direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C42⋊2C2, C122⋊4C2, C23.4C62, C62.295C23, (C4×C12)⋊9C6, C42⋊5(C3×C6), (C2×C4).13C62, (C2×C62).4C22, (C6×C12).273C22, C22.16(C2×C62), C4⋊C4⋊5(C3×C6), (C3×C4⋊C4)⋊14C6, C6.56(C3×C4○D4), (C32×C4⋊C4)⋊23C2, (C2×C12).99(C2×C6), C22⋊C4.2(C3×C6), C2.9(C32×C4○D4), (C3×C22⋊C4).12C6, (C22×C6).14(C2×C6), (C3×C6).173(C4○D4), (C2×C6).101(C22×C6), (C32×C22⋊C4).8C2, SmallGroup(288,823)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C42⋊2C2
G = < a,b,c,d,e | a3=b3=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, ede=c2d-1 >
Subgroups: 252 in 180 conjugacy classes, 120 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C3×C6, C3×C6, C2×C12, C22×C6, C42⋊2C2, C3×C12, C62, C62, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C6×C12, C2×C62, C3×C42⋊2C2, C122, C32×C22⋊C4, C32×C4⋊C4, C32×C42⋊2C2
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C4○D4, C3×C6, C22×C6, C42⋊2C2, C62, C3×C4○D4, C2×C62, C3×C42⋊2C2, C32×C4○D4, C32×C42⋊2C2
►On 144 points
Generators in S144
(1 127 119)(2 128 120)(3 125 117)(4 126 118)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(9 89 81)(10 90 82)(11 91 83)(12 92 84)(17 85 77)(18 86 78)(19 87 79)(20 88 80)(25 105 33)(26 106 34)(27 107 35)(28 108 36)(29 97 37)(30 98 38)(31 99 39)(32 100 40)(41 121 49)(42 122 50)(43 123 51)(44 124 52)(45 113 53)(46 114 54)(47 115 55)(48 116 56)(57 73 65)(58 74 66)(59 75 67)(60 76 68)(61 141 133)(62 142 134)(63 143 135)(64 144 136)(69 137 129)(70 138 130)(71 139 131)(72 140 132)(93 109 101)(94 110 102)(95 111 103)(96 112 104)
(1 115 43)(2 116 44)(3 113 41)(4 114 42)(5 9 77)(6 10 78)(7 11 79)(8 12 80)(13 81 85)(14 82 86)(15 83 87)(16 84 88)(17 21 89)(18 22 90)(19 23 91)(20 24 92)(25 93 97)(26 94 98)(27 95 99)(28 96 100)(29 33 101)(30 34 102)(31 35 103)(32 36 104)(37 105 109)(38 106 110)(39 107 111)(40 108 112)(45 49 117)(46 50 118)(47 51 119)(48 52 120)(53 121 125)(54 122 126)(55 123 127)(56 124 128)(57 61 129)(58 62 130)(59 63 131)(60 64 132)(65 133 137)(66 134 138)(67 135 139)(68 136 140)(69 73 141)(70 74 142)(71 75 143)(72 76 144)
(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 93 5 59)(2 94 6 60)(3 95 7 57)(4 96 8 58)(9 63 115 97)(10 64 116 98)(11 61 113 99)(12 62 114 100)(13 67 119 101)(14 68 120 102)(15 65 117 103)(16 66 118 104)(17 71 123 105)(18 72 124 106)(19 69 121 107)(20 70 122 108)(21 75 127 109)(22 76 128 110)(23 73 125 111)(24 74 126 112)(25 77 131 43)(26 78 132 44)(27 79 129 41)(28 80 130 42)(29 81 135 47)(30 82 136 48)(31 83 133 45)(32 84 134 46)(33 85 139 51)(34 86 140 52)(35 87 137 49)(36 88 138 50)(37 89 143 55)(38 90 144 56)(39 91 141 53)(40 92 142 54)
(2 6)(4 8)(10 116)(12 114)(14 120)(16 118)(18 124)(20 122)(22 128)(24 126)(25 129)(26 28)(27 131)(29 133)(30 32)(31 135)(33 137)(34 36)(35 139)(37 141)(38 40)(39 143)(42 80)(44 78)(46 84)(48 82)(50 88)(52 86)(54 92)(56 90)(57 93)(58 60)(59 95)(61 97)(62 64)(63 99)(65 101)(66 68)(67 103)(69 105)(70 72)(71 107)(73 109)(74 76)(75 111)(94 96)(98 100)(102 104)(106 108)(110 112)(130 132)(134 136)(138 140)(142 144)
G:=sub<Sym(144)| (1,127,119)(2,128,120)(3,125,117)(4,126,118)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(9,89,81)(10,90,82)(11,91,83)(12,92,84)(17,85,77)(18,86,78)(19,87,79)(20,88,80)(25,105,33)(26,106,34)(27,107,35)(28,108,36)(29,97,37)(30,98,38)(31,99,39)(32,100,40)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(93,109,101)(94,110,102)(95,111,103)(96,112,104), (1,115,43)(2,116,44)(3,113,41)(4,114,42)(5,9,77)(6,10,78)(7,11,79)(8,12,80)(13,81,85)(14,82,86)(15,83,87)(16,84,88)(17,21,89)(18,22,90)(19,23,91)(20,24,92)(25,93,97)(26,94,98)(27,95,99)(28,96,100)(29,33,101)(30,34,102)(31,35,103)(32,36,104)(37,105,109)(38,106,110)(39,107,111)(40,108,112)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,73,141)(70,74,142)(71,75,143)(72,76,144), (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), (1,93,5,59)(2,94,6,60)(3,95,7,57)(4,96,8,58)(9,63,115,97)(10,64,116,98)(11,61,113,99)(12,62,114,100)(13,67,119,101)(14,68,120,102)(15,65,117,103)(16,66,118,104)(17,71,123,105)(18,72,124,106)(19,69,121,107)(20,70,122,108)(21,75,127,109)(22,76,128,110)(23,73,125,111)(24,74,126,112)(25,77,131,43)(26,78,132,44)(27,79,129,41)(28,80,130,42)(29,81,135,47)(30,82,136,48)(31,83,133,45)(32,84,134,46)(33,85,139,51)(34,86,140,52)(35,87,137,49)(36,88,138,50)(37,89,143,55)(38,90,144,56)(39,91,141,53)(40,92,142,54), (2,6)(4,8)(10,116)(12,114)(14,120)(16,118)(18,124)(20,122)(22,128)(24,126)(25,129)(26,28)(27,131)(29,133)(30,32)(31,135)(33,137)(34,36)(35,139)(37,141)(38,40)(39,143)(42,80)(44,78)(46,84)(48,82)(50,88)(52,86)(54,92)(56,90)(57,93)(58,60)(59,95)(61,97)(62,64)(63,99)(65,101)(66,68)(67,103)(69,105)(70,72)(71,107)(73,109)(74,76)(75,111)(94,96)(98,100)(102,104)(106,108)(110,112)(130,132)(134,136)(138,140)(142,144)>;
G:=Group( (1,127,119)(2,128,120)(3,125,117)(4,126,118)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(9,89,81)(10,90,82)(11,91,83)(12,92,84)(17,85,77)(18,86,78)(19,87,79)(20,88,80)(25,105,33)(26,106,34)(27,107,35)(28,108,36)(29,97,37)(30,98,38)(31,99,39)(32,100,40)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(93,109,101)(94,110,102)(95,111,103)(96,112,104), (1,115,43)(2,116,44)(3,113,41)(4,114,42)(5,9,77)(6,10,78)(7,11,79)(8,12,80)(13,81,85)(14,82,86)(15,83,87)(16,84,88)(17,21,89)(18,22,90)(19,23,91)(20,24,92)(25,93,97)(26,94,98)(27,95,99)(28,96,100)(29,33,101)(30,34,102)(31,35,103)(32,36,104)(37,105,109)(38,106,110)(39,107,111)(40,108,112)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,73,141)(70,74,142)(71,75,143)(72,76,144), (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), (1,93,5,59)(2,94,6,60)(3,95,7,57)(4,96,8,58)(9,63,115,97)(10,64,116,98)(11,61,113,99)(12,62,114,100)(13,67,119,101)(14,68,120,102)(15,65,117,103)(16,66,118,104)(17,71,123,105)(18,72,124,106)(19,69,121,107)(20,70,122,108)(21,75,127,109)(22,76,128,110)(23,73,125,111)(24,74,126,112)(25,77,131,43)(26,78,132,44)(27,79,129,41)(28,80,130,42)(29,81,135,47)(30,82,136,48)(31,83,133,45)(32,84,134,46)(33,85,139,51)(34,86,140,52)(35,87,137,49)(36,88,138,50)(37,89,143,55)(38,90,144,56)(39,91,141,53)(40,92,142,54), (2,6)(4,8)(10,116)(12,114)(14,120)(16,118)(18,124)(20,122)(22,128)(24,126)(25,129)(26,28)(27,131)(29,133)(30,32)(31,135)(33,137)(34,36)(35,139)(37,141)(38,40)(39,143)(42,80)(44,78)(46,84)(48,82)(50,88)(52,86)(54,92)(56,90)(57,93)(58,60)(59,95)(61,97)(62,64)(63,99)(65,101)(66,68)(67,103)(69,105)(70,72)(71,107)(73,109)(74,76)(75,111)(94,96)(98,100)(102,104)(106,108)(110,112)(130,132)(134,136)(138,140)(142,144) );
G=PermutationGroup([[(1,127,119),(2,128,120),(3,125,117),(4,126,118),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(9,89,81),(10,90,82),(11,91,83),(12,92,84),(17,85,77),(18,86,78),(19,87,79),(20,88,80),(25,105,33),(26,106,34),(27,107,35),(28,108,36),(29,97,37),(30,98,38),(31,99,39),(32,100,40),(41,121,49),(42,122,50),(43,123,51),(44,124,52),(45,113,53),(46,114,54),(47,115,55),(48,116,56),(57,73,65),(58,74,66),(59,75,67),(60,76,68),(61,141,133),(62,142,134),(63,143,135),(64,144,136),(69,137,129),(70,138,130),(71,139,131),(72,140,132),(93,109,101),(94,110,102),(95,111,103),(96,112,104)], [(1,115,43),(2,116,44),(3,113,41),(4,114,42),(5,9,77),(6,10,78),(7,11,79),(8,12,80),(13,81,85),(14,82,86),(15,83,87),(16,84,88),(17,21,89),(18,22,90),(19,23,91),(20,24,92),(25,93,97),(26,94,98),(27,95,99),(28,96,100),(29,33,101),(30,34,102),(31,35,103),(32,36,104),(37,105,109),(38,106,110),(39,107,111),(40,108,112),(45,49,117),(46,50,118),(47,51,119),(48,52,120),(53,121,125),(54,122,126),(55,123,127),(56,124,128),(57,61,129),(58,62,130),(59,63,131),(60,64,132),(65,133,137),(66,134,138),(67,135,139),(68,136,140),(69,73,141),(70,74,142),(71,75,143),(72,76,144)], [(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)], [(1,93,5,59),(2,94,6,60),(3,95,7,57),(4,96,8,58),(9,63,115,97),(10,64,116,98),(11,61,113,99),(12,62,114,100),(13,67,119,101),(14,68,120,102),(15,65,117,103),(16,66,118,104),(17,71,123,105),(18,72,124,106),(19,69,121,107),(20,70,122,108),(21,75,127,109),(22,76,128,110),(23,73,125,111),(24,74,126,112),(25,77,131,43),(26,78,132,44),(27,79,129,41),(28,80,130,42),(29,81,135,47),(30,82,136,48),(31,83,133,45),(32,84,134,46),(33,85,139,51),(34,86,140,52),(35,87,137,49),(36,88,138,50),(37,89,143,55),(38,90,144,56),(39,91,141,53),(40,92,142,54)], [(2,6),(4,8),(10,116),(12,114),(14,120),(16,118),(18,124),(20,122),(22,128),(24,126),(25,129),(26,28),(27,131),(29,133),(30,32),(31,135),(33,137),(34,36),(35,139),(37,141),(38,40),(39,143),(42,80),(44,78),(46,84),(48,82),(50,88),(52,86),(54,92),(56,90),(57,93),(58,60),(59,95),(61,97),(62,64),(63,99),(65,101),(66,68),(67,103),(69,105),(70,72),(71,107),(73,109),(74,76),(75,111),(94,96),(98,100),(102,104),(106,108),(110,112),(130,132),(134,136),(138,140),(142,144)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | ··· | 3H | 4A | ··· | 4F | 4G | 4H | 4I | 6A | ··· | 6X | 6Y | ··· | 6AF | 12A | ··· | 12AV | 12AW | ··· | 12BT |
| order | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C4○D4 | C3×C4○D4 |
| kernel | C32×C42⋊2C2 | C122 | C32×C22⋊C4 | C32×C4⋊C4 | C3×C42⋊2C2 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C3×C6 | C6 |
| # reps | 1 | 1 | 3 | 3 | 8 | 8 | 24 | 24 | 6 | 48 |
Matrix representation of C32×C42⋊2C2 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 8 | 3 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 11 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[8,5,0,0,3,5,0,0,0,0,5,0,0,0,0,5],[12,1,0,0,11,1,0,0,0,0,0,1,0,0,1,0],[12,1,0,0,0,1,0,0,0,0,1,0,0,0,0,12] >;
C32×C42⋊2C2 in GAP, Magma, Sage, TeX
C_3^2\times C_4^2\rtimes_2C_2
% in TeX
G:=Group("C3^2xC4^2:2C2"); // GroupNames label
G:=SmallGroup(288,823);
// by ID
G=gap.SmallGroup(288,823);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,1520,3110,394]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^4=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,c*d=d*c,e*c*e=c*d^2,e*d*e=c^2*d^-1>;
// generators/relations