direct product, metabelian, nilpotent (class 3), monomial
Aliases: C32×C8.C22, C8.C62, D4.3C62, Q8.6C62, C62.100D4, (C3×Q16)⋊6C6, Q16⋊2(C3×C6), (C6×Q8)⋊15C6, C6.96(C6×D4), C24.14(C2×C6), (C3×SD16)⋊6C6, SD16⋊2(C3×C6), C12.81(C3×D4), (C2×C4).7C62, C4.6(C2×C62), (C3×C12).182D4, M4(2)⋊2(C3×C6), (C3×M4(2))⋊4C6, C4.15(D4×C32), (C3×C24).40C22, C12.60(C22×C6), (C32×Q16)⋊10C2, C22.6(D4×C32), (C3×C12).190C23, (C6×C12).276C22, (C32×SD16)⋊10C2, (C32×M4(2))⋊6C2, (D4×C32).34C22, (Q8×C32).37C22, (Q8×C3×C6)⋊18C2, C2.16(D4×C3×C6), (C2×Q8)⋊6(C3×C6), C4○D4.4(C3×C6), (C2×C6).35(C3×D4), (C2×C12).77(C2×C6), (C3×C4○D4).19C6, (C3×D4).18(C2×C6), (C3×C6).313(C2×D4), (C3×Q8).31(C2×C6), (C32×C4○D4).8C2, SmallGroup(288,834)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C8.C22
G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >
Subgroups: 252 in 180 conjugacy classes, 120 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C32, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×C6, C3×C6, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×C12, C3×C12, C62, C62, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C3×C24, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8×C32, Q8×C32, C3×C8.C22, C32×M4(2), C32×SD16, C32×Q16, Q8×C3×C6, C32×C4○D4, C32×C8.C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, C8.C22, C62, C6×D4, D4×C32, C2×C62, C3×C8.C22, D4×C3×C6, C32×C8.C22
►On 144 points
Generators in S144
(1 31 17)(2 32 18)(3 25 19)(4 26 20)(5 27 21)(6 28 22)(7 29 23)(8 30 24)(9 36 131)(10 37 132)(11 38 133)(12 39 134)(13 40 135)(14 33 136)(15 34 129)(16 35 130)(41 96 49)(42 89 50)(43 90 51)(44 91 52)(45 92 53)(46 93 54)(47 94 55)(48 95 56)(57 78 65)(58 79 66)(59 80 67)(60 73 68)(61 74 69)(62 75 70)(63 76 71)(64 77 72)(81 123 115)(82 124 116)(83 125 117)(84 126 118)(85 127 119)(86 128 120)(87 121 113)(88 122 114)(97 141 105)(98 142 106)(99 143 107)(100 144 108)(101 137 109)(102 138 110)(103 139 111)(104 140 112)
(1 83 42)(2 84 43)(3 85 44)(4 86 45)(5 87 46)(6 88 47)(7 81 48)(8 82 41)(9 139 77)(10 140 78)(11 141 79)(12 142 80)(13 143 73)(14 144 74)(15 137 75)(16 138 76)(17 117 50)(18 118 51)(19 119 52)(20 120 53)(21 113 54)(22 114 55)(23 115 56)(24 116 49)(25 127 91)(26 128 92)(27 121 93)(28 122 94)(29 123 95)(30 124 96)(31 125 89)(32 126 90)(33 108 69)(34 109 70)(35 110 71)(36 111 72)(37 112 65)(38 105 66)(39 106 67)(40 107 68)(57 132 104)(58 133 97)(59 134 98)(60 135 99)(61 136 100)(62 129 101)(63 130 102)(64 131 103)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(25 29)(26 32)(28 30)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(57 59)(58 62)(61 63)(65 67)(66 70)(69 71)(74 76)(75 79)(78 80)(81 85)(82 88)(84 86)(90 92)(91 95)(94 96)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)(114 116)(115 119)(118 120)(122 124)(123 127)(126 128)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)
(1 62)(2 59)(3 64)(4 61)(5 58)(6 63)(7 60)(8 57)(9 127)(10 124)(11 121)(12 126)(13 123)(14 128)(15 125)(16 122)(17 70)(18 67)(19 72)(20 69)(21 66)(22 71)(23 68)(24 65)(25 77)(26 74)(27 79)(28 76)(29 73)(30 78)(31 75)(32 80)(33 120)(34 117)(35 114)(36 119)(37 116)(38 113)(39 118)(40 115)(41 104)(42 101)(43 98)(44 103)(45 100)(46 97)(47 102)(48 99)(49 112)(50 109)(51 106)(52 111)(53 108)(54 105)(55 110)(56 107)(81 135)(82 132)(83 129)(84 134)(85 131)(86 136)(87 133)(88 130)(89 137)(90 142)(91 139)(92 144)(93 141)(94 138)(95 143)(96 140)
G:=sub<Sym(144)| (1,31,17)(2,32,18)(3,25,19)(4,26,20)(5,27,21)(6,28,22)(7,29,23)(8,30,24)(9,36,131)(10,37,132)(11,38,133)(12,39,134)(13,40,135)(14,33,136)(15,34,129)(16,35,130)(41,96,49)(42,89,50)(43,90,51)(44,91,52)(45,92,53)(46,93,54)(47,94,55)(48,95,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(81,123,115)(82,124,116)(83,125,117)(84,126,118)(85,127,119)(86,128,120)(87,121,113)(88,122,114)(97,141,105)(98,142,106)(99,143,107)(100,144,108)(101,137,109)(102,138,110)(103,139,111)(104,140,112), (1,83,42)(2,84,43)(3,85,44)(4,86,45)(5,87,46)(6,88,47)(7,81,48)(8,82,41)(9,139,77)(10,140,78)(11,141,79)(12,142,80)(13,143,73)(14,144,74)(15,137,75)(16,138,76)(17,117,50)(18,118,51)(19,119,52)(20,120,53)(21,113,54)(22,114,55)(23,115,56)(24,116,49)(25,127,91)(26,128,92)(27,121,93)(28,122,94)(29,123,95)(30,124,96)(31,125,89)(32,126,90)(33,108,69)(34,109,70)(35,110,71)(36,111,72)(37,112,65)(38,105,66)(39,106,67)(40,107,68)(57,132,104)(58,133,97)(59,134,98)(60,135,99)(61,136,100)(62,129,101)(63,130,102)(64,131,103), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,59)(58,62)(61,63)(65,67)(66,70)(69,71)(74,76)(75,79)(78,80)(81,85)(82,88)(84,86)(90,92)(91,95)(94,96)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)(114,116)(115,119)(118,120)(122,124)(123,127)(126,128)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,127)(10,124)(11,121)(12,126)(13,123)(14,128)(15,125)(16,122)(17,70)(18,67)(19,72)(20,69)(21,66)(22,71)(23,68)(24,65)(25,77)(26,74)(27,79)(28,76)(29,73)(30,78)(31,75)(32,80)(33,120)(34,117)(35,114)(36,119)(37,116)(38,113)(39,118)(40,115)(41,104)(42,101)(43,98)(44,103)(45,100)(46,97)(47,102)(48,99)(49,112)(50,109)(51,106)(52,111)(53,108)(54,105)(55,110)(56,107)(81,135)(82,132)(83,129)(84,134)(85,131)(86,136)(87,133)(88,130)(89,137)(90,142)(91,139)(92,144)(93,141)(94,138)(95,143)(96,140)>;
G:=Group( (1,31,17)(2,32,18)(3,25,19)(4,26,20)(5,27,21)(6,28,22)(7,29,23)(8,30,24)(9,36,131)(10,37,132)(11,38,133)(12,39,134)(13,40,135)(14,33,136)(15,34,129)(16,35,130)(41,96,49)(42,89,50)(43,90,51)(44,91,52)(45,92,53)(46,93,54)(47,94,55)(48,95,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(81,123,115)(82,124,116)(83,125,117)(84,126,118)(85,127,119)(86,128,120)(87,121,113)(88,122,114)(97,141,105)(98,142,106)(99,143,107)(100,144,108)(101,137,109)(102,138,110)(103,139,111)(104,140,112), (1,83,42)(2,84,43)(3,85,44)(4,86,45)(5,87,46)(6,88,47)(7,81,48)(8,82,41)(9,139,77)(10,140,78)(11,141,79)(12,142,80)(13,143,73)(14,144,74)(15,137,75)(16,138,76)(17,117,50)(18,118,51)(19,119,52)(20,120,53)(21,113,54)(22,114,55)(23,115,56)(24,116,49)(25,127,91)(26,128,92)(27,121,93)(28,122,94)(29,123,95)(30,124,96)(31,125,89)(32,126,90)(33,108,69)(34,109,70)(35,110,71)(36,111,72)(37,112,65)(38,105,66)(39,106,67)(40,107,68)(57,132,104)(58,133,97)(59,134,98)(60,135,99)(61,136,100)(62,129,101)(63,130,102)(64,131,103), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,59)(58,62)(61,63)(65,67)(66,70)(69,71)(74,76)(75,79)(78,80)(81,85)(82,88)(84,86)(90,92)(91,95)(94,96)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)(114,116)(115,119)(118,120)(122,124)(123,127)(126,128)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142), (1,62)(2,59)(3,64)(4,61)(5,58)(6,63)(7,60)(8,57)(9,127)(10,124)(11,121)(12,126)(13,123)(14,128)(15,125)(16,122)(17,70)(18,67)(19,72)(20,69)(21,66)(22,71)(23,68)(24,65)(25,77)(26,74)(27,79)(28,76)(29,73)(30,78)(31,75)(32,80)(33,120)(34,117)(35,114)(36,119)(37,116)(38,113)(39,118)(40,115)(41,104)(42,101)(43,98)(44,103)(45,100)(46,97)(47,102)(48,99)(49,112)(50,109)(51,106)(52,111)(53,108)(54,105)(55,110)(56,107)(81,135)(82,132)(83,129)(84,134)(85,131)(86,136)(87,133)(88,130)(89,137)(90,142)(91,139)(92,144)(93,141)(94,138)(95,143)(96,140) );
G=PermutationGroup([[(1,31,17),(2,32,18),(3,25,19),(4,26,20),(5,27,21),(6,28,22),(7,29,23),(8,30,24),(9,36,131),(10,37,132),(11,38,133),(12,39,134),(13,40,135),(14,33,136),(15,34,129),(16,35,130),(41,96,49),(42,89,50),(43,90,51),(44,91,52),(45,92,53),(46,93,54),(47,94,55),(48,95,56),(57,78,65),(58,79,66),(59,80,67),(60,73,68),(61,74,69),(62,75,70),(63,76,71),(64,77,72),(81,123,115),(82,124,116),(83,125,117),(84,126,118),(85,127,119),(86,128,120),(87,121,113),(88,122,114),(97,141,105),(98,142,106),(99,143,107),(100,144,108),(101,137,109),(102,138,110),(103,139,111),(104,140,112)], [(1,83,42),(2,84,43),(3,85,44),(4,86,45),(5,87,46),(6,88,47),(7,81,48),(8,82,41),(9,139,77),(10,140,78),(11,141,79),(12,142,80),(13,143,73),(14,144,74),(15,137,75),(16,138,76),(17,117,50),(18,118,51),(19,119,52),(20,120,53),(21,113,54),(22,114,55),(23,115,56),(24,116,49),(25,127,91),(26,128,92),(27,121,93),(28,122,94),(29,123,95),(30,124,96),(31,125,89),(32,126,90),(33,108,69),(34,109,70),(35,110,71),(36,111,72),(37,112,65),(38,105,66),(39,106,67),(40,107,68),(57,132,104),(58,133,97),(59,134,98),(60,135,99),(61,136,100),(62,129,101),(63,130,102),(64,131,103)], [(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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24),(25,29),(26,32),(28,30),(33,35),(34,38),(37,39),(41,47),(43,45),(44,48),(49,55),(51,53),(52,56),(57,59),(58,62),(61,63),(65,67),(66,70),(69,71),(74,76),(75,79),(78,80),(81,85),(82,88),(84,86),(90,92),(91,95),(94,96),(97,101),(98,104),(100,102),(105,109),(106,112),(108,110),(114,116),(115,119),(118,120),(122,124),(123,127),(126,128),(129,133),(130,136),(132,134),(137,141),(138,144),(140,142)], [(1,62),(2,59),(3,64),(4,61),(5,58),(6,63),(7,60),(8,57),(9,127),(10,124),(11,121),(12,126),(13,123),(14,128),(15,125),(16,122),(17,70),(18,67),(19,72),(20,69),(21,66),(22,71),(23,68),(24,65),(25,77),(26,74),(27,79),(28,76),(29,73),(30,78),(31,75),(32,80),(33,120),(34,117),(35,114),(36,119),(37,116),(38,113),(39,118),(40,115),(41,104),(42,101),(43,98),(44,103),(45,100),(46,97),(47,102),(48,99),(49,112),(50,109),(51,106),(52,111),(53,108),(54,105),(55,110),(56,107),(81,135),(82,132),(83,129),(84,134),(85,131),(86,136),(87,133),(88,130),(89,137),(90,142),(91,139),(92,144),(93,141),(94,138),(95,143),(96,140)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6H | 6I | ··· | 6P | 6Q | ··· | 6X | 8A | 8B | 12A | ··· | 12P | 12Q | ··· | 12AN | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C8.C22 | C3×C8.C22 |
| kernel | C32×C8.C22 | C32×M4(2) | C32×SD16 | C32×Q16 | Q8×C3×C6 | C32×C4○D4 | C3×C8.C22 | C3×M4(2) | C3×SD16 | C3×Q16 | C6×Q8 | C3×C4○D4 | C3×C12 | C62 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 8 | 16 | 16 | 8 | 8 | 1 | 1 | 8 | 8 | 1 | 8 |
Matrix representation of C32×C8.C22 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 17 | 2 | 0 | 0 | 0 | 0 |
| 1 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 50 | 5 | 28 |
| 0 | 0 | 68 | 0 | 23 | 45 |
| 0 | 0 | 22 | 0 | 45 | 50 |
| 0 | 0 | 51 | 51 | 23 | 23 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 56 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 71 | 72 | 72 | 72 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,1,0,0,0,0,2,56,0,0,0,0,0,0,5,68,22,51,0,0,50,0,0,51,0,0,5,23,45,23,0,0,28,45,50,23],[1,56,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,71,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1,72,0] >;
C32×C8.C22 in GAP, Magma, Sage, TeX
C_3^2\times C_8.C_2^2
% in TeX
G:=Group("C3^2xC8.C2^2"); // GroupNames label
G:=SmallGroup(288,834);
// by ID
G=gap.SmallGroup(288,834);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,1016,3110,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations