metabelian, supersoluble, monomial
Aliases: C24.3S3, C6.9D12, C32⋊5Q16, C3⋊1Dic12, C12.48D6, C8.(C3⋊S3), (C3×C24).1C2, (C3×C6).24D4, C2.5(C12⋊S3), C32⋊4Q8.1C2, (C3×C12).34C22, C4.10(C2×C3⋊S3), SmallGroup(144,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊5Q16
G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 178 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C3, C4, C4, C6, C8, Q8, C32, Dic3, C12, Q16, C3×C6, C24, Dic6, C3⋊Dic3, C3×C12, Dic12, C3×C24, C32⋊4Q8, C32⋊5Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊S3, D12, C2×C3⋊S3, Dic12, C12⋊S3, C32⋊5Q16
►Regular action on 144 points
Generators in S144
(1 121 82)(2 122 83)(3 123 84)(4 124 85)(5 125 86)(6 126 87)(7 127 88)(8 128 81)(9 49 28)(10 50 29)(11 51 30)(12 52 31)(13 53 32)(14 54 25)(15 55 26)(16 56 27)(17 59 95)(18 60 96)(19 61 89)(20 62 90)(21 63 91)(22 64 92)(23 57 93)(24 58 94)(33 44 139)(34 45 140)(35 46 141)(36 47 142)(37 48 143)(38 41 144)(39 42 137)(40 43 138)(65 131 118)(66 132 119)(67 133 120)(68 134 113)(69 135 114)(70 136 115)(71 129 116)(72 130 117)(73 98 105)(74 99 106)(75 100 107)(76 101 108)(77 102 109)(78 103 110)(79 104 111)(80 97 112)
(1 104 116)(2 97 117)(3 98 118)(4 99 119)(5 100 120)(6 101 113)(7 102 114)(8 103 115)(9 42 59)(10 43 60)(11 44 61)(12 45 62)(13 46 63)(14 47 64)(15 48 57)(16 41 58)(17 28 39)(18 29 40)(19 30 33)(20 31 34)(21 32 35)(22 25 36)(23 26 37)(24 27 38)(49 137 95)(50 138 96)(51 139 89)(52 140 90)(53 141 91)(54 142 92)(55 143 93)(56 144 94)(65 123 105)(66 124 106)(67 125 107)(68 126 108)(69 127 109)(70 128 110)(71 121 111)(72 122 112)(73 131 84)(74 132 85)(75 133 86)(76 134 87)(77 135 88)(78 136 81)(79 129 82)(80 130 83)
(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 38 5 34)(2 37 6 33)(3 36 7 40)(4 35 8 39)(9 74 13 78)(10 73 14 77)(11 80 15 76)(12 79 16 75)(17 119 21 115)(18 118 22 114)(19 117 23 113)(20 116 24 120)(25 102 29 98)(26 101 30 97)(27 100 31 104)(28 99 32 103)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 106 53 110)(50 105 54 109)(51 112 55 108)(52 111 56 107)(57 134 61 130)(58 133 62 129)(59 132 63 136)(60 131 64 135)(65 92 69 96)(66 91 70 95)(67 90 71 94)(68 89 72 93)(121 144 125 140)(122 143 126 139)(123 142 127 138)(124 141 128 137)
G:=sub<Sym(144)| (1,121,82)(2,122,83)(3,123,84)(4,124,85)(5,125,86)(6,126,87)(7,127,88)(8,128,81)(9,49,28)(10,50,29)(11,51,30)(12,52,31)(13,53,32)(14,54,25)(15,55,26)(16,56,27)(17,59,95)(18,60,96)(19,61,89)(20,62,90)(21,63,91)(22,64,92)(23,57,93)(24,58,94)(33,44,139)(34,45,140)(35,46,141)(36,47,142)(37,48,143)(38,41,144)(39,42,137)(40,43,138)(65,131,118)(66,132,119)(67,133,120)(68,134,113)(69,135,114)(70,136,115)(71,129,116)(72,130,117)(73,98,105)(74,99,106)(75,100,107)(76,101,108)(77,102,109)(78,103,110)(79,104,111)(80,97,112), (1,104,116)(2,97,117)(3,98,118)(4,99,119)(5,100,120)(6,101,113)(7,102,114)(8,103,115)(9,42,59)(10,43,60)(11,44,61)(12,45,62)(13,46,63)(14,47,64)(15,48,57)(16,41,58)(17,28,39)(18,29,40)(19,30,33)(20,31,34)(21,32,35)(22,25,36)(23,26,37)(24,27,38)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,123,105)(66,124,106)(67,125,107)(68,126,108)(69,127,109)(70,128,110)(71,121,111)(72,122,112)(73,131,84)(74,132,85)(75,133,86)(76,134,87)(77,135,88)(78,136,81)(79,129,82)(80,130,83), (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), (1,38,5,34)(2,37,6,33)(3,36,7,40)(4,35,8,39)(9,74,13,78)(10,73,14,77)(11,80,15,76)(12,79,16,75)(17,119,21,115)(18,118,22,114)(19,117,23,113)(20,116,24,120)(25,102,29,98)(26,101,30,97)(27,100,31,104)(28,99,32,103)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,106,53,110)(50,105,54,109)(51,112,55,108)(52,111,56,107)(57,134,61,130)(58,133,62,129)(59,132,63,136)(60,131,64,135)(65,92,69,96)(66,91,70,95)(67,90,71,94)(68,89,72,93)(121,144,125,140)(122,143,126,139)(123,142,127,138)(124,141,128,137)>;
G:=Group( (1,121,82)(2,122,83)(3,123,84)(4,124,85)(5,125,86)(6,126,87)(7,127,88)(8,128,81)(9,49,28)(10,50,29)(11,51,30)(12,52,31)(13,53,32)(14,54,25)(15,55,26)(16,56,27)(17,59,95)(18,60,96)(19,61,89)(20,62,90)(21,63,91)(22,64,92)(23,57,93)(24,58,94)(33,44,139)(34,45,140)(35,46,141)(36,47,142)(37,48,143)(38,41,144)(39,42,137)(40,43,138)(65,131,118)(66,132,119)(67,133,120)(68,134,113)(69,135,114)(70,136,115)(71,129,116)(72,130,117)(73,98,105)(74,99,106)(75,100,107)(76,101,108)(77,102,109)(78,103,110)(79,104,111)(80,97,112), (1,104,116)(2,97,117)(3,98,118)(4,99,119)(5,100,120)(6,101,113)(7,102,114)(8,103,115)(9,42,59)(10,43,60)(11,44,61)(12,45,62)(13,46,63)(14,47,64)(15,48,57)(16,41,58)(17,28,39)(18,29,40)(19,30,33)(20,31,34)(21,32,35)(22,25,36)(23,26,37)(24,27,38)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,123,105)(66,124,106)(67,125,107)(68,126,108)(69,127,109)(70,128,110)(71,121,111)(72,122,112)(73,131,84)(74,132,85)(75,133,86)(76,134,87)(77,135,88)(78,136,81)(79,129,82)(80,130,83), (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), (1,38,5,34)(2,37,6,33)(3,36,7,40)(4,35,8,39)(9,74,13,78)(10,73,14,77)(11,80,15,76)(12,79,16,75)(17,119,21,115)(18,118,22,114)(19,117,23,113)(20,116,24,120)(25,102,29,98)(26,101,30,97)(27,100,31,104)(28,99,32,103)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,106,53,110)(50,105,54,109)(51,112,55,108)(52,111,56,107)(57,134,61,130)(58,133,62,129)(59,132,63,136)(60,131,64,135)(65,92,69,96)(66,91,70,95)(67,90,71,94)(68,89,72,93)(121,144,125,140)(122,143,126,139)(123,142,127,138)(124,141,128,137) );
G=PermutationGroup([[(1,121,82),(2,122,83),(3,123,84),(4,124,85),(5,125,86),(6,126,87),(7,127,88),(8,128,81),(9,49,28),(10,50,29),(11,51,30),(12,52,31),(13,53,32),(14,54,25),(15,55,26),(16,56,27),(17,59,95),(18,60,96),(19,61,89),(20,62,90),(21,63,91),(22,64,92),(23,57,93),(24,58,94),(33,44,139),(34,45,140),(35,46,141),(36,47,142),(37,48,143),(38,41,144),(39,42,137),(40,43,138),(65,131,118),(66,132,119),(67,133,120),(68,134,113),(69,135,114),(70,136,115),(71,129,116),(72,130,117),(73,98,105),(74,99,106),(75,100,107),(76,101,108),(77,102,109),(78,103,110),(79,104,111),(80,97,112)], [(1,104,116),(2,97,117),(3,98,118),(4,99,119),(5,100,120),(6,101,113),(7,102,114),(8,103,115),(9,42,59),(10,43,60),(11,44,61),(12,45,62),(13,46,63),(14,47,64),(15,48,57),(16,41,58),(17,28,39),(18,29,40),(19,30,33),(20,31,34),(21,32,35),(22,25,36),(23,26,37),(24,27,38),(49,137,95),(50,138,96),(51,139,89),(52,140,90),(53,141,91),(54,142,92),(55,143,93),(56,144,94),(65,123,105),(66,124,106),(67,125,107),(68,126,108),(69,127,109),(70,128,110),(71,121,111),(72,122,112),(73,131,84),(74,132,85),(75,133,86),(76,134,87),(77,135,88),(78,136,81),(79,129,82),(80,130,83)], [(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)], [(1,38,5,34),(2,37,6,33),(3,36,7,40),(4,35,8,39),(9,74,13,78),(10,73,14,77),(11,80,15,76),(12,79,16,75),(17,119,21,115),(18,118,22,114),(19,117,23,113),(20,116,24,120),(25,102,29,98),(26,101,30,97),(27,100,31,104),(28,99,32,103),(41,86,45,82),(42,85,46,81),(43,84,47,88),(44,83,48,87),(49,106,53,110),(50,105,54,109),(51,112,55,108),(52,111,56,107),(57,134,61,130),(58,133,62,129),(59,132,63,136),(60,131,64,135),(65,92,69,96),(66,91,70,95),(67,90,71,94),(68,89,72,93),(121,144,125,140),(122,143,126,139),(123,142,127,138),(124,141,128,137)]])
C32⋊5Q16 is a maximal subgroup of
C32⋊3SD32 C32⋊3Q32 C6.D24 C32⋊5Q32 C32⋊8SD32 C32⋊7Q32 S3×Dic12 C24.3D6 D24⋊7S3 C24.78D6 C24.5D6 C24.26D6 C24.32D6 Q16×C3⋊S3 He3⋊4Q16 C24.D9 C33⋊8Q16 C33⋊12Q16
C32⋊5Q16 is a maximal quotient of
C6.4Dic12 C24⋊1Dic3 C24.D9 He3⋊5Q16 C33⋊8Q16 C33⋊12Q16
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D6 | Q16 | D12 | Dic12 |
| kernel | C32⋊5Q16 | C3×C24 | C32⋊4Q8 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 8 | 16 |
Matrix representation of C32⋊5Q16 ►in GL4(𝔽73) generated by
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 23 | 5 |
| 0 | 0 | 68 | 18 |
| 50 | 68 | 0 | 0 |
| 18 | 23 | 0 | 0 |
| 0 | 0 | 2 | 20 |
| 0 | 0 | 18 | 71 |
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,72,0,0,0,0,0,72,0,0,1,72],[66,14,0,0,59,7,0,0,0,0,23,68,0,0,5,18],[50,18,0,0,68,23,0,0,0,0,2,18,0,0,20,71] >;
C32⋊5Q16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_5Q_{16} % in TeX
G:=Group("C3^2:5Q16"); // GroupNames label
G:=SmallGroup(144,89);
// by ID
G=gap.SmallGroup(144,89);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,79,218,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations