direct product, metabelian, nilpotent (class 2), monomial
Aliases: C22⋊C4×C3×C6, C62.142D4, C23.9C62, C62.284C23, (C2×C4)⋊3C62, (C2×C62)⋊7C4, C6.81(C6×D4), C22⋊3(C6×C12), (C22×C12)⋊7C6, (C22×C6)⋊5C12, C62⋊22(C2×C4), C23⋊3(C3×C12), C24.2(C3×C6), (C6×C12)⋊30C22, (C23×C6).11C6, C6.36(C22×C12), C22.4(C2×C62), (C22×C62).1C2, (C2×C62).85C22, C22.12(D4×C32), (C2×C6×C12)⋊5C2, C2.1(D4×C3×C6), C2.1(C2×C6×C12), (C2×C6)⋊10(C2×C12), (C2×C12)⋊11(C2×C6), (C22×C4)⋊3(C3×C6), (C2×C6).70(C3×D4), (C3×C6).298(C2×D4), (C2×C6).90(C22×C6), (C22×C6).77(C2×C6), (C3×C6).128(C22×C4), SmallGroup(288,812)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C3×C6
G = < a,b,c,d,e | a3=b6=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 564 in 396 conjugacy classes, 228 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, C23, C32, C12, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×C6, C3×C6, C3×C6, C2×C12, C2×C12, C22×C6, C22×C6, C2×C22⋊C4, C3×C12, C62, C62, C62, C3×C22⋊C4, C22×C12, C23×C6, C6×C12, C6×C12, C2×C62, C2×C62, C2×C62, C6×C22⋊C4, C32×C22⋊C4, C2×C6×C12, C22×C62, C22⋊C4×C3×C6
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C32, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C3×C6, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C3×C12, C62, C3×C22⋊C4, C22×C12, C6×D4, C6×C12, D4×C32, C2×C62, C6×C22⋊C4, C32×C22⋊C4, C2×C6×C12, D4×C3×C6, C22⋊C4×C3×C6
►On 144 points
Generators in S144
(1 76 80)(2 77 81)(3 78 82)(4 73 83)(5 74 84)(6 75 79)(7 144 138)(8 139 133)(9 140 134)(10 141 135)(11 142 136)(12 143 137)(13 57 31)(14 58 32)(15 59 33)(16 60 34)(17 55 35)(18 56 36)(19 27 49)(20 28 50)(21 29 51)(22 30 52)(23 25 53)(24 26 54)(37 102 45)(38 97 46)(39 98 47)(40 99 48)(41 100 43)(42 101 44)(61 111 67)(62 112 68)(63 113 69)(64 114 70)(65 109 71)(66 110 72)(85 95 103)(86 96 104)(87 91 105)(88 92 106)(89 93 107)(90 94 108)(115 129 123)(116 130 124)(117 131 125)(118 132 126)(119 127 121)(120 128 122)
(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 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(25 77)(26 78)(27 73)(28 74)(29 75)(30 76)(31 111)(32 112)(33 113)(34 114)(35 109)(36 110)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 107)(44 108)(45 103)(46 104)(47 105)(48 106)(49 83)(50 84)(51 79)(52 80)(53 81)(54 82)(55 65)(56 66)(57 61)(58 62)(59 63)(60 64)(91 98)(92 99)(93 100)(94 101)(95 102)(96 97)(115 134)(116 135)(117 136)(118 137)(119 138)(120 133)(121 144)(122 139)(123 140)(124 141)(125 142)(126 143)
(1 63)(2 64)(3 65)(4 66)(5 61)(6 62)(7 44)(8 45)(9 46)(10 47)(11 48)(12 43)(13 50)(14 51)(15 52)(16 53)(17 54)(18 49)(19 56)(20 57)(21 58)(22 59)(23 60)(24 55)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(67 84)(68 79)(69 80)(70 81)(71 82)(72 83)(73 110)(74 111)(75 112)(76 113)(77 114)(78 109)(85 122)(86 123)(87 124)(88 125)(89 126)(90 121)(91 116)(92 117)(93 118)(94 119)(95 120)(96 115)(97 134)(98 135)(99 136)(100 137)(101 138)(102 133)(103 128)(104 129)(105 130)(106 131)(107 132)(108 127)
(1 138 22 94)(2 133 23 95)(3 134 24 96)(4 135 19 91)(5 136 20 92)(6 137 21 93)(7 30 108 76)(8 25 103 77)(9 26 104 78)(10 27 105 73)(11 28 106 74)(12 29 107 75)(13 125 67 40)(14 126 68 41)(15 121 69 42)(16 122 70 37)(17 123 71 38)(18 124 72 39)(31 131 111 48)(32 132 112 43)(33 127 113 44)(34 128 114 45)(35 129 109 46)(36 130 110 47)(49 87 83 141)(50 88 84 142)(51 89 79 143)(52 90 80 144)(53 85 81 139)(54 86 82 140)(55 115 65 97)(56 116 66 98)(57 117 61 99)(58 118 62 100)(59 119 63 101)(60 120 64 102)
G:=sub<Sym(144)| (1,76,80)(2,77,81)(3,78,82)(4,73,83)(5,74,84)(6,75,79)(7,144,138)(8,139,133)(9,140,134)(10,141,135)(11,142,136)(12,143,137)(13,57,31)(14,58,32)(15,59,33)(16,60,34)(17,55,35)(18,56,36)(19,27,49)(20,28,50)(21,29,51)(22,30,52)(23,25,53)(24,26,54)(37,102,45)(38,97,46)(39,98,47)(40,99,48)(41,100,43)(42,101,44)(61,111,67)(62,112,68)(63,113,69)(64,114,70)(65,109,71)(66,110,72)(85,95,103)(86,96,104)(87,91,105)(88,92,106)(89,93,107)(90,94,108)(115,129,123)(116,130,124)(117,131,125)(118,132,126)(119,127,121)(120,128,122), (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), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(25,77)(26,78)(27,73)(28,74)(29,75)(30,76)(31,111)(32,112)(33,113)(34,114)(35,109)(36,110)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,107)(44,108)(45,103)(46,104)(47,105)(48,106)(49,83)(50,84)(51,79)(52,80)(53,81)(54,82)(55,65)(56,66)(57,61)(58,62)(59,63)(60,64)(91,98)(92,99)(93,100)(94,101)(95,102)(96,97)(115,134)(116,135)(117,136)(118,137)(119,138)(120,133)(121,144)(122,139)(123,140)(124,141)(125,142)(126,143), (1,63)(2,64)(3,65)(4,66)(5,61)(6,62)(7,44)(8,45)(9,46)(10,47)(11,48)(12,43)(13,50)(14,51)(15,52)(16,53)(17,54)(18,49)(19,56)(20,57)(21,58)(22,59)(23,60)(24,55)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(67,84)(68,79)(69,80)(70,81)(71,82)(72,83)(73,110)(74,111)(75,112)(76,113)(77,114)(78,109)(85,122)(86,123)(87,124)(88,125)(89,126)(90,121)(91,116)(92,117)(93,118)(94,119)(95,120)(96,115)(97,134)(98,135)(99,136)(100,137)(101,138)(102,133)(103,128)(104,129)(105,130)(106,131)(107,132)(108,127), (1,138,22,94)(2,133,23,95)(3,134,24,96)(4,135,19,91)(5,136,20,92)(6,137,21,93)(7,30,108,76)(8,25,103,77)(9,26,104,78)(10,27,105,73)(11,28,106,74)(12,29,107,75)(13,125,67,40)(14,126,68,41)(15,121,69,42)(16,122,70,37)(17,123,71,38)(18,124,72,39)(31,131,111,48)(32,132,112,43)(33,127,113,44)(34,128,114,45)(35,129,109,46)(36,130,110,47)(49,87,83,141)(50,88,84,142)(51,89,79,143)(52,90,80,144)(53,85,81,139)(54,86,82,140)(55,115,65,97)(56,116,66,98)(57,117,61,99)(58,118,62,100)(59,119,63,101)(60,120,64,102)>;
G:=Group( (1,76,80)(2,77,81)(3,78,82)(4,73,83)(5,74,84)(6,75,79)(7,144,138)(8,139,133)(9,140,134)(10,141,135)(11,142,136)(12,143,137)(13,57,31)(14,58,32)(15,59,33)(16,60,34)(17,55,35)(18,56,36)(19,27,49)(20,28,50)(21,29,51)(22,30,52)(23,25,53)(24,26,54)(37,102,45)(38,97,46)(39,98,47)(40,99,48)(41,100,43)(42,101,44)(61,111,67)(62,112,68)(63,113,69)(64,114,70)(65,109,71)(66,110,72)(85,95,103)(86,96,104)(87,91,105)(88,92,106)(89,93,107)(90,94,108)(115,129,123)(116,130,124)(117,131,125)(118,132,126)(119,127,121)(120,128,122), (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), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(25,77)(26,78)(27,73)(28,74)(29,75)(30,76)(31,111)(32,112)(33,113)(34,114)(35,109)(36,110)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,107)(44,108)(45,103)(46,104)(47,105)(48,106)(49,83)(50,84)(51,79)(52,80)(53,81)(54,82)(55,65)(56,66)(57,61)(58,62)(59,63)(60,64)(91,98)(92,99)(93,100)(94,101)(95,102)(96,97)(115,134)(116,135)(117,136)(118,137)(119,138)(120,133)(121,144)(122,139)(123,140)(124,141)(125,142)(126,143), (1,63)(2,64)(3,65)(4,66)(5,61)(6,62)(7,44)(8,45)(9,46)(10,47)(11,48)(12,43)(13,50)(14,51)(15,52)(16,53)(17,54)(18,49)(19,56)(20,57)(21,58)(22,59)(23,60)(24,55)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(67,84)(68,79)(69,80)(70,81)(71,82)(72,83)(73,110)(74,111)(75,112)(76,113)(77,114)(78,109)(85,122)(86,123)(87,124)(88,125)(89,126)(90,121)(91,116)(92,117)(93,118)(94,119)(95,120)(96,115)(97,134)(98,135)(99,136)(100,137)(101,138)(102,133)(103,128)(104,129)(105,130)(106,131)(107,132)(108,127), (1,138,22,94)(2,133,23,95)(3,134,24,96)(4,135,19,91)(5,136,20,92)(6,137,21,93)(7,30,108,76)(8,25,103,77)(9,26,104,78)(10,27,105,73)(11,28,106,74)(12,29,107,75)(13,125,67,40)(14,126,68,41)(15,121,69,42)(16,122,70,37)(17,123,71,38)(18,124,72,39)(31,131,111,48)(32,132,112,43)(33,127,113,44)(34,128,114,45)(35,129,109,46)(36,130,110,47)(49,87,83,141)(50,88,84,142)(51,89,79,143)(52,90,80,144)(53,85,81,139)(54,86,82,140)(55,115,65,97)(56,116,66,98)(57,117,61,99)(58,118,62,100)(59,119,63,101)(60,120,64,102) );
G=PermutationGroup([[(1,76,80),(2,77,81),(3,78,82),(4,73,83),(5,74,84),(6,75,79),(7,144,138),(8,139,133),(9,140,134),(10,141,135),(11,142,136),(12,143,137),(13,57,31),(14,58,32),(15,59,33),(16,60,34),(17,55,35),(18,56,36),(19,27,49),(20,28,50),(21,29,51),(22,30,52),(23,25,53),(24,26,54),(37,102,45),(38,97,46),(39,98,47),(40,99,48),(41,100,43),(42,101,44),(61,111,67),(62,112,68),(63,113,69),(64,114,70),(65,109,71),(66,110,72),(85,95,103),(86,96,104),(87,91,105),(88,92,106),(89,93,107),(90,94,108),(115,129,123),(116,130,124),(117,131,125),(118,132,126),(119,127,121),(120,128,122)], [(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)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(25,77),(26,78),(27,73),(28,74),(29,75),(30,76),(31,111),(32,112),(33,113),(34,114),(35,109),(36,110),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,107),(44,108),(45,103),(46,104),(47,105),(48,106),(49,83),(50,84),(51,79),(52,80),(53,81),(54,82),(55,65),(56,66),(57,61),(58,62),(59,63),(60,64),(91,98),(92,99),(93,100),(94,101),(95,102),(96,97),(115,134),(116,135),(117,136),(118,137),(119,138),(120,133),(121,144),(122,139),(123,140),(124,141),(125,142),(126,143)], [(1,63),(2,64),(3,65),(4,66),(5,61),(6,62),(7,44),(8,45),(9,46),(10,47),(11,48),(12,43),(13,50),(14,51),(15,52),(16,53),(17,54),(18,49),(19,56),(20,57),(21,58),(22,59),(23,60),(24,55),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(67,84),(68,79),(69,80),(70,81),(71,82),(72,83),(73,110),(74,111),(75,112),(76,113),(77,114),(78,109),(85,122),(86,123),(87,124),(88,125),(89,126),(90,121),(91,116),(92,117),(93,118),(94,119),(95,120),(96,115),(97,134),(98,135),(99,136),(100,137),(101,138),(102,133),(103,128),(104,129),(105,130),(106,131),(107,132),(108,127)], [(1,138,22,94),(2,133,23,95),(3,134,24,96),(4,135,19,91),(5,136,20,92),(6,137,21,93),(7,30,108,76),(8,25,103,77),(9,26,104,78),(10,27,105,73),(11,28,106,74),(12,29,107,75),(13,125,67,40),(14,126,68,41),(15,121,69,42),(16,122,70,37),(17,123,71,38),(18,124,72,39),(31,131,111,48),(32,132,112,43),(33,127,113,44),(34,128,114,45),(35,129,109,46),(36,130,110,47),(49,87,83,141),(50,88,84,142),(51,89,79,143),(52,90,80,144),(53,85,81,139),(54,86,82,140),(55,115,65,97),(56,116,66,98),(57,117,61,99),(58,118,62,100),(59,119,63,101),(60,120,64,102)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | ··· | 3H | 4A | ··· | 4H | 6A | ··· | 6BD | 6BE | ··· | 6CJ | 12A | ··· | 12BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | C3×D4 |
| kernel | C22⋊C4×C3×C6 | C32×C22⋊C4 | C2×C6×C12 | C22×C62 | C6×C22⋊C4 | C2×C62 | C3×C22⋊C4 | C22×C12 | C23×C6 | C22×C6 | C62 | C2×C6 |
| # reps | 1 | 4 | 2 | 1 | 8 | 8 | 32 | 16 | 8 | 64 | 4 | 32 |
Matrix representation of C22⋊C4×C3×C6 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 8 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[4,0,0,0,0,4,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,8,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,12,0,0,0,0,8,0,0,0,2,5] >;
C22⋊C4×C3×C6 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_3\times C_6
% in TeX
G:=Group("C2^2:C4xC3xC6"); // GroupNames label
G:=SmallGroup(288,812);
// by ID
G=gap.SmallGroup(288,812);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^2=d^2=e^4=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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations