metabelian, supersoluble, monomial
Aliases: He3⋊6Q16, C12.9(S3×C6), C32⋊4C8.C6, C32⋊7Q16⋊C3, (C3×C12).12D6, (Q8×He3).1C2, C32⋊3(C3×Q16), (C2×He3).29D4, He3⋊3C8.2C2, He3⋊3Q8.3C2, (Q8×C32).1S3, (Q8×C32).1C6, C32⋊4Q8.2C6, C2.6(He3⋊6D4), C32⋊4(C3⋊Q16), Q8.3(C32⋊C6), (C4×He3).10C22, (C3×C12).3(C2×C6), (C3×C6).14(C3×D4), C6.26(C3×C3⋊D4), C4.3(C2×C32⋊C6), C3.2(C3×C3⋊Q16), (C3×Q8).23(C3×S3), (C3×C6).27(C3⋊D4), SmallGroup(432,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊6Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 289 in 75 conjugacy classes, 26 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C32, C32, Dic3, C12, C12, Q16, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C32⋊4Q8, Q8×C32, Q8×C32, C32⋊C12, C4×He3, C4×He3, C3×C3⋊Q16, C32⋊7Q16, He3⋊3C8, He3⋊3Q8, Q8×He3, He3⋊6Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊D4, C3×D4, S3×C6, C3⋊Q16, C3×Q16, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, C3×C3⋊Q16, He3⋊6D4, He3⋊6Q16
►On 144 points
Generators in S144
(9 39 120)(10 113 40)(11 33 114)(12 115 34)(13 35 116)(14 117 36)(15 37 118)(16 119 38)(25 64 121)(26 122 57)(27 58 123)(28 124 59)(29 60 125)(30 126 61)(31 62 127)(32 128 63)(41 55 106)(42 107 56)(43 49 108)(44 109 50)(45 51 110)(46 111 52)(47 53 112)(48 105 54)(65 132 100)(66 101 133)(67 134 102)(68 103 135)(69 136 104)(70 97 129)(71 130 98)(72 99 131)
(1 139 77)(2 78 140)(3 141 79)(4 80 142)(5 143 73)(6 74 144)(7 137 75)(8 76 138)(9 120 39)(10 40 113)(11 114 33)(12 34 115)(13 116 35)(14 36 117)(15 118 37)(16 38 119)(17 81 92)(18 93 82)(19 83 94)(20 95 84)(21 85 96)(22 89 86)(23 87 90)(24 91 88)(25 64 121)(26 122 57)(27 58 123)(28 124 59)(29 60 125)(30 126 61)(31 62 127)(32 128 63)(41 55 106)(42 107 56)(43 49 108)(44 109 50)(45 51 110)(46 111 52)(47 53 112)(48 105 54)(65 100 132)(66 133 101)(67 102 134)(68 135 103)(69 104 136)(70 129 97)(71 98 130)(72 131 99)
(1 15 43)(2 16 44)(3 9 45)(4 10 46)(5 11 47)(6 12 48)(7 13 41)(8 14 42)(17 132 62)(18 133 63)(19 134 64)(20 135 57)(21 136 58)(22 129 59)(23 130 60)(24 131 61)(25 94 102)(26 95 103)(27 96 104)(28 89 97)(29 90 98)(30 91 99)(31 92 100)(32 93 101)(33 112 73)(34 105 74)(35 106 75)(36 107 76)(37 108 77)(38 109 78)(39 110 79)(40 111 80)(49 139 118)(50 140 119)(51 141 120)(52 142 113)(53 143 114)(54 144 115)(55 137 116)(56 138 117)(65 127 81)(66 128 82)(67 121 83)(68 122 84)(69 123 85)(70 124 86)(71 125 87)(72 126 88)
(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 87 5 83)(2 86 6 82)(3 85 7 81)(4 84 8 88)(9 69 13 65)(10 68 14 72)(11 67 15 71)(12 66 16 70)(17 79 21 75)(18 78 22 74)(19 77 23 73)(20 76 24 80)(25 49 29 53)(26 56 30 52)(27 55 31 51)(28 54 32 50)(33 134 37 130)(34 133 38 129)(35 132 39 136)(36 131 40 135)(41 127 45 123)(42 126 46 122)(43 125 47 121)(44 124 48 128)(57 107 61 111)(58 106 62 110)(59 105 63 109)(60 112 64 108)(89 144 93 140)(90 143 94 139)(91 142 95 138)(92 141 96 137)(97 115 101 119)(98 114 102 118)(99 113 103 117)(100 120 104 116)
G:=sub<Sym(144)| (9,39,120)(10,113,40)(11,33,114)(12,115,34)(13,35,116)(14,117,36)(15,37,118)(16,119,38)(25,64,121)(26,122,57)(27,58,123)(28,124,59)(29,60,125)(30,126,61)(31,62,127)(32,128,63)(41,55,106)(42,107,56)(43,49,108)(44,109,50)(45,51,110)(46,111,52)(47,53,112)(48,105,54)(65,132,100)(66,101,133)(67,134,102)(68,103,135)(69,136,104)(70,97,129)(71,130,98)(72,99,131), (1,139,77)(2,78,140)(3,141,79)(4,80,142)(5,143,73)(6,74,144)(7,137,75)(8,76,138)(9,120,39)(10,40,113)(11,114,33)(12,34,115)(13,116,35)(14,36,117)(15,118,37)(16,38,119)(17,81,92)(18,93,82)(19,83,94)(20,95,84)(21,85,96)(22,89,86)(23,87,90)(24,91,88)(25,64,121)(26,122,57)(27,58,123)(28,124,59)(29,60,125)(30,126,61)(31,62,127)(32,128,63)(41,55,106)(42,107,56)(43,49,108)(44,109,50)(45,51,110)(46,111,52)(47,53,112)(48,105,54)(65,100,132)(66,133,101)(67,102,134)(68,135,103)(69,104,136)(70,129,97)(71,98,130)(72,131,99), (1,15,43)(2,16,44)(3,9,45)(4,10,46)(5,11,47)(6,12,48)(7,13,41)(8,14,42)(17,132,62)(18,133,63)(19,134,64)(20,135,57)(21,136,58)(22,129,59)(23,130,60)(24,131,61)(25,94,102)(26,95,103)(27,96,104)(28,89,97)(29,90,98)(30,91,99)(31,92,100)(32,93,101)(33,112,73)(34,105,74)(35,106,75)(36,107,76)(37,108,77)(38,109,78)(39,110,79)(40,111,80)(49,139,118)(50,140,119)(51,141,120)(52,142,113)(53,143,114)(54,144,115)(55,137,116)(56,138,117)(65,127,81)(66,128,82)(67,121,83)(68,122,84)(69,123,85)(70,124,86)(71,125,87)(72,126,88), (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), (1,87,5,83)(2,86,6,82)(3,85,7,81)(4,84,8,88)(9,69,13,65)(10,68,14,72)(11,67,15,71)(12,66,16,70)(17,79,21,75)(18,78,22,74)(19,77,23,73)(20,76,24,80)(25,49,29,53)(26,56,30,52)(27,55,31,51)(28,54,32,50)(33,134,37,130)(34,133,38,129)(35,132,39,136)(36,131,40,135)(41,127,45,123)(42,126,46,122)(43,125,47,121)(44,124,48,128)(57,107,61,111)(58,106,62,110)(59,105,63,109)(60,112,64,108)(89,144,93,140)(90,143,94,139)(91,142,95,138)(92,141,96,137)(97,115,101,119)(98,114,102,118)(99,113,103,117)(100,120,104,116)>;
G:=Group( (9,39,120)(10,113,40)(11,33,114)(12,115,34)(13,35,116)(14,117,36)(15,37,118)(16,119,38)(25,64,121)(26,122,57)(27,58,123)(28,124,59)(29,60,125)(30,126,61)(31,62,127)(32,128,63)(41,55,106)(42,107,56)(43,49,108)(44,109,50)(45,51,110)(46,111,52)(47,53,112)(48,105,54)(65,132,100)(66,101,133)(67,134,102)(68,103,135)(69,136,104)(70,97,129)(71,130,98)(72,99,131), (1,139,77)(2,78,140)(3,141,79)(4,80,142)(5,143,73)(6,74,144)(7,137,75)(8,76,138)(9,120,39)(10,40,113)(11,114,33)(12,34,115)(13,116,35)(14,36,117)(15,118,37)(16,38,119)(17,81,92)(18,93,82)(19,83,94)(20,95,84)(21,85,96)(22,89,86)(23,87,90)(24,91,88)(25,64,121)(26,122,57)(27,58,123)(28,124,59)(29,60,125)(30,126,61)(31,62,127)(32,128,63)(41,55,106)(42,107,56)(43,49,108)(44,109,50)(45,51,110)(46,111,52)(47,53,112)(48,105,54)(65,100,132)(66,133,101)(67,102,134)(68,135,103)(69,104,136)(70,129,97)(71,98,130)(72,131,99), (1,15,43)(2,16,44)(3,9,45)(4,10,46)(5,11,47)(6,12,48)(7,13,41)(8,14,42)(17,132,62)(18,133,63)(19,134,64)(20,135,57)(21,136,58)(22,129,59)(23,130,60)(24,131,61)(25,94,102)(26,95,103)(27,96,104)(28,89,97)(29,90,98)(30,91,99)(31,92,100)(32,93,101)(33,112,73)(34,105,74)(35,106,75)(36,107,76)(37,108,77)(38,109,78)(39,110,79)(40,111,80)(49,139,118)(50,140,119)(51,141,120)(52,142,113)(53,143,114)(54,144,115)(55,137,116)(56,138,117)(65,127,81)(66,128,82)(67,121,83)(68,122,84)(69,123,85)(70,124,86)(71,125,87)(72,126,88), (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), (1,87,5,83)(2,86,6,82)(3,85,7,81)(4,84,8,88)(9,69,13,65)(10,68,14,72)(11,67,15,71)(12,66,16,70)(17,79,21,75)(18,78,22,74)(19,77,23,73)(20,76,24,80)(25,49,29,53)(26,56,30,52)(27,55,31,51)(28,54,32,50)(33,134,37,130)(34,133,38,129)(35,132,39,136)(36,131,40,135)(41,127,45,123)(42,126,46,122)(43,125,47,121)(44,124,48,128)(57,107,61,111)(58,106,62,110)(59,105,63,109)(60,112,64,108)(89,144,93,140)(90,143,94,139)(91,142,95,138)(92,141,96,137)(97,115,101,119)(98,114,102,118)(99,113,103,117)(100,120,104,116) );
G=PermutationGroup([[(9,39,120),(10,113,40),(11,33,114),(12,115,34),(13,35,116),(14,117,36),(15,37,118),(16,119,38),(25,64,121),(26,122,57),(27,58,123),(28,124,59),(29,60,125),(30,126,61),(31,62,127),(32,128,63),(41,55,106),(42,107,56),(43,49,108),(44,109,50),(45,51,110),(46,111,52),(47,53,112),(48,105,54),(65,132,100),(66,101,133),(67,134,102),(68,103,135),(69,136,104),(70,97,129),(71,130,98),(72,99,131)], [(1,139,77),(2,78,140),(3,141,79),(4,80,142),(5,143,73),(6,74,144),(7,137,75),(8,76,138),(9,120,39),(10,40,113),(11,114,33),(12,34,115),(13,116,35),(14,36,117),(15,118,37),(16,38,119),(17,81,92),(18,93,82),(19,83,94),(20,95,84),(21,85,96),(22,89,86),(23,87,90),(24,91,88),(25,64,121),(26,122,57),(27,58,123),(28,124,59),(29,60,125),(30,126,61),(31,62,127),(32,128,63),(41,55,106),(42,107,56),(43,49,108),(44,109,50),(45,51,110),(46,111,52),(47,53,112),(48,105,54),(65,100,132),(66,133,101),(67,102,134),(68,135,103),(69,104,136),(70,129,97),(71,98,130),(72,131,99)], [(1,15,43),(2,16,44),(3,9,45),(4,10,46),(5,11,47),(6,12,48),(7,13,41),(8,14,42),(17,132,62),(18,133,63),(19,134,64),(20,135,57),(21,136,58),(22,129,59),(23,130,60),(24,131,61),(25,94,102),(26,95,103),(27,96,104),(28,89,97),(29,90,98),(30,91,99),(31,92,100),(32,93,101),(33,112,73),(34,105,74),(35,106,75),(36,107,76),(37,108,77),(38,109,78),(39,110,79),(40,111,80),(49,139,118),(50,140,119),(51,141,120),(52,142,113),(53,143,114),(54,144,115),(55,137,116),(56,138,117),(65,127,81),(66,128,82),(67,121,83),(68,122,84),(69,123,85),(70,124,86),(71,125,87),(72,126,88)], [(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)], [(1,87,5,83),(2,86,6,82),(3,85,7,81),(4,84,8,88),(9,69,13,65),(10,68,14,72),(11,67,15,71),(12,66,16,70),(17,79,21,75),(18,78,22,74),(19,77,23,73),(20,76,24,80),(25,49,29,53),(26,56,30,52),(27,55,31,51),(28,54,32,50),(33,134,37,130),(34,133,38,129),(35,132,39,136),(36,131,40,135),(41,127,45,123),(42,126,46,122),(43,125,47,121),(44,124,48,128),(57,107,61,111),(58,106,62,110),(59,105,63,109),(60,112,64,108),(89,144,93,140),(90,143,94,139),(91,142,95,138),(92,141,96,137),(97,115,101,119),(98,114,102,118),(99,113,103,117),(100,120,104,116)]])
41 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | ··· | 12P | 12Q | 12R | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 4 | 36 | 2 | 3 | 3 | 6 | 6 | 6 | 18 | 18 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 36 | 36 | 18 | 18 | 18 | 18 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | + | - | - | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | He3⋊6Q16 | S3 | D4 | D6 | Q16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×Q16 | C3×C3⋊D4 | C3⋊Q16 | C3×C3⋊Q16 | C32⋊C6 | C2×C32⋊C6 | He3⋊6D4 |
| kernel | He3⋊6Q16 | He3⋊3C8 | He3⋊3Q8 | Q8×He3 | C32⋊7Q16 | C32⋊4C8 | C32⋊4Q8 | Q8×C32 | C1 | Q8×C32 | C2×He3 | C3×C12 | He3 | C3×Q8 | C3×C6 | C3×C6 | C12 | C32 | C6 | C32 | C3 | Q8 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of He3⋊6Q16 ►in GL10(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 57 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 57 | 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 31 | 45 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 31 | 45 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 42 |
| 60 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 61 | 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 61 | 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(10,GF(73))| [0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,57,0,0,0,0,0,0,0,0,57,0,0,0,0,0,0,0,0,32,0,32,0,0,0,0,0,0,32,0,32,0,0,0,0,0,0,0,0,0,0,0,31,3,0,0,0,0,0,0,0,0,45,42,0,0,0,0,0,0,0,0,0,0,31,3,0,0,0,0,0,0,0,0,45,42,0,0,0,0,0,0,0,0,0,0,31,3,0,0,0,0,0,0,0,0,45,42],[60,0,61,0,0,0,0,0,0,0,0,60,0,61,0,0,0,0,0,0,2,0,13,0,0,0,0,0,0,0,0,2,0,13,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72] >;
He3⋊6Q16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_6Q_{16} % in TeX
G:=Group("He3:6Q16"); // GroupNames label
G:=SmallGroup(432,160);
// by ID
G=gap.SmallGroup(432,160);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,4037,2035,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations