non-abelian, supersoluble, monomial
Aliases: He3⋊5Q16, C32⋊4Dic12, (C3×C24).4S3, C24.6(C3⋊S3), (C3×C6).25D12, (C3×C12).48D6, C8.(He3⋊C2), (C8×He3).2C2, (C2×He3).24D4, He3⋊4Q8.3C2, C2.5(He3⋊5D4), C6.29(C12⋊S3), C3.2(C32⋊5Q16), (C4×He3).37C22, C12.81(C2×C3⋊S3), C4.10(C2×He3⋊C2), SmallGroup(432,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊5Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 377 in 99 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, Dic3, C12, C12, Q16, C3×C6, C24, C24, Dic6, C3×Q8, He3, C3×Dic3, C3×C12, Dic12, C3×Q16, C2×He3, C3×C24, C3×Dic6, He3⋊3C4, C4×He3, C3×Dic12, C8×He3, He3⋊4Q8, He3⋊5Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊S3, D12, C2×C3⋊S3, Dic12, He3⋊C2, C12⋊S3, C2×He3⋊C2, C32⋊5Q16, He3⋊5D4, He3⋊5Q16
►On 144 points
Generators in S144
(25 38 62)(26 39 63)(27 40 64)(28 33 57)(29 34 58)(30 35 59)(31 36 60)(32 37 61)(49 95 137)(50 96 138)(51 89 139)(52 90 140)(53 91 141)(54 92 142)(55 93 143)(56 94 144)(65 123 104)(66 124 97)(67 125 98)(68 126 99)(69 127 100)(70 128 101)(71 121 102)(72 122 103)(73 112 120)(74 105 113)(75 106 114)(76 107 115)(77 108 116)(78 109 117)(79 110 118)(80 111 119)
(1 86 134)(2 87 135)(3 88 136)(4 81 129)(5 82 130)(6 83 131)(7 84 132)(8 85 133)(9 43 20)(10 44 21)(11 45 22)(12 46 23)(13 47 24)(14 48 17)(15 41 18)(16 42 19)(25 62 38)(26 63 39)(27 64 40)(28 57 33)(29 58 34)(30 59 35)(31 60 36)(32 61 37)(49 95 137)(50 96 138)(51 89 139)(52 90 140)(53 91 141)(54 92 142)(55 93 143)(56 94 144)(65 104 123)(66 97 124)(67 98 125)(68 99 126)(69 100 127)(70 101 128)(71 102 121)(72 103 122)(73 112 120)(74 105 113)(75 106 114)(76 107 115)(77 108 116)(78 109 117)(79 110 118)(80 111 119)
(1 102 118)(2 103 119)(3 104 120)(4 97 113)(5 98 114)(6 99 115)(7 100 116)(8 101 117)(9 35 56)(10 36 49)(11 37 50)(12 38 51)(13 39 52)(14 40 53)(15 33 54)(16 34 55)(17 64 141)(18 57 142)(19 58 143)(20 59 144)(21 60 137)(22 61 138)(23 62 139)(24 63 140)(25 89 46)(26 90 47)(27 91 48)(28 92 41)(29 93 42)(30 94 43)(31 95 44)(32 96 45)(65 112 136)(66 105 129)(67 106 130)(68 107 131)(69 108 132)(70 109 133)(71 110 134)(72 111 135)(73 88 123)(74 81 124)(75 82 125)(76 83 126)(77 84 127)(78 85 128)(79 86 121)(80 87 122)
(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 20 5 24)(2 19 6 23)(3 18 7 22)(4 17 8 21)(9 82 13 86)(10 81 14 85)(11 88 15 84)(12 87 16 83)(25 111 29 107)(26 110 30 106)(27 109 31 105)(28 108 32 112)(33 77 37 73)(34 76 38 80)(35 75 39 79)(36 74 40 78)(41 132 45 136)(42 131 46 135)(43 130 47 134)(44 129 48 133)(49 124 53 128)(50 123 54 127)(51 122 55 126)(52 121 56 125)(57 116 61 120)(58 115 62 119)(59 114 63 118)(60 113 64 117)(65 92 69 96)(66 91 70 95)(67 90 71 94)(68 89 72 93)(97 141 101 137)(98 140 102 144)(99 139 103 143)(100 138 104 142)
G:=sub<Sym(144)| (25,38,62)(26,39,63)(27,40,64)(28,33,57)(29,34,58)(30,35,59)(31,36,60)(32,37,61)(49,95,137)(50,96,138)(51,89,139)(52,90,140)(53,91,141)(54,92,142)(55,93,143)(56,94,144)(65,123,104)(66,124,97)(67,125,98)(68,126,99)(69,127,100)(70,128,101)(71,121,102)(72,122,103)(73,112,120)(74,105,113)(75,106,114)(76,107,115)(77,108,116)(78,109,117)(79,110,118)(80,111,119), (1,86,134)(2,87,135)(3,88,136)(4,81,129)(5,82,130)(6,83,131)(7,84,132)(8,85,133)(9,43,20)(10,44,21)(11,45,22)(12,46,23)(13,47,24)(14,48,17)(15,41,18)(16,42,19)(25,62,38)(26,63,39)(27,64,40)(28,57,33)(29,58,34)(30,59,35)(31,60,36)(32,61,37)(49,95,137)(50,96,138)(51,89,139)(52,90,140)(53,91,141)(54,92,142)(55,93,143)(56,94,144)(65,104,123)(66,97,124)(67,98,125)(68,99,126)(69,100,127)(70,101,128)(71,102,121)(72,103,122)(73,112,120)(74,105,113)(75,106,114)(76,107,115)(77,108,116)(78,109,117)(79,110,118)(80,111,119), (1,102,118)(2,103,119)(3,104,120)(4,97,113)(5,98,114)(6,99,115)(7,100,116)(8,101,117)(9,35,56)(10,36,49)(11,37,50)(12,38,51)(13,39,52)(14,40,53)(15,33,54)(16,34,55)(17,64,141)(18,57,142)(19,58,143)(20,59,144)(21,60,137)(22,61,138)(23,62,139)(24,63,140)(25,89,46)(26,90,47)(27,91,48)(28,92,41)(29,93,42)(30,94,43)(31,95,44)(32,96,45)(65,112,136)(66,105,129)(67,106,130)(68,107,131)(69,108,132)(70,109,133)(71,110,134)(72,111,135)(73,88,123)(74,81,124)(75,82,125)(76,83,126)(77,84,127)(78,85,128)(79,86,121)(80,87,122), (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), (1,20,5,24)(2,19,6,23)(3,18,7,22)(4,17,8,21)(9,82,13,86)(10,81,14,85)(11,88,15,84)(12,87,16,83)(25,111,29,107)(26,110,30,106)(27,109,31,105)(28,108,32,112)(33,77,37,73)(34,76,38,80)(35,75,39,79)(36,74,40,78)(41,132,45,136)(42,131,46,135)(43,130,47,134)(44,129,48,133)(49,124,53,128)(50,123,54,127)(51,122,55,126)(52,121,56,125)(57,116,61,120)(58,115,62,119)(59,114,63,118)(60,113,64,117)(65,92,69,96)(66,91,70,95)(67,90,71,94)(68,89,72,93)(97,141,101,137)(98,140,102,144)(99,139,103,143)(100,138,104,142)>;
G:=Group( (25,38,62)(26,39,63)(27,40,64)(28,33,57)(29,34,58)(30,35,59)(31,36,60)(32,37,61)(49,95,137)(50,96,138)(51,89,139)(52,90,140)(53,91,141)(54,92,142)(55,93,143)(56,94,144)(65,123,104)(66,124,97)(67,125,98)(68,126,99)(69,127,100)(70,128,101)(71,121,102)(72,122,103)(73,112,120)(74,105,113)(75,106,114)(76,107,115)(77,108,116)(78,109,117)(79,110,118)(80,111,119), (1,86,134)(2,87,135)(3,88,136)(4,81,129)(5,82,130)(6,83,131)(7,84,132)(8,85,133)(9,43,20)(10,44,21)(11,45,22)(12,46,23)(13,47,24)(14,48,17)(15,41,18)(16,42,19)(25,62,38)(26,63,39)(27,64,40)(28,57,33)(29,58,34)(30,59,35)(31,60,36)(32,61,37)(49,95,137)(50,96,138)(51,89,139)(52,90,140)(53,91,141)(54,92,142)(55,93,143)(56,94,144)(65,104,123)(66,97,124)(67,98,125)(68,99,126)(69,100,127)(70,101,128)(71,102,121)(72,103,122)(73,112,120)(74,105,113)(75,106,114)(76,107,115)(77,108,116)(78,109,117)(79,110,118)(80,111,119), (1,102,118)(2,103,119)(3,104,120)(4,97,113)(5,98,114)(6,99,115)(7,100,116)(8,101,117)(9,35,56)(10,36,49)(11,37,50)(12,38,51)(13,39,52)(14,40,53)(15,33,54)(16,34,55)(17,64,141)(18,57,142)(19,58,143)(20,59,144)(21,60,137)(22,61,138)(23,62,139)(24,63,140)(25,89,46)(26,90,47)(27,91,48)(28,92,41)(29,93,42)(30,94,43)(31,95,44)(32,96,45)(65,112,136)(66,105,129)(67,106,130)(68,107,131)(69,108,132)(70,109,133)(71,110,134)(72,111,135)(73,88,123)(74,81,124)(75,82,125)(76,83,126)(77,84,127)(78,85,128)(79,86,121)(80,87,122), (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), (1,20,5,24)(2,19,6,23)(3,18,7,22)(4,17,8,21)(9,82,13,86)(10,81,14,85)(11,88,15,84)(12,87,16,83)(25,111,29,107)(26,110,30,106)(27,109,31,105)(28,108,32,112)(33,77,37,73)(34,76,38,80)(35,75,39,79)(36,74,40,78)(41,132,45,136)(42,131,46,135)(43,130,47,134)(44,129,48,133)(49,124,53,128)(50,123,54,127)(51,122,55,126)(52,121,56,125)(57,116,61,120)(58,115,62,119)(59,114,63,118)(60,113,64,117)(65,92,69,96)(66,91,70,95)(67,90,71,94)(68,89,72,93)(97,141,101,137)(98,140,102,144)(99,139,103,143)(100,138,104,142) );
G=PermutationGroup([[(25,38,62),(26,39,63),(27,40,64),(28,33,57),(29,34,58),(30,35,59),(31,36,60),(32,37,61),(49,95,137),(50,96,138),(51,89,139),(52,90,140),(53,91,141),(54,92,142),(55,93,143),(56,94,144),(65,123,104),(66,124,97),(67,125,98),(68,126,99),(69,127,100),(70,128,101),(71,121,102),(72,122,103),(73,112,120),(74,105,113),(75,106,114),(76,107,115),(77,108,116),(78,109,117),(79,110,118),(80,111,119)], [(1,86,134),(2,87,135),(3,88,136),(4,81,129),(5,82,130),(6,83,131),(7,84,132),(8,85,133),(9,43,20),(10,44,21),(11,45,22),(12,46,23),(13,47,24),(14,48,17),(15,41,18),(16,42,19),(25,62,38),(26,63,39),(27,64,40),(28,57,33),(29,58,34),(30,59,35),(31,60,36),(32,61,37),(49,95,137),(50,96,138),(51,89,139),(52,90,140),(53,91,141),(54,92,142),(55,93,143),(56,94,144),(65,104,123),(66,97,124),(67,98,125),(68,99,126),(69,100,127),(70,101,128),(71,102,121),(72,103,122),(73,112,120),(74,105,113),(75,106,114),(76,107,115),(77,108,116),(78,109,117),(79,110,118),(80,111,119)], [(1,102,118),(2,103,119),(3,104,120),(4,97,113),(5,98,114),(6,99,115),(7,100,116),(8,101,117),(9,35,56),(10,36,49),(11,37,50),(12,38,51),(13,39,52),(14,40,53),(15,33,54),(16,34,55),(17,64,141),(18,57,142),(19,58,143),(20,59,144),(21,60,137),(22,61,138),(23,62,139),(24,63,140),(25,89,46),(26,90,47),(27,91,48),(28,92,41),(29,93,42),(30,94,43),(31,95,44),(32,96,45),(65,112,136),(66,105,129),(67,106,130),(68,107,131),(69,108,132),(70,109,133),(71,110,134),(72,111,135),(73,88,123),(74,81,124),(75,82,125),(76,83,126),(77,84,127),(78,85,128),(79,86,121),(80,87,122)], [(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)], [(1,20,5,24),(2,19,6,23),(3,18,7,22),(4,17,8,21),(9,82,13,86),(10,81,14,85),(11,88,15,84),(12,87,16,83),(25,111,29,107),(26,110,30,106),(27,109,31,105),(28,108,32,112),(33,77,37,73),(34,76,38,80),(35,75,39,79),(36,74,40,78),(41,132,45,136),(42,131,46,135),(43,130,47,134),(44,129,48,133),(49,124,53,128),(50,123,54,127),(51,122,55,126),(52,121,56,125),(57,116,61,120),(58,115,62,119),(59,114,63,118),(60,113,64,117),(65,92,69,96),(66,91,70,95),(67,90,71,94),(68,89,72,93),(97,141,101,137),(98,140,102,144),(99,139,103,143),(100,138,104,142)]])
53 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 | ··· | 12J | 12K | 12L | 12M | 12N | 24A | 24B | 24C | 24D | 24E | ··· | 24T |
| 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 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 36 | 36 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
53 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | S3 | D4 | D6 | Q16 | D12 | Dic12 | He3⋊C2 | C2×He3⋊C2 | He3⋊5D4 | He3⋊5Q16 |
| kernel | He3⋊5Q16 | C8×He3 | He3⋊4Q8 | C3×C24 | C2×He3 | C3×C12 | He3 | C3×C6 | C32 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 8 | 16 | 4 | 4 | 2 | 4 |
Matrix representation of He3⋊5Q16 ►in GL5(𝔽73)
| 0 | 72 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 64 | 0 |
| 0 | 0 | 65 | 0 | 8 |
| 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 63 | 0 |
| 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 23 | 68 | 0 | 0 | 0 |
| 5 | 18 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 34 | 65 | 0 | 0 | 0 |
| 26 | 39 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 72 | 0 |
G:=sub<GL(5,GF(73))| [0,1,0,0,0,72,72,0,0,0,0,0,1,1,65,0,0,0,64,0,0,0,0,0,8],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,63,72,72,0,0,0,1,0],[23,5,0,0,0,68,18,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[34,26,0,0,0,65,39,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,72,0] >;
He3⋊5Q16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_5Q_{16} % in TeX
G:=Group("He3:5Q16"); // GroupNames label
G:=SmallGroup(432,177);
// by ID
G=gap.SmallGroup(432,177);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,1124,4037,537]);
// 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,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations