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 [×4], C4, C4 [×2], C6 [×4], C8, Q8 [×2], C32, Dic3 [×8], C12 [×4], Q16, C3×C6, C24 [×4], Dic6 [×8], C3⋊Dic3 [×2], C3×C12, Dic12 [×4], C3×C24, C32⋊4Q8 [×2], C32⋊5Q16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], Q16, C3⋊S3, D12 [×4], C2×C3⋊S3, Dic12 [×4], C12⋊S3, C32⋊5Q16
►Regular action on 144 points
Generators in S144
(1 68 82)(2 69 83)(3 70 84)(4 71 85)(5 72 86)(6 65 87)(7 66 88)(8 67 81)(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 59 95)(18 60 96)(19 61 89)(20 62 90)(21 63 91)(22 64 92)(23 57 93)(24 58 94)(25 138 41)(26 139 42)(27 140 43)(28 141 44)(29 142 45)(30 143 46)(31 144 47)(32 137 48)(73 108 132)(74 109 133)(75 110 134)(76 111 135)(77 112 136)(78 105 129)(79 106 130)(80 107 131)(97 124 117)(98 125 118)(99 126 119)(100 127 120)(101 128 113)(102 121 114)(103 122 115)(104 123 116)
(1 114 134)(2 115 135)(3 116 136)(4 117 129)(5 118 130)(6 119 131)(7 120 132)(8 113 133)(9 47 24)(10 48 17)(11 41 18)(12 42 19)(13 43 20)(14 44 21)(15 45 22)(16 46 23)(25 60 37)(26 61 38)(27 62 39)(28 63 40)(29 64 33)(30 57 34)(31 58 35)(32 59 36)(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 99 80)(66 100 73)(67 101 74)(68 102 75)(69 103 76)(70 104 77)(71 97 78)(72 98 79)(81 128 109)(82 121 110)(83 122 111)(84 123 112)(85 124 105)(86 125 106)(87 126 107)(88 127 108)
(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 47 5 43)(2 46 6 42)(3 45 7 41)(4 44 8 48)(9 118 13 114)(10 117 14 113)(11 116 15 120)(12 115 16 119)(17 129 21 133)(18 136 22 132)(19 135 23 131)(20 134 24 130)(25 84 29 88)(26 83 30 87)(27 82 31 86)(28 81 32 85)(33 127 37 123)(34 126 38 122)(35 125 39 121)(36 124 40 128)(49 97 53 101)(50 104 54 100)(51 103 55 99)(52 102 56 98)(57 107 61 111)(58 106 62 110)(59 105 63 109)(60 112 64 108)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)(73 96 77 92)(74 95 78 91)(75 94 79 90)(76 93 80 89)
G:=sub<Sym(144)| (1,68,82)(2,69,83)(3,70,84)(4,71,85)(5,72,86)(6,65,87)(7,66,88)(8,67,81)(9,35,56)(10,36,49)(11,37,50)(12,38,51)(13,39,52)(14,40,53)(15,33,54)(16,34,55)(17,59,95)(18,60,96)(19,61,89)(20,62,90)(21,63,91)(22,64,92)(23,57,93)(24,58,94)(25,138,41)(26,139,42)(27,140,43)(28,141,44)(29,142,45)(30,143,46)(31,144,47)(32,137,48)(73,108,132)(74,109,133)(75,110,134)(76,111,135)(77,112,136)(78,105,129)(79,106,130)(80,107,131)(97,124,117)(98,125,118)(99,126,119)(100,127,120)(101,128,113)(102,121,114)(103,122,115)(104,123,116), (1,114,134)(2,115,135)(3,116,136)(4,117,129)(5,118,130)(6,119,131)(7,120,132)(8,113,133)(9,47,24)(10,48,17)(11,41,18)(12,42,19)(13,43,20)(14,44,21)(15,45,22)(16,46,23)(25,60,37)(26,61,38)(27,62,39)(28,63,40)(29,64,33)(30,57,34)(31,58,35)(32,59,36)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,99,80)(66,100,73)(67,101,74)(68,102,75)(69,103,76)(70,104,77)(71,97,78)(72,98,79)(81,128,109)(82,121,110)(83,122,111)(84,123,112)(85,124,105)(86,125,106)(87,126,107)(88,127,108), (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), (1,47,5,43)(2,46,6,42)(3,45,7,41)(4,44,8,48)(9,118,13,114)(10,117,14,113)(11,116,15,120)(12,115,16,119)(17,129,21,133)(18,136,22,132)(19,135,23,131)(20,134,24,130)(25,84,29,88)(26,83,30,87)(27,82,31,86)(28,81,32,85)(33,127,37,123)(34,126,38,122)(35,125,39,121)(36,124,40,128)(49,97,53,101)(50,104,54,100)(51,103,55,99)(52,102,56,98)(57,107,61,111)(58,106,62,110)(59,105,63,109)(60,112,64,108)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(73,96,77,92)(74,95,78,91)(75,94,79,90)(76,93,80,89)>;
G:=Group( (1,68,82)(2,69,83)(3,70,84)(4,71,85)(5,72,86)(6,65,87)(7,66,88)(8,67,81)(9,35,56)(10,36,49)(11,37,50)(12,38,51)(13,39,52)(14,40,53)(15,33,54)(16,34,55)(17,59,95)(18,60,96)(19,61,89)(20,62,90)(21,63,91)(22,64,92)(23,57,93)(24,58,94)(25,138,41)(26,139,42)(27,140,43)(28,141,44)(29,142,45)(30,143,46)(31,144,47)(32,137,48)(73,108,132)(74,109,133)(75,110,134)(76,111,135)(77,112,136)(78,105,129)(79,106,130)(80,107,131)(97,124,117)(98,125,118)(99,126,119)(100,127,120)(101,128,113)(102,121,114)(103,122,115)(104,123,116), (1,114,134)(2,115,135)(3,116,136)(4,117,129)(5,118,130)(6,119,131)(7,120,132)(8,113,133)(9,47,24)(10,48,17)(11,41,18)(12,42,19)(13,43,20)(14,44,21)(15,45,22)(16,46,23)(25,60,37)(26,61,38)(27,62,39)(28,63,40)(29,64,33)(30,57,34)(31,58,35)(32,59,36)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,99,80)(66,100,73)(67,101,74)(68,102,75)(69,103,76)(70,104,77)(71,97,78)(72,98,79)(81,128,109)(82,121,110)(83,122,111)(84,123,112)(85,124,105)(86,125,106)(87,126,107)(88,127,108), (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), (1,47,5,43)(2,46,6,42)(3,45,7,41)(4,44,8,48)(9,118,13,114)(10,117,14,113)(11,116,15,120)(12,115,16,119)(17,129,21,133)(18,136,22,132)(19,135,23,131)(20,134,24,130)(25,84,29,88)(26,83,30,87)(27,82,31,86)(28,81,32,85)(33,127,37,123)(34,126,38,122)(35,125,39,121)(36,124,40,128)(49,97,53,101)(50,104,54,100)(51,103,55,99)(52,102,56,98)(57,107,61,111)(58,106,62,110)(59,105,63,109)(60,112,64,108)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(73,96,77,92)(74,95,78,91)(75,94,79,90)(76,93,80,89) );
G=PermutationGroup([(1,68,82),(2,69,83),(3,70,84),(4,71,85),(5,72,86),(6,65,87),(7,66,88),(8,67,81),(9,35,56),(10,36,49),(11,37,50),(12,38,51),(13,39,52),(14,40,53),(15,33,54),(16,34,55),(17,59,95),(18,60,96),(19,61,89),(20,62,90),(21,63,91),(22,64,92),(23,57,93),(24,58,94),(25,138,41),(26,139,42),(27,140,43),(28,141,44),(29,142,45),(30,143,46),(31,144,47),(32,137,48),(73,108,132),(74,109,133),(75,110,134),(76,111,135),(77,112,136),(78,105,129),(79,106,130),(80,107,131),(97,124,117),(98,125,118),(99,126,119),(100,127,120),(101,128,113),(102,121,114),(103,122,115),(104,123,116)], [(1,114,134),(2,115,135),(3,116,136),(4,117,129),(5,118,130),(6,119,131),(7,120,132),(8,113,133),(9,47,24),(10,48,17),(11,41,18),(12,42,19),(13,43,20),(14,44,21),(15,45,22),(16,46,23),(25,60,37),(26,61,38),(27,62,39),(28,63,40),(29,64,33),(30,57,34),(31,58,35),(32,59,36),(49,137,95),(50,138,96),(51,139,89),(52,140,90),(53,141,91),(54,142,92),(55,143,93),(56,144,94),(65,99,80),(66,100,73),(67,101,74),(68,102,75),(69,103,76),(70,104,77),(71,97,78),(72,98,79),(81,128,109),(82,121,110),(83,122,111),(84,123,112),(85,124,105),(86,125,106),(87,126,107),(88,127,108)], [(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)], [(1,47,5,43),(2,46,6,42),(3,45,7,41),(4,44,8,48),(9,118,13,114),(10,117,14,113),(11,116,15,120),(12,115,16,119),(17,129,21,133),(18,136,22,132),(19,135,23,131),(20,134,24,130),(25,84,29,88),(26,83,30,87),(27,82,31,86),(28,81,32,85),(33,127,37,123),(34,126,38,122),(35,125,39,121),(36,124,40,128),(49,97,53,101),(50,104,54,100),(51,103,55,99),(52,102,56,98),(57,107,61,111),(58,106,62,110),(59,105,63,109),(60,112,64,108),(65,139,69,143),(66,138,70,142),(67,137,71,141),(68,144,72,140),(73,96,77,92),(74,95,78,91),(75,94,79,90),(76,93,80,89)])
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