direct product, metabelian, nilpotent (class 3), monomial
Aliases: Q16×He3, C12.24C62, C8.(C2×He3), C24.6(C3×C6), (C3×C24).3C6, (C32×Q16)⋊C3, C2.5(D4×He3), (C8×He3).3C2, Q8.2(C2×He3), (Q8×He3).3C2, C32⋊6(C3×Q16), (C2×He3).43D4, C6.33(D4×C32), C4.3(C22×He3), (Q8×C32).8C6, C3.2(C32×Q16), (C3×Q16).2C32, (C4×He3).52C22, (C3×C6).36(C3×D4), (C3×C12).21(C2×C6), (C3×Q8).18(C3×C6), SmallGroup(432,222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×He3
G = < a,b,c,d,e | a8=c3=d3=e3=1, b2=a4, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 209 in 99 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, C12, C12, Q16, C3×C6, C24, C24, C3×Q8, C3×Q8, He3, C3×C12, C3×C12, C3×Q16, C3×Q16, C2×He3, C3×C24, Q8×C32, C4×He3, C4×He3, C32×Q16, C8×He3, Q8×He3, Q16×He3
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, Q16, C3×C6, C3×D4, He3, C62, C3×Q16, C2×He3, D4×C32, C22×He3, C32×Q16, D4×He3, Q16×He3
►On 144 points
Generators in S144
(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 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 43 21 47)(18 42 22 46)(19 41 23 45)(20 48 24 44)(25 54 29 50)(26 53 30 49)(27 52 31 56)(28 51 32 55)(33 96 37 92)(34 95 38 91)(35 94 39 90)(36 93 40 89)(57 141 61 137)(58 140 62 144)(59 139 63 143)(60 138 64 142)(65 105 69 109)(66 112 70 108)(67 111 71 107)(68 110 72 106)(73 128 77 124)(74 127 78 123)(75 126 79 122)(76 125 80 121)(81 136 85 132)(82 135 86 131)(83 134 87 130)(84 133 88 129)(97 115 101 119)(98 114 102 118)(99 113 103 117)(100 120 104 116)
(1 82 139)(2 83 140)(3 84 141)(4 85 142)(5 86 143)(6 87 144)(7 88 137)(8 81 138)(9 132 60)(10 133 61)(11 134 62)(12 135 63)(13 136 64)(14 129 57)(15 130 58)(16 131 59)(17 53 67)(18 54 68)(19 55 69)(20 56 70)(21 49 71)(22 50 72)(23 51 65)(24 52 66)(25 106 46)(26 107 47)(27 108 48)(28 109 41)(29 110 42)(30 111 43)(31 112 44)(32 105 45)(33 120 76)(34 113 77)(35 114 78)(36 115 79)(37 116 80)(38 117 73)(39 118 74)(40 119 75)(89 97 126)(90 98 127)(91 99 128)(92 100 121)(93 101 122)(94 102 123)(95 103 124)(96 104 125)
(1 39 26)(2 40 27)(3 33 28)(4 34 29)(5 35 30)(6 36 31)(7 37 32)(8 38 25)(9 95 50)(10 96 51)(11 89 52)(12 90 53)(13 91 54)(14 92 55)(15 93 56)(16 94 49)(17 63 127)(18 64 128)(19 57 121)(20 58 122)(21 59 123)(22 60 124)(23 61 125)(24 62 126)(41 141 76)(42 142 77)(43 143 78)(44 144 79)(45 137 80)(46 138 73)(47 139 74)(48 140 75)(65 133 104)(66 134 97)(67 135 98)(68 136 99)(69 129 100)(70 130 101)(71 131 102)(72 132 103)(81 117 106)(82 118 107)(83 119 108)(84 120 109)(85 113 110)(86 114 111)(87 115 112)(88 116 105)
(1 139 107)(2 140 108)(3 141 109)(4 142 110)(5 143 111)(6 144 112)(7 137 105)(8 138 106)(9 60 72)(10 61 65)(11 62 66)(12 63 67)(13 64 68)(14 57 69)(15 58 70)(16 59 71)(17 98 53)(18 99 54)(19 100 55)(20 101 56)(21 102 49)(22 103 50)(23 104 51)(24 97 52)(25 46 117)(26 47 118)(27 48 119)(28 41 120)(29 42 113)(30 43 114)(31 44 115)(32 45 116)(33 76 84)(34 77 85)(35 78 86)(36 79 87)(37 80 88)(38 73 81)(39 74 82)(40 75 83)(89 126 134)(90 127 135)(91 128 136)(92 121 129)(93 122 130)(94 123 131)(95 124 132)(96 125 133)
G:=sub<Sym(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), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,43,21,47)(18,42,22,46)(19,41,23,45)(20,48,24,44)(25,54,29,50)(26,53,30,49)(27,52,31,56)(28,51,32,55)(33,96,37,92)(34,95,38,91)(35,94,39,90)(36,93,40,89)(57,141,61,137)(58,140,62,144)(59,139,63,143)(60,138,64,142)(65,105,69,109)(66,112,70,108)(67,111,71,107)(68,110,72,106)(73,128,77,124)(74,127,78,123)(75,126,79,122)(76,125,80,121)(81,136,85,132)(82,135,86,131)(83,134,87,130)(84,133,88,129)(97,115,101,119)(98,114,102,118)(99,113,103,117)(100,120,104,116), (1,82,139)(2,83,140)(3,84,141)(4,85,142)(5,86,143)(6,87,144)(7,88,137)(8,81,138)(9,132,60)(10,133,61)(11,134,62)(12,135,63)(13,136,64)(14,129,57)(15,130,58)(16,131,59)(17,53,67)(18,54,68)(19,55,69)(20,56,70)(21,49,71)(22,50,72)(23,51,65)(24,52,66)(25,106,46)(26,107,47)(27,108,48)(28,109,41)(29,110,42)(30,111,43)(31,112,44)(32,105,45)(33,120,76)(34,113,77)(35,114,78)(36,115,79)(37,116,80)(38,117,73)(39,118,74)(40,119,75)(89,97,126)(90,98,127)(91,99,128)(92,100,121)(93,101,122)(94,102,123)(95,103,124)(96,104,125), (1,39,26)(2,40,27)(3,33,28)(4,34,29)(5,35,30)(6,36,31)(7,37,32)(8,38,25)(9,95,50)(10,96,51)(11,89,52)(12,90,53)(13,91,54)(14,92,55)(15,93,56)(16,94,49)(17,63,127)(18,64,128)(19,57,121)(20,58,122)(21,59,123)(22,60,124)(23,61,125)(24,62,126)(41,141,76)(42,142,77)(43,143,78)(44,144,79)(45,137,80)(46,138,73)(47,139,74)(48,140,75)(65,133,104)(66,134,97)(67,135,98)(68,136,99)(69,129,100)(70,130,101)(71,131,102)(72,132,103)(81,117,106)(82,118,107)(83,119,108)(84,120,109)(85,113,110)(86,114,111)(87,115,112)(88,116,105), (1,139,107)(2,140,108)(3,141,109)(4,142,110)(5,143,111)(6,144,112)(7,137,105)(8,138,106)(9,60,72)(10,61,65)(11,62,66)(12,63,67)(13,64,68)(14,57,69)(15,58,70)(16,59,71)(17,98,53)(18,99,54)(19,100,55)(20,101,56)(21,102,49)(22,103,50)(23,104,51)(24,97,52)(25,46,117)(26,47,118)(27,48,119)(28,41,120)(29,42,113)(30,43,114)(31,44,115)(32,45,116)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(89,126,134)(90,127,135)(91,128,136)(92,121,129)(93,122,130)(94,123,131)(95,124,132)(96,125,133)>;
G:=Group( (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), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,43,21,47)(18,42,22,46)(19,41,23,45)(20,48,24,44)(25,54,29,50)(26,53,30,49)(27,52,31,56)(28,51,32,55)(33,96,37,92)(34,95,38,91)(35,94,39,90)(36,93,40,89)(57,141,61,137)(58,140,62,144)(59,139,63,143)(60,138,64,142)(65,105,69,109)(66,112,70,108)(67,111,71,107)(68,110,72,106)(73,128,77,124)(74,127,78,123)(75,126,79,122)(76,125,80,121)(81,136,85,132)(82,135,86,131)(83,134,87,130)(84,133,88,129)(97,115,101,119)(98,114,102,118)(99,113,103,117)(100,120,104,116), (1,82,139)(2,83,140)(3,84,141)(4,85,142)(5,86,143)(6,87,144)(7,88,137)(8,81,138)(9,132,60)(10,133,61)(11,134,62)(12,135,63)(13,136,64)(14,129,57)(15,130,58)(16,131,59)(17,53,67)(18,54,68)(19,55,69)(20,56,70)(21,49,71)(22,50,72)(23,51,65)(24,52,66)(25,106,46)(26,107,47)(27,108,48)(28,109,41)(29,110,42)(30,111,43)(31,112,44)(32,105,45)(33,120,76)(34,113,77)(35,114,78)(36,115,79)(37,116,80)(38,117,73)(39,118,74)(40,119,75)(89,97,126)(90,98,127)(91,99,128)(92,100,121)(93,101,122)(94,102,123)(95,103,124)(96,104,125), (1,39,26)(2,40,27)(3,33,28)(4,34,29)(5,35,30)(6,36,31)(7,37,32)(8,38,25)(9,95,50)(10,96,51)(11,89,52)(12,90,53)(13,91,54)(14,92,55)(15,93,56)(16,94,49)(17,63,127)(18,64,128)(19,57,121)(20,58,122)(21,59,123)(22,60,124)(23,61,125)(24,62,126)(41,141,76)(42,142,77)(43,143,78)(44,144,79)(45,137,80)(46,138,73)(47,139,74)(48,140,75)(65,133,104)(66,134,97)(67,135,98)(68,136,99)(69,129,100)(70,130,101)(71,131,102)(72,132,103)(81,117,106)(82,118,107)(83,119,108)(84,120,109)(85,113,110)(86,114,111)(87,115,112)(88,116,105), (1,139,107)(2,140,108)(3,141,109)(4,142,110)(5,143,111)(6,144,112)(7,137,105)(8,138,106)(9,60,72)(10,61,65)(11,62,66)(12,63,67)(13,64,68)(14,57,69)(15,58,70)(16,59,71)(17,98,53)(18,99,54)(19,100,55)(20,101,56)(21,102,49)(22,103,50)(23,104,51)(24,97,52)(25,46,117)(26,47,118)(27,48,119)(28,41,120)(29,42,113)(30,43,114)(31,44,115)(32,45,116)(33,76,84)(34,77,85)(35,78,86)(36,79,87)(37,80,88)(38,73,81)(39,74,82)(40,75,83)(89,126,134)(90,127,135)(91,128,136)(92,121,129)(93,122,130)(94,123,131)(95,124,132)(96,125,133) );
G=PermutationGroup([[(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)], [(1,12,5,16),(2,11,6,15),(3,10,7,14),(4,9,8,13),(17,43,21,47),(18,42,22,46),(19,41,23,45),(20,48,24,44),(25,54,29,50),(26,53,30,49),(27,52,31,56),(28,51,32,55),(33,96,37,92),(34,95,38,91),(35,94,39,90),(36,93,40,89),(57,141,61,137),(58,140,62,144),(59,139,63,143),(60,138,64,142),(65,105,69,109),(66,112,70,108),(67,111,71,107),(68,110,72,106),(73,128,77,124),(74,127,78,123),(75,126,79,122),(76,125,80,121),(81,136,85,132),(82,135,86,131),(83,134,87,130),(84,133,88,129),(97,115,101,119),(98,114,102,118),(99,113,103,117),(100,120,104,116)], [(1,82,139),(2,83,140),(3,84,141),(4,85,142),(5,86,143),(6,87,144),(7,88,137),(8,81,138),(9,132,60),(10,133,61),(11,134,62),(12,135,63),(13,136,64),(14,129,57),(15,130,58),(16,131,59),(17,53,67),(18,54,68),(19,55,69),(20,56,70),(21,49,71),(22,50,72),(23,51,65),(24,52,66),(25,106,46),(26,107,47),(27,108,48),(28,109,41),(29,110,42),(30,111,43),(31,112,44),(32,105,45),(33,120,76),(34,113,77),(35,114,78),(36,115,79),(37,116,80),(38,117,73),(39,118,74),(40,119,75),(89,97,126),(90,98,127),(91,99,128),(92,100,121),(93,101,122),(94,102,123),(95,103,124),(96,104,125)], [(1,39,26),(2,40,27),(3,33,28),(4,34,29),(5,35,30),(6,36,31),(7,37,32),(8,38,25),(9,95,50),(10,96,51),(11,89,52),(12,90,53),(13,91,54),(14,92,55),(15,93,56),(16,94,49),(17,63,127),(18,64,128),(19,57,121),(20,58,122),(21,59,123),(22,60,124),(23,61,125),(24,62,126),(41,141,76),(42,142,77),(43,143,78),(44,144,79),(45,137,80),(46,138,73),(47,139,74),(48,140,75),(65,133,104),(66,134,97),(67,135,98),(68,136,99),(69,129,100),(70,130,101),(71,131,102),(72,132,103),(81,117,106),(82,118,107),(83,119,108),(84,120,109),(85,113,110),(86,114,111),(87,115,112),(88,116,105)], [(1,139,107),(2,140,108),(3,141,109),(4,142,110),(5,143,111),(6,144,112),(7,137,105),(8,138,106),(9,60,72),(10,61,65),(11,62,66),(12,63,67),(13,64,68),(14,57,69),(15,58,70),(16,59,71),(17,98,53),(18,99,54),(19,100,55),(20,101,56),(21,102,49),(22,103,50),(23,104,51),(24,97,52),(25,46,117),(26,47,118),(27,48,119),(28,41,120),(29,42,113),(30,43,114),(31,44,115),(32,45,116),(33,76,84),(34,77,85),(35,78,86),(36,79,87),(37,80,88),(38,73,81),(39,74,82),(40,75,83),(89,126,134),(90,127,135),(91,128,136),(92,121,129),(93,122,130),(94,123,131),(95,124,132),(96,125,133)]])
77 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6J | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12N | 12O | ··· | 12AD | 24A | 24B | 24C | 24D | 24E | ··· | 24T |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 4 | 4 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | Q16 | C3×D4 | C3×Q16 | He3 | C2×He3 | C2×He3 | D4×He3 | Q16×He3 |
| kernel | Q16×He3 | C8×He3 | Q8×He3 | C32×Q16 | C3×C24 | Q8×C32 | C2×He3 | He3 | C3×C6 | C32 | Q16 | C8 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 2 | 8 | 16 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of Q16×He3 ►in GL5(𝔽73)
| 57 | 16 | 0 | 0 | 0 |
| 57 | 57 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 60 | 66 | 0 | 0 | 0 |
| 66 | 13 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 63 | 0 |
| 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 0 | 65 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 64 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 17 |
| 0 | 0 | 72 | 0 | 65 |
| 0 | 0 | 8 | 8 | 72 |
G:=sub<GL(5,GF(73))| [57,57,0,0,0,16,57,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[60,66,0,0,0,66,13,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,63,9,65,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[64,0,0,0,0,0,64,0,0,0,0,0,1,72,8,0,0,0,0,8,0,0,17,65,72] >;
Q16×He3 in GAP, Magma, Sage, TeX
Q_{16}\times {\rm He}_3 % in TeX
G:=Group("Q16xHe3"); // GroupNames label
G:=SmallGroup(432,222);
// by ID
G=gap.SmallGroup(432,222);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,1520,605,8824,4421,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=c^3=d^3=e^3=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations