61 |
| 70 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 75 | 0 |
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,35],[61,0,0,0,1,0,0,0,1],[1,0,0,0,35,0,0,0,61],[70,0,0,0,0,75,0,1,0] >;
C32×C3⋊C16 in GAP, Magma, Sage, TeX
C_3^2\times C_3\rtimes C_{16} % in TeX
G:=Group("C3^2xC3:C16"); // GroupNames label
G:=SmallGroup(432,229);
// by ID
G=gap.SmallGroup(432,229);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-2,-2,-3,126,80,102,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations