direct product, metacyclic, nilpotent (class 3), monomial
Aliases: C32×Q16, C24.4C6, C4.3C62, C8.(C3×C6), (C3×C24).3C2, (C3×C6).43D4, C6.22(C3×D4), (C3×Q8).8C6, Q8.2(C3×C6), C12.25(C2×C6), C2.5(D4×C32), (Q8×C32).3C2, (C3×C12).52C22, SmallGroup(144,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×Q16
G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 66 in 54 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C3, C4, C4, C6, C8, Q8, C32, C12, C12, Q16, C3×C6, C24, C3×Q8, C3×C12, C3×C12, C3×Q16, C3×C24, Q8×C32, C32×Q16
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, Q16, C3×C6, C3×D4, C62, C3×Q16, D4×C32, C32×Q16
►Regular action on 144 points
Generators in S144
(1 106 39)(2 107 40)(3 108 33)(4 109 34)(5 110 35)(6 111 36)(7 112 37)(8 105 38)(9 67 17)(10 68 18)(11 69 19)(12 70 20)(13 71 21)(14 72 22)(15 65 23)(16 66 24)(25 52 129)(26 53 130)(27 54 131)(28 55 132)(29 56 133)(30 49 134)(31 50 135)(32 51 136)(41 142 77)(42 143 78)(43 144 79)(44 137 80)(45 138 73)(46 139 74)(47 140 75)(48 141 76)(57 123 115)(58 124 116)(59 125 117)(60 126 118)(61 127 119)(62 128 120)(63 121 113)(64 122 114)(81 97 89)(82 98 90)(83 99 91)(84 100 92)(85 101 93)(86 102 94)(87 103 95)(88 104 96)
(1 131 46)(2 132 47)(3 133 48)(4 134 41)(5 135 42)(6 136 43)(7 129 44)(8 130 45)(9 88 59)(10 81 60)(11 82 61)(12 83 62)(13 84 63)(14 85 64)(15 86 57)(16 87 58)(17 96 117)(18 89 118)(19 90 119)(20 91 120)(21 92 113)(22 93 114)(23 94 115)(24 95 116)(25 137 112)(26 138 105)(27 139 106)(28 140 107)(29 141 108)(30 142 109)(31 143 110)(32 144 111)(33 56 76)(34 49 77)(35 50 78)(36 51 79)(37 52 80)(38 53 73)(39 54 74)(40 55 75)(65 102 123)(66 103 124)(67 104 125)(68 97 126)(69 98 127)(70 99 128)(71 100 121)(72 101 122)
(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 82 5 86)(2 81 6 85)(3 88 7 84)(4 87 8 83)(9 44 13 48)(10 43 14 47)(11 42 15 46)(12 41 16 45)(17 80 21 76)(18 79 22 75)(19 78 23 74)(20 77 24 73)(25 121 29 125)(26 128 30 124)(27 127 31 123)(28 126 32 122)(33 96 37 92)(34 95 38 91)(35 94 39 90)(36 93 40 89)(49 116 53 120)(50 115 54 119)(51 114 55 118)(52 113 56 117)(57 131 61 135)(58 130 62 134)(59 129 63 133)(60 136 64 132)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)(97 111 101 107)(98 110 102 106)(99 109 103 105)(100 108 104 112)
G:=sub<Sym(144)| (1,106,39)(2,107,40)(3,108,33)(4,109,34)(5,110,35)(6,111,36)(7,112,37)(8,105,38)(9,67,17)(10,68,18)(11,69,19)(12,70,20)(13,71,21)(14,72,22)(15,65,23)(16,66,24)(25,52,129)(26,53,130)(27,54,131)(28,55,132)(29,56,133)(30,49,134)(31,50,135)(32,51,136)(41,142,77)(42,143,78)(43,144,79)(44,137,80)(45,138,73)(46,139,74)(47,140,75)(48,141,76)(57,123,115)(58,124,116)(59,125,117)(60,126,118)(61,127,119)(62,128,120)(63,121,113)(64,122,114)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(85,101,93)(86,102,94)(87,103,95)(88,104,96), (1,131,46)(2,132,47)(3,133,48)(4,134,41)(5,135,42)(6,136,43)(7,129,44)(8,130,45)(9,88,59)(10,81,60)(11,82,61)(12,83,62)(13,84,63)(14,85,64)(15,86,57)(16,87,58)(17,96,117)(18,89,118)(19,90,119)(20,91,120)(21,92,113)(22,93,114)(23,94,115)(24,95,116)(25,137,112)(26,138,105)(27,139,106)(28,140,107)(29,141,108)(30,142,109)(31,143,110)(32,144,111)(33,56,76)(34,49,77)(35,50,78)(36,51,79)(37,52,80)(38,53,73)(39,54,74)(40,55,75)(65,102,123)(66,103,124)(67,104,125)(68,97,126)(69,98,127)(70,99,128)(71,100,121)(72,101,122), (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), (1,82,5,86)(2,81,6,85)(3,88,7,84)(4,87,8,83)(9,44,13,48)(10,43,14,47)(11,42,15,46)(12,41,16,45)(17,80,21,76)(18,79,22,75)(19,78,23,74)(20,77,24,73)(25,121,29,125)(26,128,30,124)(27,127,31,123)(28,126,32,122)(33,96,37,92)(34,95,38,91)(35,94,39,90)(36,93,40,89)(49,116,53,120)(50,115,54,119)(51,114,55,118)(52,113,56,117)(57,131,61,135)(58,130,62,134)(59,129,63,133)(60,136,64,132)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(97,111,101,107)(98,110,102,106)(99,109,103,105)(100,108,104,112)>;
G:=Group( (1,106,39)(2,107,40)(3,108,33)(4,109,34)(5,110,35)(6,111,36)(7,112,37)(8,105,38)(9,67,17)(10,68,18)(11,69,19)(12,70,20)(13,71,21)(14,72,22)(15,65,23)(16,66,24)(25,52,129)(26,53,130)(27,54,131)(28,55,132)(29,56,133)(30,49,134)(31,50,135)(32,51,136)(41,142,77)(42,143,78)(43,144,79)(44,137,80)(45,138,73)(46,139,74)(47,140,75)(48,141,76)(57,123,115)(58,124,116)(59,125,117)(60,126,118)(61,127,119)(62,128,120)(63,121,113)(64,122,114)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(85,101,93)(86,102,94)(87,103,95)(88,104,96), (1,131,46)(2,132,47)(3,133,48)(4,134,41)(5,135,42)(6,136,43)(7,129,44)(8,130,45)(9,88,59)(10,81,60)(11,82,61)(12,83,62)(13,84,63)(14,85,64)(15,86,57)(16,87,58)(17,96,117)(18,89,118)(19,90,119)(20,91,120)(21,92,113)(22,93,114)(23,94,115)(24,95,116)(25,137,112)(26,138,105)(27,139,106)(28,140,107)(29,141,108)(30,142,109)(31,143,110)(32,144,111)(33,56,76)(34,49,77)(35,50,78)(36,51,79)(37,52,80)(38,53,73)(39,54,74)(40,55,75)(65,102,123)(66,103,124)(67,104,125)(68,97,126)(69,98,127)(70,99,128)(71,100,121)(72,101,122), (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), (1,82,5,86)(2,81,6,85)(3,88,7,84)(4,87,8,83)(9,44,13,48)(10,43,14,47)(11,42,15,46)(12,41,16,45)(17,80,21,76)(18,79,22,75)(19,78,23,74)(20,77,24,73)(25,121,29,125)(26,128,30,124)(27,127,31,123)(28,126,32,122)(33,96,37,92)(34,95,38,91)(35,94,39,90)(36,93,40,89)(49,116,53,120)(50,115,54,119)(51,114,55,118)(52,113,56,117)(57,131,61,135)(58,130,62,134)(59,129,63,133)(60,136,64,132)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(97,111,101,107)(98,110,102,106)(99,109,103,105)(100,108,104,112) );
G=PermutationGroup([[(1,106,39),(2,107,40),(3,108,33),(4,109,34),(5,110,35),(6,111,36),(7,112,37),(8,105,38),(9,67,17),(10,68,18),(11,69,19),(12,70,20),(13,71,21),(14,72,22),(15,65,23),(16,66,24),(25,52,129),(26,53,130),(27,54,131),(28,55,132),(29,56,133),(30,49,134),(31,50,135),(32,51,136),(41,142,77),(42,143,78),(43,144,79),(44,137,80),(45,138,73),(46,139,74),(47,140,75),(48,141,76),(57,123,115),(58,124,116),(59,125,117),(60,126,118),(61,127,119),(62,128,120),(63,121,113),(64,122,114),(81,97,89),(82,98,90),(83,99,91),(84,100,92),(85,101,93),(86,102,94),(87,103,95),(88,104,96)], [(1,131,46),(2,132,47),(3,133,48),(4,134,41),(5,135,42),(6,136,43),(7,129,44),(8,130,45),(9,88,59),(10,81,60),(11,82,61),(12,83,62),(13,84,63),(14,85,64),(15,86,57),(16,87,58),(17,96,117),(18,89,118),(19,90,119),(20,91,120),(21,92,113),(22,93,114),(23,94,115),(24,95,116),(25,137,112),(26,138,105),(27,139,106),(28,140,107),(29,141,108),(30,142,109),(31,143,110),(32,144,111),(33,56,76),(34,49,77),(35,50,78),(36,51,79),(37,52,80),(38,53,73),(39,54,74),(40,55,75),(65,102,123),(66,103,124),(67,104,125),(68,97,126),(69,98,127),(70,99,128),(71,100,121),(72,101,122)], [(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)], [(1,82,5,86),(2,81,6,85),(3,88,7,84),(4,87,8,83),(9,44,13,48),(10,43,14,47),(11,42,15,46),(12,41,16,45),(17,80,21,76),(18,79,22,75),(19,78,23,74),(20,77,24,73),(25,121,29,125),(26,128,30,124),(27,127,31,123),(28,126,32,122),(33,96,37,92),(34,95,38,91),(35,94,39,90),(36,93,40,89),(49,116,53,120),(50,115,54,119),(51,114,55,118),(52,113,56,117),(57,131,61,135),(58,130,62,134),(59,129,63,133),(60,136,64,132),(65,139,69,143),(66,138,70,142),(67,137,71,141),(68,144,72,140),(97,111,101,107),(98,110,102,106),(99,109,103,105),(100,108,104,112)]])
C32×Q16 is a maximal subgroup of
C32⋊10SD32 C32⋊7Q32 C24.35D6 C24.28D6
63 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 4C | 6A | ··· | 6H | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12X | 24A | ··· | 24P |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | Q16 | C3×D4 | C3×Q16 |
| kernel | C32×Q16 | C3×C24 | Q8×C32 | C3×Q16 | C24 | C3×Q8 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 2 | 8 | 16 |
Matrix representation of C32×Q16 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 72 | 0 | 0 |
| 0 | 16 | 57 |
| 0 | 16 | 16 |
| 1 | 0 | 0 |
| 0 | 23 | 28 |
| 0 | 28 | 50 |
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[8,0,0,0,8,0,0,0,8],[72,0,0,0,16,16,0,57,16],[1,0,0,0,23,28,0,28,50] >;
C32×Q16 in GAP, Magma, Sage, TeX
C_3^2\times Q_{16} % in TeX
G:=Group("C3^2xQ16"); // GroupNames label
G:=SmallGroup(144,108);
// by ID
G=gap.SmallGroup(144,108);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,432,457,439,3244,1630,88]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations