direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×2- 1+4, D4.4C62, Q8.7C62, C62.158C23, (C6×Q8)⋊16C6, (C2×C4).8C62, C6.25(C23×C6), (C3×C6).72C24, C4.10(C2×C62), C12.64(C22×C6), C2.5(C22×C62), C22.3(C2×C62), (C6×C12).277C22, (C3×C12).193C23, (D4×C32).35C22, (Q8×C32).38C22, (Q8×C3×C6)⋊19C2, C4○D4⋊6(C3×C6), (C2×Q8)⋊7(C3×C6), (C3×C4○D4)⋊13C6, (C2×C12).78(C2×C6), (C3×D4).23(C2×C6), (C3×Q8).36(C2×C6), (C32×C4○D4)⋊14C2, (C2×C6).13(C22×C6), SmallGroup(288,1023)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×2- 1+4
G = < a,b,c,d,e,f | a3=b3=c4=d2=1, e2=f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >
Subgroups: 468 in 438 conjugacy classes, 408 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C2×Q8, C4○D4, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, 2- 1+4, C3×C12, C62, C6×Q8, C3×C4○D4, C6×C12, D4×C32, Q8×C32, C3×2- 1+4, Q8×C3×C6, C32×C4○D4, C32×2- 1+4
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C24, C3×C6, C22×C6, 2- 1+4, C62, C23×C6, C2×C62, C3×2- 1+4, C22×C62, C32×2- 1+4
►On 144 points
Generators in S144
(1 47 43)(2 48 44)(3 45 41)(4 46 42)(5 94 9)(6 95 10)(7 96 11)(8 93 12)(13 89 70)(14 90 71)(15 91 72)(16 92 69)(17 56 102)(18 53 103)(19 54 104)(20 55 101)(21 80 25)(22 77 26)(23 78 27)(24 79 28)(29 111 33)(30 112 34)(31 109 35)(32 110 36)(37 49 139)(38 50 140)(39 51 137)(40 52 138)(57 65 61)(58 66 62)(59 67 63)(60 68 64)(73 135 131)(74 136 132)(75 133 129)(76 134 130)(81 143 85)(82 144 86)(83 141 87)(84 142 88)(97 113 105)(98 114 106)(99 115 107)(100 116 108)(117 125 121)(118 126 122)(119 127 123)(120 128 124)
(1 75 24)(2 76 21)(3 73 22)(4 74 23)(5 114 62)(6 115 63)(7 116 64)(8 113 61)(9 98 66)(10 99 67)(11 100 68)(12 97 65)(13 102 118)(14 103 119)(15 104 120)(16 101 117)(17 126 89)(18 127 90)(19 128 91)(20 125 92)(25 44 130)(26 41 131)(27 42 132)(28 43 129)(29 138 84)(30 139 81)(31 140 82)(32 137 83)(33 52 88)(34 49 85)(35 50 86)(36 51 87)(37 143 112)(38 144 109)(39 141 110)(40 142 111)(45 135 77)(46 136 78)(47 133 79)(48 134 80)(53 123 71)(54 124 72)(55 121 69)(56 122 70)(57 93 105)(58 94 106)(59 95 107)(60 96 108)
(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)
(2 4)(5 7)(9 11)(13 15)(17 19)(21 23)(25 27)(29 31)(33 35)(38 40)(42 44)(46 48)(50 52)(54 56)(58 60)(62 64)(66 68)(70 72)(74 76)(78 80)(82 84)(86 88)(89 91)(94 96)(98 100)(102 104)(106 108)(109 111)(114 116)(118 120)(122 124)(126 128)(130 132)(134 136)(138 140)(142 144)
(1 30 3 32)(2 31 4 29)(5 70 7 72)(6 71 8 69)(9 89 11 91)(10 90 12 92)(13 96 15 94)(14 93 16 95)(17 100 19 98)(18 97 20 99)(21 82 23 84)(22 83 24 81)(25 86 27 88)(26 87 28 85)(33 44 35 42)(34 41 36 43)(37 135 39 133)(38 136 40 134)(45 110 47 112)(46 111 48 109)(49 131 51 129)(50 132 52 130)(53 113 55 115)(54 114 56 116)(57 117 59 119)(58 118 60 120)(61 121 63 123)(62 122 64 124)(65 125 67 127)(66 126 68 128)(73 137 75 139)(74 138 76 140)(77 141 79 143)(78 142 80 144)(101 107 103 105)(102 108 104 106)
(1 117 3 119)(2 118 4 120)(5 52 7 50)(6 49 8 51)(9 40 11 38)(10 37 12 39)(13 74 15 76)(14 75 16 73)(17 78 19 80)(18 79 20 77)(21 102 23 104)(22 103 24 101)(25 56 27 54)(26 53 28 55)(29 60 31 58)(30 57 32 59)(33 64 35 62)(34 61 36 63)(41 123 43 121)(42 124 44 122)(45 127 47 125)(46 128 48 126)(65 110 67 112)(66 111 68 109)(69 131 71 129)(70 132 72 130)(81 105 83 107)(82 106 84 108)(85 113 87 115)(86 114 88 116)(89 136 91 134)(90 133 92 135)(93 137 95 139)(94 138 96 140)(97 141 99 143)(98 142 100 144)
G:=sub<Sym(144)| (1,47,43)(2,48,44)(3,45,41)(4,46,42)(5,94,9)(6,95,10)(7,96,11)(8,93,12)(13,89,70)(14,90,71)(15,91,72)(16,92,69)(17,56,102)(18,53,103)(19,54,104)(20,55,101)(21,80,25)(22,77,26)(23,78,27)(24,79,28)(29,111,33)(30,112,34)(31,109,35)(32,110,36)(37,49,139)(38,50,140)(39,51,137)(40,52,138)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(73,135,131)(74,136,132)(75,133,129)(76,134,130)(81,143,85)(82,144,86)(83,141,87)(84,142,88)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(117,125,121)(118,126,122)(119,127,123)(120,128,124), (1,75,24)(2,76,21)(3,73,22)(4,74,23)(5,114,62)(6,115,63)(7,116,64)(8,113,61)(9,98,66)(10,99,67)(11,100,68)(12,97,65)(13,102,118)(14,103,119)(15,104,120)(16,101,117)(17,126,89)(18,127,90)(19,128,91)(20,125,92)(25,44,130)(26,41,131)(27,42,132)(28,43,129)(29,138,84)(30,139,81)(31,140,82)(32,137,83)(33,52,88)(34,49,85)(35,50,86)(36,51,87)(37,143,112)(38,144,109)(39,141,110)(40,142,111)(45,135,77)(46,136,78)(47,133,79)(48,134,80)(53,123,71)(54,124,72)(55,121,69)(56,122,70)(57,93,105)(58,94,106)(59,95,107)(60,96,108), (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), (2,4)(5,7)(9,11)(13,15)(17,19)(21,23)(25,27)(29,31)(33,35)(38,40)(42,44)(46,48)(50,52)(54,56)(58,60)(62,64)(66,68)(70,72)(74,76)(78,80)(82,84)(86,88)(89,91)(94,96)(98,100)(102,104)(106,108)(109,111)(114,116)(118,120)(122,124)(126,128)(130,132)(134,136)(138,140)(142,144), (1,30,3,32)(2,31,4,29)(5,70,7,72)(6,71,8,69)(9,89,11,91)(10,90,12,92)(13,96,15,94)(14,93,16,95)(17,100,19,98)(18,97,20,99)(21,82,23,84)(22,83,24,81)(25,86,27,88)(26,87,28,85)(33,44,35,42)(34,41,36,43)(37,135,39,133)(38,136,40,134)(45,110,47,112)(46,111,48,109)(49,131,51,129)(50,132,52,130)(53,113,55,115)(54,114,56,116)(57,117,59,119)(58,118,60,120)(61,121,63,123)(62,122,64,124)(65,125,67,127)(66,126,68,128)(73,137,75,139)(74,138,76,140)(77,141,79,143)(78,142,80,144)(101,107,103,105)(102,108,104,106), (1,117,3,119)(2,118,4,120)(5,52,7,50)(6,49,8,51)(9,40,11,38)(10,37,12,39)(13,74,15,76)(14,75,16,73)(17,78,19,80)(18,79,20,77)(21,102,23,104)(22,103,24,101)(25,56,27,54)(26,53,28,55)(29,60,31,58)(30,57,32,59)(33,64,35,62)(34,61,36,63)(41,123,43,121)(42,124,44,122)(45,127,47,125)(46,128,48,126)(65,110,67,112)(66,111,68,109)(69,131,71,129)(70,132,72,130)(81,105,83,107)(82,106,84,108)(85,113,87,115)(86,114,88,116)(89,136,91,134)(90,133,92,135)(93,137,95,139)(94,138,96,140)(97,141,99,143)(98,142,100,144)>;
G:=Group( (1,47,43)(2,48,44)(3,45,41)(4,46,42)(5,94,9)(6,95,10)(7,96,11)(8,93,12)(13,89,70)(14,90,71)(15,91,72)(16,92,69)(17,56,102)(18,53,103)(19,54,104)(20,55,101)(21,80,25)(22,77,26)(23,78,27)(24,79,28)(29,111,33)(30,112,34)(31,109,35)(32,110,36)(37,49,139)(38,50,140)(39,51,137)(40,52,138)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(73,135,131)(74,136,132)(75,133,129)(76,134,130)(81,143,85)(82,144,86)(83,141,87)(84,142,88)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(117,125,121)(118,126,122)(119,127,123)(120,128,124), (1,75,24)(2,76,21)(3,73,22)(4,74,23)(5,114,62)(6,115,63)(7,116,64)(8,113,61)(9,98,66)(10,99,67)(11,100,68)(12,97,65)(13,102,118)(14,103,119)(15,104,120)(16,101,117)(17,126,89)(18,127,90)(19,128,91)(20,125,92)(25,44,130)(26,41,131)(27,42,132)(28,43,129)(29,138,84)(30,139,81)(31,140,82)(32,137,83)(33,52,88)(34,49,85)(35,50,86)(36,51,87)(37,143,112)(38,144,109)(39,141,110)(40,142,111)(45,135,77)(46,136,78)(47,133,79)(48,134,80)(53,123,71)(54,124,72)(55,121,69)(56,122,70)(57,93,105)(58,94,106)(59,95,107)(60,96,108), (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), (2,4)(5,7)(9,11)(13,15)(17,19)(21,23)(25,27)(29,31)(33,35)(38,40)(42,44)(46,48)(50,52)(54,56)(58,60)(62,64)(66,68)(70,72)(74,76)(78,80)(82,84)(86,88)(89,91)(94,96)(98,100)(102,104)(106,108)(109,111)(114,116)(118,120)(122,124)(126,128)(130,132)(134,136)(138,140)(142,144), (1,30,3,32)(2,31,4,29)(5,70,7,72)(6,71,8,69)(9,89,11,91)(10,90,12,92)(13,96,15,94)(14,93,16,95)(17,100,19,98)(18,97,20,99)(21,82,23,84)(22,83,24,81)(25,86,27,88)(26,87,28,85)(33,44,35,42)(34,41,36,43)(37,135,39,133)(38,136,40,134)(45,110,47,112)(46,111,48,109)(49,131,51,129)(50,132,52,130)(53,113,55,115)(54,114,56,116)(57,117,59,119)(58,118,60,120)(61,121,63,123)(62,122,64,124)(65,125,67,127)(66,126,68,128)(73,137,75,139)(74,138,76,140)(77,141,79,143)(78,142,80,144)(101,107,103,105)(102,108,104,106), (1,117,3,119)(2,118,4,120)(5,52,7,50)(6,49,8,51)(9,40,11,38)(10,37,12,39)(13,74,15,76)(14,75,16,73)(17,78,19,80)(18,79,20,77)(21,102,23,104)(22,103,24,101)(25,56,27,54)(26,53,28,55)(29,60,31,58)(30,57,32,59)(33,64,35,62)(34,61,36,63)(41,123,43,121)(42,124,44,122)(45,127,47,125)(46,128,48,126)(65,110,67,112)(66,111,68,109)(69,131,71,129)(70,132,72,130)(81,105,83,107)(82,106,84,108)(85,113,87,115)(86,114,88,116)(89,136,91,134)(90,133,92,135)(93,137,95,139)(94,138,96,140)(97,141,99,143)(98,142,100,144) );
G=PermutationGroup([[(1,47,43),(2,48,44),(3,45,41),(4,46,42),(5,94,9),(6,95,10),(7,96,11),(8,93,12),(13,89,70),(14,90,71),(15,91,72),(16,92,69),(17,56,102),(18,53,103),(19,54,104),(20,55,101),(21,80,25),(22,77,26),(23,78,27),(24,79,28),(29,111,33),(30,112,34),(31,109,35),(32,110,36),(37,49,139),(38,50,140),(39,51,137),(40,52,138),(57,65,61),(58,66,62),(59,67,63),(60,68,64),(73,135,131),(74,136,132),(75,133,129),(76,134,130),(81,143,85),(82,144,86),(83,141,87),(84,142,88),(97,113,105),(98,114,106),(99,115,107),(100,116,108),(117,125,121),(118,126,122),(119,127,123),(120,128,124)], [(1,75,24),(2,76,21),(3,73,22),(4,74,23),(5,114,62),(6,115,63),(7,116,64),(8,113,61),(9,98,66),(10,99,67),(11,100,68),(12,97,65),(13,102,118),(14,103,119),(15,104,120),(16,101,117),(17,126,89),(18,127,90),(19,128,91),(20,125,92),(25,44,130),(26,41,131),(27,42,132),(28,43,129),(29,138,84),(30,139,81),(31,140,82),(32,137,83),(33,52,88),(34,49,85),(35,50,86),(36,51,87),(37,143,112),(38,144,109),(39,141,110),(40,142,111),(45,135,77),(46,136,78),(47,133,79),(48,134,80),(53,123,71),(54,124,72),(55,121,69),(56,122,70),(57,93,105),(58,94,106),(59,95,107),(60,96,108)], [(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)], [(2,4),(5,7),(9,11),(13,15),(17,19),(21,23),(25,27),(29,31),(33,35),(38,40),(42,44),(46,48),(50,52),(54,56),(58,60),(62,64),(66,68),(70,72),(74,76),(78,80),(82,84),(86,88),(89,91),(94,96),(98,100),(102,104),(106,108),(109,111),(114,116),(118,120),(122,124),(126,128),(130,132),(134,136),(138,140),(142,144)], [(1,30,3,32),(2,31,4,29),(5,70,7,72),(6,71,8,69),(9,89,11,91),(10,90,12,92),(13,96,15,94),(14,93,16,95),(17,100,19,98),(18,97,20,99),(21,82,23,84),(22,83,24,81),(25,86,27,88),(26,87,28,85),(33,44,35,42),(34,41,36,43),(37,135,39,133),(38,136,40,134),(45,110,47,112),(46,111,48,109),(49,131,51,129),(50,132,52,130),(53,113,55,115),(54,114,56,116),(57,117,59,119),(58,118,60,120),(61,121,63,123),(62,122,64,124),(65,125,67,127),(66,126,68,128),(73,137,75,139),(74,138,76,140),(77,141,79,143),(78,142,80,144),(101,107,103,105),(102,108,104,106)], [(1,117,3,119),(2,118,4,120),(5,52,7,50),(6,49,8,51),(9,40,11,38),(10,37,12,39),(13,74,15,76),(14,75,16,73),(17,78,19,80),(18,79,20,77),(21,102,23,104),(22,103,24,101),(25,56,27,54),(26,53,28,55),(29,60,31,58),(30,57,32,59),(33,64,35,62),(34,61,36,63),(41,123,43,121),(42,124,44,122),(45,127,47,125),(46,128,48,126),(65,110,67,112),(66,111,68,109),(69,131,71,129),(70,132,72,130),(81,105,83,107),(82,106,84,108),(85,113,87,115),(86,114,88,116),(89,136,91,134),(90,133,92,135),(93,137,95,139),(94,138,96,140),(97,141,99,143),(98,142,100,144)]])
153 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 3A | ··· | 3H | 4A | ··· | 4J | 6A | ··· | 6H | 6I | ··· | 6AV | 12A | ··· | 12CB |
| order | 1 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
153 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | 2- 1+4 | C3×2- 1+4 |
| kernel | C32×2- 1+4 | Q8×C3×C6 | C32×C4○D4 | C3×2- 1+4 | C6×Q8 | C3×C4○D4 | C32 | C3 |
| # reps | 1 | 5 | 10 | 8 | 40 | 80 | 1 | 8 |
Matrix representation of C32×2- 1+4 ►in GL5(𝔽13)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 5 |
| 0 | 0 | 0 | 11 | 3 |
| 0 | 10 | 5 | 0 | 0 |
| 0 | 11 | 3 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 7 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 1 | 0 | 0 |
| 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 11 | 12 |
| 0 | 0 | 0 | 5 | 2 |
G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,10,11,0,0,0,5,3,0,10,11,0,0,0,5,3,0,0],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,5,6,0,0,0,0,8,0,0,0,0,0,8,7,0,0,0,0,5],[1,0,0,0,0,0,2,8,0,0,0,1,11,0,0,0,0,0,11,5,0,0,0,12,2] >;
C32×2- 1+4 in GAP, Magma, Sage, TeX
C_3^2\times 2_-^{1+4} % in TeX
G:=Group("C3^2xES-(2,2)"); // GroupNames label
G:=SmallGroup(288,1023);
// by ID
G=gap.SmallGroup(288,1023);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,2045,1016,1563,772,4259]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=d^2=1,e^2=f^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c^2*e>;
// generators/relations