direct product, metabelian, supersoluble, monomial, A-group
Aliases: C22×C3⋊Dic3, C62⋊7C4, C62.35C22, (C2×C6)⋊5Dic3, C6⋊2(C2×Dic3), (C2×C6).40D6, (C2×C62).5C2, C32⋊8(C22×C4), C23.3(C3⋊S3), (C22×C6).11S3, C6.38(C22×S3), (C3×C6).37C23, C3⋊2(C22×Dic3), (C3×C6)⋊7(C2×C4), C2.2(C22×C3⋊S3), C22.11(C2×C3⋊S3), SmallGroup(144,176)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C22×C3⋊Dic3 |
Generators and relations for C22×C3⋊Dic3
G = < a,b,c,d,e | a2=b2=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 338 in 162 conjugacy classes, 107 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, C23, C32, Dic3, C2×C6, C22×C4, C3×C6, C3×C6, C2×Dic3, C22×C6, C3⋊Dic3, C62, C22×Dic3, C2×C3⋊Dic3, C2×C62, C22×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C3⋊S3, C2×Dic3, C22×S3, C3⋊Dic3, C2×C3⋊S3, C22×Dic3, C2×C3⋊Dic3, C22×C3⋊S3, C22×C3⋊Dic3
►Regular action on 144 points
Generators in S144
(1 66)(2 61)(3 62)(4 63)(5 64)(6 65)(7 117)(8 118)(9 119)(10 120)(11 115)(12 116)(13 111)(14 112)(15 113)(16 114)(17 109)(18 110)(19 58)(20 59)(21 60)(22 55)(23 56)(24 57)(25 53)(26 54)(27 49)(28 50)(29 51)(30 52)(31 70)(32 71)(33 72)(34 67)(35 68)(36 69)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(85 124)(86 125)(87 126)(88 121)(89 122)(90 123)(91 130)(92 131)(93 132)(94 127)(95 128)(96 129)(97 136)(98 137)(99 138)(100 133)(101 134)(102 135)(103 142)(104 143)(105 144)(106 139)(107 140)(108 141)
(1 45)(2 46)(3 47)(4 48)(5 43)(6 44)(7 138)(8 133)(9 134)(10 135)(11 136)(12 137)(13 132)(14 127)(15 128)(16 129)(17 130)(18 131)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 32)(26 33)(27 34)(28 35)(29 36)(30 31)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 76)(56 77)(57 78)(58 73)(59 74)(60 75)(61 82)(62 83)(63 84)(64 79)(65 80)(66 81)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(1 23 26)(2 24 27)(3 19 28)(4 20 29)(5 21 30)(6 22 25)(7 16 141)(8 17 142)(9 18 143)(10 13 144)(11 14 139)(12 15 140)(31 43 39)(32 44 40)(33 45 41)(34 46 42)(35 47 37)(36 48 38)(49 61 57)(50 62 58)(51 63 59)(52 64 60)(53 65 55)(54 66 56)(67 82 78)(68 83 73)(69 84 74)(70 79 75)(71 80 76)(72 81 77)(85 100 91)(86 101 92)(87 102 93)(88 97 94)(89 98 95)(90 99 96)(103 118 109)(104 119 110)(105 120 111)(106 115 112)(107 116 113)(108 117 114)(121 136 127)(122 137 128)(123 138 129)(124 133 130)(125 134 131)(126 135 132)
(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)
(1 121 4 124)(2 126 5 123)(3 125 6 122)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 39 16 42)(14 38 17 41)(15 37 18 40)(19 131 22 128)(20 130 23 127)(21 129 24 132)(25 137 28 134)(26 136 29 133)(27 135 30 138)(43 141 46 144)(44 140 47 143)(45 139 48 142)(49 102 52 99)(50 101 53 98)(51 100 54 97)(55 95 58 92)(56 94 59 91)(57 93 60 96)(61 87 64 90)(62 86 65 89)(63 85 66 88)(67 120 70 117)(68 119 71 116)(69 118 72 115)(73 110 76 113)(74 109 77 112)(75 114 78 111)(79 108 82 105)(80 107 83 104)(81 106 84 103)
G:=sub<Sym(144)| (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,117)(8,118)(9,119)(10,120)(11,115)(12,116)(13,111)(14,112)(15,113)(16,114)(17,109)(18,110)(19,58)(20,59)(21,60)(22,55)(23,56)(24,57)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(85,124)(86,125)(87,126)(88,121)(89,122)(90,123)(91,130)(92,131)(93,132)(94,127)(95,128)(96,129)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,142)(104,143)(105,144)(106,139)(107,140)(108,141), (1,45)(2,46)(3,47)(4,48)(5,43)(6,44)(7,138)(8,133)(9,134)(10,135)(11,136)(12,137)(13,132)(14,127)(15,128)(16,129)(17,130)(18,131)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,76)(56,77)(57,78)(58,73)(59,74)(60,75)(61,82)(62,83)(63,84)(64,79)(65,80)(66,81)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,16,141)(8,17,142)(9,18,143)(10,13,144)(11,14,139)(12,15,140)(31,43,39)(32,44,40)(33,45,41)(34,46,42)(35,47,37)(36,48,38)(49,61,57)(50,62,58)(51,63,59)(52,64,60)(53,65,55)(54,66,56)(67,82,78)(68,83,73)(69,84,74)(70,79,75)(71,80,76)(72,81,77)(85,100,91)(86,101,92)(87,102,93)(88,97,94)(89,98,95)(90,99,96)(103,118,109)(104,119,110)(105,120,111)(106,115,112)(107,116,113)(108,117,114)(121,136,127)(122,137,128)(123,138,129)(124,133,130)(125,134,131)(126,135,132), (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), (1,121,4,124)(2,126,5,123)(3,125,6,122)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,39,16,42)(14,38,17,41)(15,37,18,40)(19,131,22,128)(20,130,23,127)(21,129,24,132)(25,137,28,134)(26,136,29,133)(27,135,30,138)(43,141,46,144)(44,140,47,143)(45,139,48,142)(49,102,52,99)(50,101,53,98)(51,100,54,97)(55,95,58,92)(56,94,59,91)(57,93,60,96)(61,87,64,90)(62,86,65,89)(63,85,66,88)(67,120,70,117)(68,119,71,116)(69,118,72,115)(73,110,76,113)(74,109,77,112)(75,114,78,111)(79,108,82,105)(80,107,83,104)(81,106,84,103)>;
G:=Group( (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,117)(8,118)(9,119)(10,120)(11,115)(12,116)(13,111)(14,112)(15,113)(16,114)(17,109)(18,110)(19,58)(20,59)(21,60)(22,55)(23,56)(24,57)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(85,124)(86,125)(87,126)(88,121)(89,122)(90,123)(91,130)(92,131)(93,132)(94,127)(95,128)(96,129)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,142)(104,143)(105,144)(106,139)(107,140)(108,141), (1,45)(2,46)(3,47)(4,48)(5,43)(6,44)(7,138)(8,133)(9,134)(10,135)(11,136)(12,137)(13,132)(14,127)(15,128)(16,129)(17,130)(18,131)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,76)(56,77)(57,78)(58,73)(59,74)(60,75)(61,82)(62,83)(63,84)(64,79)(65,80)(66,81)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,16,141)(8,17,142)(9,18,143)(10,13,144)(11,14,139)(12,15,140)(31,43,39)(32,44,40)(33,45,41)(34,46,42)(35,47,37)(36,48,38)(49,61,57)(50,62,58)(51,63,59)(52,64,60)(53,65,55)(54,66,56)(67,82,78)(68,83,73)(69,84,74)(70,79,75)(71,80,76)(72,81,77)(85,100,91)(86,101,92)(87,102,93)(88,97,94)(89,98,95)(90,99,96)(103,118,109)(104,119,110)(105,120,111)(106,115,112)(107,116,113)(108,117,114)(121,136,127)(122,137,128)(123,138,129)(124,133,130)(125,134,131)(126,135,132), (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), (1,121,4,124)(2,126,5,123)(3,125,6,122)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,39,16,42)(14,38,17,41)(15,37,18,40)(19,131,22,128)(20,130,23,127)(21,129,24,132)(25,137,28,134)(26,136,29,133)(27,135,30,138)(43,141,46,144)(44,140,47,143)(45,139,48,142)(49,102,52,99)(50,101,53,98)(51,100,54,97)(55,95,58,92)(56,94,59,91)(57,93,60,96)(61,87,64,90)(62,86,65,89)(63,85,66,88)(67,120,70,117)(68,119,71,116)(69,118,72,115)(73,110,76,113)(74,109,77,112)(75,114,78,111)(79,108,82,105)(80,107,83,104)(81,106,84,103) );
G=PermutationGroup([[(1,66),(2,61),(3,62),(4,63),(5,64),(6,65),(7,117),(8,118),(9,119),(10,120),(11,115),(12,116),(13,111),(14,112),(15,113),(16,114),(17,109),(18,110),(19,58),(20,59),(21,60),(22,55),(23,56),(24,57),(25,53),(26,54),(27,49),(28,50),(29,51),(30,52),(31,70),(32,71),(33,72),(34,67),(35,68),(36,69),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(85,124),(86,125),(87,126),(88,121),(89,122),(90,123),(91,130),(92,131),(93,132),(94,127),(95,128),(96,129),(97,136),(98,137),(99,138),(100,133),(101,134),(102,135),(103,142),(104,143),(105,144),(106,139),(107,140),(108,141)], [(1,45),(2,46),(3,47),(4,48),(5,43),(6,44),(7,138),(8,133),(9,134),(10,135),(11,136),(12,137),(13,132),(14,127),(15,128),(16,129),(17,130),(18,131),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,32),(26,33),(27,34),(28,35),(29,36),(30,31),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,76),(56,77),(57,78),(58,73),(59,74),(60,75),(61,82),(62,83),(63,84),(64,79),(65,80),(66,81),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,113),(96,114),(97,115),(98,116),(99,117),(100,118),(101,119),(102,120),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144)], [(1,23,26),(2,24,27),(3,19,28),(4,20,29),(5,21,30),(6,22,25),(7,16,141),(8,17,142),(9,18,143),(10,13,144),(11,14,139),(12,15,140),(31,43,39),(32,44,40),(33,45,41),(34,46,42),(35,47,37),(36,48,38),(49,61,57),(50,62,58),(51,63,59),(52,64,60),(53,65,55),(54,66,56),(67,82,78),(68,83,73),(69,84,74),(70,79,75),(71,80,76),(72,81,77),(85,100,91),(86,101,92),(87,102,93),(88,97,94),(89,98,95),(90,99,96),(103,118,109),(104,119,110),(105,120,111),(106,115,112),(107,116,113),(108,117,114),(121,136,127),(122,137,128),(123,138,129),(124,133,130),(125,134,131),(126,135,132)], [(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)], [(1,121,4,124),(2,126,5,123),(3,125,6,122),(7,34,10,31),(8,33,11,36),(9,32,12,35),(13,39,16,42),(14,38,17,41),(15,37,18,40),(19,131,22,128),(20,130,23,127),(21,129,24,132),(25,137,28,134),(26,136,29,133),(27,135,30,138),(43,141,46,144),(44,140,47,143),(45,139,48,142),(49,102,52,99),(50,101,53,98),(51,100,54,97),(55,95,58,92),(56,94,59,91),(57,93,60,96),(61,87,64,90),(62,86,65,89),(63,85,66,88),(67,120,70,117),(68,119,71,116),(69,118,72,115),(73,110,76,113),(74,109,77,112),(75,114,78,111),(79,108,82,105),(80,107,83,104),(81,106,84,103)]])
C22×C3⋊Dic3 is a maximal subgroup of
C62.6Q8 C62.15Q8 C62⋊3C8 C2×Dic32 C62.99C23 C62.57D4 C62.115C23 C62⋊7D4 C62⋊4Q8 C62.221C23 C62⋊6Q8 C62.225C23 C62.69D4 C62.72D4 C62⋊14D4 C22×S3×Dic3 C22×C4×C3⋊S3 C62⋊4C12
C22×C3⋊Dic3 is a maximal quotient of
C62.247C23 D4.(C3⋊Dic3)
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4H | 6A | ··· | 6AB |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 |
| kernel | C22×C3⋊Dic3 | C2×C3⋊Dic3 | C2×C62 | C62 | C22×C6 | C2×C6 | C2×C6 |
| # reps | 1 | 6 | 1 | 8 | 4 | 16 | 12 |
Matrix representation of C22×C3⋊Dic3 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 5 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 8 | 10 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 4 | 2 | 0 | 0 |
| 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 2 | 8 |
| 0 | 0 | 0 | 1 | 11 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,5,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,12,1,0,0,0,0,0,4,8,0,0,0,0,10],[12,0,0,0,0,0,4,11,0,0,0,2,9,0,0,0,0,0,2,1,0,0,0,8,11] >;
C22×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_2^2\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2^2xC3:Dic3"); // GroupNames label
G:=SmallGroup(144,176);
// by ID
G=gap.SmallGroup(144,176);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,964,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^6=1,e^2=d^3,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^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations