metabelian, supersoluble, monomial
Aliases: He3⋊3C16, C32⋊2C48, C24.S3⋊C3, (C3×C6).2C24, (C3×C24).5C6, (C3×C24).6S3, C24.15(C3×S3), C32⋊2(C3⋊C16), (C3×C12).2C12, C2.(He3⋊3C8), (C4×He3).6C4, (C8×He3).4C2, (C2×He3).3C8, C8.2(C32⋊C6), (C3×C12).5Dic3, C12.9(C3×Dic3), C4.2(C32⋊C12), C6.2(C3×C3⋊C8), C3.2(C3×C3⋊C16), (C3×C6).2(C3⋊C8), SmallGroup(432,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — He3⋊3C16 |
Generators and relations for He3⋊3C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, dbd-1=b-1, cd=dc >
Smallest permutation representation of He3⋊3C16
►On 144 points
Generators in S144
(1 51 39)(2 40 52)(3 53 41)(4 42 54)(5 55 43)(6 44 56)(7 57 45)(8 46 58)(9 59 47)(10 48 60)(11 61 33)(12 34 62)(13 63 35)(14 36 64)(15 49 37)(16 38 50)(17 116 77)(18 78 117)(19 118 79)(20 80 119)(21 120 65)(22 66 121)(23 122 67)(24 68 123)(25 124 69)(26 70 125)(27 126 71)(28 72 127)(29 128 73)(30 74 113)(31 114 75)(32 76 115)(81 106 135)(82 136 107)(83 108 137)(84 138 109)(85 110 139)(86 140 111)(87 112 141)(88 142 97)(89 98 143)(90 144 99)(91 100 129)(92 130 101)(93 102 131)(94 132 103)(95 104 133)(96 134 105)
(1 97 76)(2 77 98)(3 99 78)(4 79 100)(5 101 80)(6 65 102)(7 103 66)(8 67 104)(9 105 68)(10 69 106)(11 107 70)(12 71 108)(13 109 72)(14 73 110)(15 111 74)(16 75 112)(17 143 40)(18 41 144)(19 129 42)(20 43 130)(21 131 44)(22 45 132)(23 133 46)(24 47 134)(25 135 48)(26 33 136)(27 137 34)(28 35 138)(29 139 36)(30 37 140)(31 141 38)(32 39 142)(49 86 113)(50 114 87)(51 88 115)(52 116 89)(53 90 117)(54 118 91)(55 92 119)(56 120 93)(57 94 121)(58 122 95)(59 96 123)(60 124 81)(61 82 125)(62 126 83)(63 84 127)(64 128 85)
(17 143 40)(18 144 41)(19 129 42)(20 130 43)(21 131 44)(22 132 45)(23 133 46)(24 134 47)(25 135 48)(26 136 33)(27 137 34)(28 138 35)(29 139 36)(30 140 37)(31 141 38)(32 142 39)(49 86 113)(50 87 114)(51 88 115)(52 89 116)(53 90 117)(54 91 118)(55 92 119)(56 93 120)(57 94 121)(58 95 122)(59 96 123)(60 81 124)(61 82 125)(62 83 126)(63 84 127)(64 85 128)
(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)| (1,51,39)(2,40,52)(3,53,41)(4,42,54)(5,55,43)(6,44,56)(7,57,45)(8,46,58)(9,59,47)(10,48,60)(11,61,33)(12,34,62)(13,63,35)(14,36,64)(15,49,37)(16,38,50)(17,116,77)(18,78,117)(19,118,79)(20,80,119)(21,120,65)(22,66,121)(23,122,67)(24,68,123)(25,124,69)(26,70,125)(27,126,71)(28,72,127)(29,128,73)(30,74,113)(31,114,75)(32,76,115)(81,106,135)(82,136,107)(83,108,137)(84,138,109)(85,110,139)(86,140,111)(87,112,141)(88,142,97)(89,98,143)(90,144,99)(91,100,129)(92,130,101)(93,102,131)(94,132,103)(95,104,133)(96,134,105), (1,97,76)(2,77,98)(3,99,78)(4,79,100)(5,101,80)(6,65,102)(7,103,66)(8,67,104)(9,105,68)(10,69,106)(11,107,70)(12,71,108)(13,109,72)(14,73,110)(15,111,74)(16,75,112)(17,143,40)(18,41,144)(19,129,42)(20,43,130)(21,131,44)(22,45,132)(23,133,46)(24,47,134)(25,135,48)(26,33,136)(27,137,34)(28,35,138)(29,139,36)(30,37,140)(31,141,38)(32,39,142)(49,86,113)(50,114,87)(51,88,115)(52,116,89)(53,90,117)(54,118,91)(55,92,119)(56,120,93)(57,94,121)(58,122,95)(59,96,123)(60,124,81)(61,82,125)(62,126,83)(63,84,127)(64,128,85), (17,143,40)(18,144,41)(19,129,42)(20,130,43)(21,131,44)(22,132,45)(23,133,46)(24,134,47)(25,135,48)(26,136,33)(27,137,34)(28,138,35)(29,139,36)(30,140,37)(31,141,38)(32,142,39)(49,86,113)(50,87,114)(51,88,115)(52,89,116)(53,90,117)(54,91,118)(55,92,119)(56,93,120)(57,94,121)(58,95,122)(59,96,123)(60,81,124)(61,82,125)(62,83,126)(63,84,127)(64,85,128), (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( (1,51,39)(2,40,52)(3,53,41)(4,42,54)(5,55,43)(6,44,56)(7,57,45)(8,46,58)(9,59,47)(10,48,60)(11,61,33)(12,34,62)(13,63,35)(14,36,64)(15,49,37)(16,38,50)(17,116,77)(18,78,117)(19,118,79)(20,80,119)(21,120,65)(22,66,121)(23,122,67)(24,68,123)(25,124,69)(26,70,125)(27,126,71)(28,72,127)(29,128,73)(30,74,113)(31,114,75)(32,76,115)(81,106,135)(82,136,107)(83,108,137)(84,138,109)(85,110,139)(86,140,111)(87,112,141)(88,142,97)(89,98,143)(90,144,99)(91,100,129)(92,130,101)(93,102,131)(94,132,103)(95,104,133)(96,134,105), (1,97,76)(2,77,98)(3,99,78)(4,79,100)(5,101,80)(6,65,102)(7,103,66)(8,67,104)(9,105,68)(10,69,106)(11,107,70)(12,71,108)(13,109,72)(14,73,110)(15,111,74)(16,75,112)(17,143,40)(18,41,144)(19,129,42)(20,43,130)(21,131,44)(22,45,132)(23,133,46)(24,47,134)(25,135,48)(26,33,136)(27,137,34)(28,35,138)(29,139,36)(30,37,140)(31,141,38)(32,39,142)(49,86,113)(50,114,87)(51,88,115)(52,116,89)(53,90,117)(54,118,91)(55,92,119)(56,120,93)(57,94,121)(58,122,95)(59,96,123)(60,124,81)(61,82,125)(62,126,83)(63,84,127)(64,128,85), (17,143,40)(18,144,41)(19,129,42)(20,130,43)(21,131,44)(22,132,45)(23,133,46)(24,134,47)(25,135,48)(26,136,33)(27,137,34)(28,138,35)(29,139,36)(30,140,37)(31,141,38)(32,142,39)(49,86,113)(50,87,114)(51,88,115)(52,89,116)(53,90,117)(54,91,118)(55,92,119)(56,93,120)(57,94,121)(58,95,122)(59,96,123)(60,81,124)(61,82,125)(62,83,126)(63,84,127)(64,85,128), (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([[(1,51,39),(2,40,52),(3,53,41),(4,42,54),(5,55,43),(6,44,56),(7,57,45),(8,46,58),(9,59,47),(10,48,60),(11,61,33),(12,34,62),(13,63,35),(14,36,64),(15,49,37),(16,38,50),(17,116,77),(18,78,117),(19,118,79),(20,80,119),(21,120,65),(22,66,121),(23,122,67),(24,68,123),(25,124,69),(26,70,125),(27,126,71),(28,72,127),(29,128,73),(30,74,113),(31,114,75),(32,76,115),(81,106,135),(82,136,107),(83,108,137),(84,138,109),(85,110,139),(86,140,111),(87,112,141),(88,142,97),(89,98,143),(90,144,99),(91,100,129),(92,130,101),(93,102,131),(94,132,103),(95,104,133),(96,134,105)], [(1,97,76),(2,77,98),(3,99,78),(4,79,100),(5,101,80),(6,65,102),(7,103,66),(8,67,104),(9,105,68),(10,69,106),(11,107,70),(12,71,108),(13,109,72),(14,73,110),(15,111,74),(16,75,112),(17,143,40),(18,41,144),(19,129,42),(20,43,130),(21,131,44),(22,45,132),(23,133,46),(24,47,134),(25,135,48),(26,33,136),(27,137,34),(28,35,138),(29,139,36),(30,37,140),(31,141,38),(32,39,142),(49,86,113),(50,114,87),(51,88,115),(52,116,89),(53,90,117),(54,118,91),(55,92,119),(56,120,93),(57,94,121),(58,122,95),(59,96,123),(60,124,81),(61,82,125),(62,126,83),(63,84,127),(64,128,85)], [(17,143,40),(18,144,41),(19,129,42),(20,130,43),(21,131,44),(22,132,45),(23,133,46),(24,134,47),(25,135,48),(26,136,33),(27,137,34),(28,138,35),(29,139,36),(30,140,37),(31,141,38),(32,142,39),(49,86,113),(50,87,114),(51,88,115),(52,89,116),(53,90,117),(54,91,118),(55,92,119),(56,93,120),(57,94,121),(58,95,122),(59,96,123),(60,81,124),(61,82,125),(62,83,126),(63,84,127),(64,85,128)], [(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)]])
80 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12L | 16A | ··· | 16H | 24A | 24B | 24C | 24D | 24E | ··· | 24L | 24M | ··· | 24X | 48A | ··· | 48P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| 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 | C32⋊C6 | C32⋊C12 | He3⋊3C8 | He3⋊3C16 |
| kernel | He3⋊3C16 | C8×He3 | C24.S3 | C4×He3 | C3×C24 | C2×He3 | C3×C12 | He3 | C3×C6 | C32 | C3×C24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of He3⋊3C16 ►in GL6(𝔽97)
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 |
| 96 | 1 | 0 | 0 | 0 | 0 |
| 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 1 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 1 |
| 0 | 0 | 0 | 0 | 96 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 1 |
| 0 | 0 | 0 | 0 | 96 | 0 |
| 89 | 0 | 0 | 0 | 0 | 0 |
| 89 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 89 | 0 |
| 0 | 0 | 0 | 0 | 89 | 8 |
| 0 | 0 | 89 | 0 | 0 | 0 |
| 0 | 0 | 89 | 8 | 0 | 0 |
G:=sub<GL(6,GF(97))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[89,89,0,0,0,0,0,8,0,0,0,0,0,0,0,0,89,89,0,0,0,0,0,8,0,0,89,89,0,0,0,0,0,8,0,0] >;
He3⋊3C16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_3C_{16} % in TeX
G:=Group("He3:3C16"); // GroupNames label
G:=SmallGroup(432,30);
// by ID
G=gap.SmallGroup(432,30);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-2,-3,-3,42,58,80,4037,4044,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations
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