direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×2- (1+4), D4.4C62, Q8.7C62, C62.158C23, (C6×Q8)⋊16C6, (C2×C4).8C62, C4.10(C2×C62), (C3×C6).72C24, C6.25(C23×C6), C12.64(C22×C6), C22.3(C2×C62), C2.5(C22×C62), (C3×C12).193C23, (C6×C12).277C22, (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
Subgroups: 468 in 438 conjugacy classes, 408 normal (6 characteristic)
C1, C2, C2 [×5], C3 [×4], C4 [×10], C22 [×5], C6 [×4], C6 [×20], C2×C4 [×15], D4 [×10], Q8 [×10], C32, C12 [×40], C2×C6 [×20], C2×Q8 [×5], C4○D4 [×10], C3×C6, C3×C6 [×5], C2×C12 [×60], C3×D4 [×40], C3×Q8 [×40], 2- (1+4), C3×C12 [×10], C62 [×5], C6×Q8 [×20], C3×C4○D4 [×40], C6×C12 [×15], D4×C32 [×10], Q8×C32 [×10], C3×2- (1+4) [×4], Q8×C3×C6 [×5], C32×C4○D4 [×10], C32×2- (1+4)
Quotients:
C1, C2 [×15], C3 [×4], C22 [×35], C6 [×60], C23 [×15], C32, C2×C6 [×140], C24, C3×C6 [×15], C22×C6 [×60], 2- (1+4), C62 [×35], C23×C6 [×4], C2×C62 [×15], C3×2- (1+4) [×4], C22×C62, C32×2- (1+4)
Generators and relations
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 >
►On 144 points
Generators in S144
(1 45 41)(2 46 42)(3 47 43)(4 48 44)(5 96 9)(6 93 10)(7 94 11)(8 95 12)(13 91 72)(14 92 69)(15 89 70)(16 90 71)(17 54 104)(18 55 101)(19 56 102)(20 53 103)(21 78 25)(22 79 26)(23 80 27)(24 77 28)(29 111 33)(30 112 34)(31 109 35)(32 110 36)(37 49 137)(38 50 138)(39 51 139)(40 52 140)(57 65 61)(58 66 62)(59 67 63)(60 68 64)(73 133 129)(74 134 130)(75 135 131)(76 136 132)(81 141 85)(82 142 86)(83 143 87)(84 144 88)(97 115 107)(98 116 108)(99 113 105)(100 114 106)(117 125 121)(118 126 122)(119 127 123)(120 128 124)
(1 73 24)(2 74 21)(3 75 22)(4 76 23)(5 114 64)(6 115 61)(7 116 62)(8 113 63)(9 100 68)(10 97 65)(11 98 66)(12 99 67)(13 104 118)(14 101 119)(15 102 120)(16 103 117)(17 126 91)(18 127 92)(19 128 89)(20 125 90)(25 42 130)(26 43 131)(27 44 132)(28 41 129)(29 138 82)(30 139 83)(31 140 84)(32 137 81)(33 50 86)(34 51 87)(35 52 88)(36 49 85)(37 141 110)(38 142 111)(39 143 112)(40 144 109)(45 133 77)(46 134 78)(47 135 79)(48 136 80)(53 121 71)(54 122 72)(55 123 69)(56 124 70)(57 93 107)(58 94 108)(59 95 105)(60 96 106)
(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 72 7 70)(6 69 8 71)(9 91 11 89)(10 92 12 90)(13 94 15 96)(14 95 16 93)(17 98 19 100)(18 99 20 97)(21 84 23 82)(22 81 24 83)(25 88 27 86)(26 85 28 87)(33 42 35 44)(34 43 36 41)(37 133 39 135)(38 134 40 136)(45 112 47 110)(46 109 48 111)(49 129 51 131)(50 130 52 132)(53 115 55 113)(54 116 56 114)(57 119 59 117)(58 120 60 118)(61 123 63 121)(62 124 64 122)(65 127 67 125)(66 128 68 126)(73 139 75 137)(74 140 76 138)(77 143 79 141)(78 144 80 142)(101 105 103 107)(102 106 104 108)
(1 117 3 119)(2 118 4 120)(5 50 7 52)(6 51 8 49)(9 38 11 40)(10 39 12 37)(13 76 15 74)(14 73 16 75)(17 80 19 78)(18 77 20 79)(21 104 23 102)(22 101 24 103)(25 54 27 56)(26 55 28 53)(29 58 31 60)(30 59 32 57)(33 62 35 64)(34 63 36 61)(41 121 43 123)(42 122 44 124)(45 125 47 127)(46 126 48 128)(65 112 67 110)(66 109 68 111)(69 129 71 131)(70 130 72 132)(81 107 83 105)(82 108 84 106)(85 115 87 113)(86 116 88 114)(89 134 91 136)(90 135 92 133)(93 139 95 137)(94 140 96 138)(97 143 99 141)(98 144 100 142)
G:=sub<Sym(144)| (1,45,41)(2,46,42)(3,47,43)(4,48,44)(5,96,9)(6,93,10)(7,94,11)(8,95,12)(13,91,72)(14,92,69)(15,89,70)(16,90,71)(17,54,104)(18,55,101)(19,56,102)(20,53,103)(21,78,25)(22,79,26)(23,80,27)(24,77,28)(29,111,33)(30,112,34)(31,109,35)(32,110,36)(37,49,137)(38,50,138)(39,51,139)(40,52,140)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(73,133,129)(74,134,130)(75,135,131)(76,136,132)(81,141,85)(82,142,86)(83,143,87)(84,144,88)(97,115,107)(98,116,108)(99,113,105)(100,114,106)(117,125,121)(118,126,122)(119,127,123)(120,128,124), (1,73,24)(2,74,21)(3,75,22)(4,76,23)(5,114,64)(6,115,61)(7,116,62)(8,113,63)(9,100,68)(10,97,65)(11,98,66)(12,99,67)(13,104,118)(14,101,119)(15,102,120)(16,103,117)(17,126,91)(18,127,92)(19,128,89)(20,125,90)(25,42,130)(26,43,131)(27,44,132)(28,41,129)(29,138,82)(30,139,83)(31,140,84)(32,137,81)(33,50,86)(34,51,87)(35,52,88)(36,49,85)(37,141,110)(38,142,111)(39,143,112)(40,144,109)(45,133,77)(46,134,78)(47,135,79)(48,136,80)(53,121,71)(54,122,72)(55,123,69)(56,124,70)(57,93,107)(58,94,108)(59,95,105)(60,96,106), (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)(5,72,7,70)(6,69,8,71)(9,91,11,89)(10,92,12,90)(13,94,15,96)(14,95,16,93)(17,98,19,100)(18,99,20,97)(21,84,23,82)(22,81,24,83)(25,88,27,86)(26,85,28,87)(33,42,35,44)(34,43,36,41)(37,133,39,135)(38,134,40,136)(45,112,47,110)(46,109,48,111)(49,129,51,131)(50,130,52,132)(53,115,55,113)(54,116,56,114)(57,119,59,117)(58,120,60,118)(61,123,63,121)(62,124,64,122)(65,127,67,125)(66,128,68,126)(73,139,75,137)(74,140,76,138)(77,143,79,141)(78,144,80,142)(101,105,103,107)(102,106,104,108), (1,117,3,119)(2,118,4,120)(5,50,7,52)(6,51,8,49)(9,38,11,40)(10,39,12,37)(13,76,15,74)(14,73,16,75)(17,80,19,78)(18,77,20,79)(21,104,23,102)(22,101,24,103)(25,54,27,56)(26,55,28,53)(29,58,31,60)(30,59,32,57)(33,62,35,64)(34,63,36,61)(41,121,43,123)(42,122,44,124)(45,125,47,127)(46,126,48,128)(65,112,67,110)(66,109,68,111)(69,129,71,131)(70,130,72,132)(81,107,83,105)(82,108,84,106)(85,115,87,113)(86,116,88,114)(89,134,91,136)(90,135,92,133)(93,139,95,137)(94,140,96,138)(97,143,99,141)(98,144,100,142)>;
G:=Group( (1,45,41)(2,46,42)(3,47,43)(4,48,44)(5,96,9)(6,93,10)(7,94,11)(8,95,12)(13,91,72)(14,92,69)(15,89,70)(16,90,71)(17,54,104)(18,55,101)(19,56,102)(20,53,103)(21,78,25)(22,79,26)(23,80,27)(24,77,28)(29,111,33)(30,112,34)(31,109,35)(32,110,36)(37,49,137)(38,50,138)(39,51,139)(40,52,140)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(73,133,129)(74,134,130)(75,135,131)(76,136,132)(81,141,85)(82,142,86)(83,143,87)(84,144,88)(97,115,107)(98,116,108)(99,113,105)(100,114,106)(117,125,121)(118,126,122)(119,127,123)(120,128,124), (1,73,24)(2,74,21)(3,75,22)(4,76,23)(5,114,64)(6,115,61)(7,116,62)(8,113,63)(9,100,68)(10,97,65)(11,98,66)(12,99,67)(13,104,118)(14,101,119)(15,102,120)(16,103,117)(17,126,91)(18,127,92)(19,128,89)(20,125,90)(25,42,130)(26,43,131)(27,44,132)(28,41,129)(29,138,82)(30,139,83)(31,140,84)(32,137,81)(33,50,86)(34,51,87)(35,52,88)(36,49,85)(37,141,110)(38,142,111)(39,143,112)(40,144,109)(45,133,77)(46,134,78)(47,135,79)(48,136,80)(53,121,71)(54,122,72)(55,123,69)(56,124,70)(57,93,107)(58,94,108)(59,95,105)(60,96,106), (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)(5,72,7,70)(6,69,8,71)(9,91,11,89)(10,92,12,90)(13,94,15,96)(14,95,16,93)(17,98,19,100)(18,99,20,97)(21,84,23,82)(22,81,24,83)(25,88,27,86)(26,85,28,87)(33,42,35,44)(34,43,36,41)(37,133,39,135)(38,134,40,136)(45,112,47,110)(46,109,48,111)(49,129,51,131)(50,130,52,132)(53,115,55,113)(54,116,56,114)(57,119,59,117)(58,120,60,118)(61,123,63,121)(62,124,64,122)(65,127,67,125)(66,128,68,126)(73,139,75,137)(74,140,76,138)(77,143,79,141)(78,144,80,142)(101,105,103,107)(102,106,104,108), (1,117,3,119)(2,118,4,120)(5,50,7,52)(6,51,8,49)(9,38,11,40)(10,39,12,37)(13,76,15,74)(14,73,16,75)(17,80,19,78)(18,77,20,79)(21,104,23,102)(22,101,24,103)(25,54,27,56)(26,55,28,53)(29,58,31,60)(30,59,32,57)(33,62,35,64)(34,63,36,61)(41,121,43,123)(42,122,44,124)(45,125,47,127)(46,126,48,128)(65,112,67,110)(66,109,68,111)(69,129,71,131)(70,130,72,132)(81,107,83,105)(82,108,84,106)(85,115,87,113)(86,116,88,114)(89,134,91,136)(90,135,92,133)(93,139,95,137)(94,140,96,138)(97,143,99,141)(98,144,100,142) );
G=PermutationGroup([(1,45,41),(2,46,42),(3,47,43),(4,48,44),(5,96,9),(6,93,10),(7,94,11),(8,95,12),(13,91,72),(14,92,69),(15,89,70),(16,90,71),(17,54,104),(18,55,101),(19,56,102),(20,53,103),(21,78,25),(22,79,26),(23,80,27),(24,77,28),(29,111,33),(30,112,34),(31,109,35),(32,110,36),(37,49,137),(38,50,138),(39,51,139),(40,52,140),(57,65,61),(58,66,62),(59,67,63),(60,68,64),(73,133,129),(74,134,130),(75,135,131),(76,136,132),(81,141,85),(82,142,86),(83,143,87),(84,144,88),(97,115,107),(98,116,108),(99,113,105),(100,114,106),(117,125,121),(118,126,122),(119,127,123),(120,128,124)], [(1,73,24),(2,74,21),(3,75,22),(4,76,23),(5,114,64),(6,115,61),(7,116,62),(8,113,63),(9,100,68),(10,97,65),(11,98,66),(12,99,67),(13,104,118),(14,101,119),(15,102,120),(16,103,117),(17,126,91),(18,127,92),(19,128,89),(20,125,90),(25,42,130),(26,43,131),(27,44,132),(28,41,129),(29,138,82),(30,139,83),(31,140,84),(32,137,81),(33,50,86),(34,51,87),(35,52,88),(36,49,85),(37,141,110),(38,142,111),(39,143,112),(40,144,109),(45,133,77),(46,134,78),(47,135,79),(48,136,80),(53,121,71),(54,122,72),(55,123,69),(56,124,70),(57,93,107),(58,94,108),(59,95,105),(60,96,106)], [(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),(5,72,7,70),(6,69,8,71),(9,91,11,89),(10,92,12,90),(13,94,15,96),(14,95,16,93),(17,98,19,100),(18,99,20,97),(21,84,23,82),(22,81,24,83),(25,88,27,86),(26,85,28,87),(33,42,35,44),(34,43,36,41),(37,133,39,135),(38,134,40,136),(45,112,47,110),(46,109,48,111),(49,129,51,131),(50,130,52,132),(53,115,55,113),(54,116,56,114),(57,119,59,117),(58,120,60,118),(61,123,63,121),(62,124,64,122),(65,127,67,125),(66,128,68,126),(73,139,75,137),(74,140,76,138),(77,143,79,141),(78,144,80,142),(101,105,103,107),(102,106,104,108)], [(1,117,3,119),(2,118,4,120),(5,50,7,52),(6,51,8,49),(9,38,11,40),(10,39,12,37),(13,76,15,74),(14,73,16,75),(17,80,19,78),(18,77,20,79),(21,104,23,102),(22,101,24,103),(25,54,27,56),(26,55,28,53),(29,58,31,60),(30,59,32,57),(33,62,35,64),(34,63,36,61),(41,121,43,123),(42,122,44,124),(45,125,47,127),(46,126,48,128),(65,112,67,110),(66,109,68,111),(69,129,71,131),(70,130,72,132),(81,107,83,105),(82,108,84,106),(85,115,87,113),(86,116,88,114),(89,134,91,136),(90,135,92,133),(93,139,95,137),(94,140,96,138),(97,143,99,141),(98,144,100,142)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀