direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C33, (C2×C66)⋊C3, (C2×C6)⋊C33, (C2×C22)⋊C32, C22⋊(C3×C33), SmallGroup(396,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C33 |
Generators and relations for A4×C33
G = < a,b,c,d | a33=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C33
►On 132 points
Generators in S132
(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)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 100)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 34)(30 35)(31 36)(32 37)(33 38)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)
(1 12 23)(2 13 24)(3 14 25)(4 15 26)(5 16 27)(6 17 28)(7 18 29)(8 19 30)(9 20 31)(10 21 32)(11 22 33)(34 112 71)(35 113 72)(36 114 73)(37 115 74)(38 116 75)(39 117 76)(40 118 77)(41 119 78)(42 120 79)(43 121 80)(44 122 81)(45 123 82)(46 124 83)(47 125 84)(48 126 85)(49 127 86)(50 128 87)(51 129 88)(52 130 89)(53 131 90)(54 132 91)(55 100 92)(56 101 93)(57 102 94)(58 103 95)(59 104 96)(60 105 97)(61 106 98)(62 107 99)(63 108 67)(64 109 68)(65 110 69)(66 111 70)
G:=sub<Sym(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), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,100), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,34)(30,35)(31,36)(32,37)(33,38)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118), (1,12,23)(2,13,24)(3,14,25)(4,15,26)(5,16,27)(6,17,28)(7,18,29)(8,19,30)(9,20,31)(10,21,32)(11,22,33)(34,112,71)(35,113,72)(36,114,73)(37,115,74)(38,116,75)(39,117,76)(40,118,77)(41,119,78)(42,120,79)(43,121,80)(44,122,81)(45,123,82)(46,124,83)(47,125,84)(48,126,85)(49,127,86)(50,128,87)(51,129,88)(52,130,89)(53,131,90)(54,132,91)(55,100,92)(56,101,93)(57,102,94)(58,103,95)(59,104,96)(60,105,97)(61,106,98)(62,107,99)(63,108,67)(64,109,68)(65,110,69)(66,111,70)>;
G:=Group( (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), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,100), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,34)(30,35)(31,36)(32,37)(33,38)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118), (1,12,23)(2,13,24)(3,14,25)(4,15,26)(5,16,27)(6,17,28)(7,18,29)(8,19,30)(9,20,31)(10,21,32)(11,22,33)(34,112,71)(35,113,72)(36,114,73)(37,115,74)(38,116,75)(39,117,76)(40,118,77)(41,119,78)(42,120,79)(43,121,80)(44,122,81)(45,123,82)(46,124,83)(47,125,84)(48,126,85)(49,127,86)(50,128,87)(51,129,88)(52,130,89)(53,131,90)(54,132,91)(55,100,92)(56,101,93)(57,102,94)(58,103,95)(59,104,96)(60,105,97)(61,106,98)(62,107,99)(63,108,67)(64,109,68)(65,110,69)(66,111,70) );
G=PermutationGroup([[(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)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,108),(42,109),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,128),(62,129),(63,130),(64,131),(65,132),(66,100)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,34),(30,35),(31,36),(32,37),(33,38),(67,119),(68,120),(69,121),(70,122),(71,123),(72,124),(73,125),(74,126),(75,127),(76,128),(77,129),(78,130),(79,131),(80,132),(81,100),(82,101),(83,102),(84,103),(85,104),(86,105),(87,106),(88,107),(89,108),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114),(96,115),(97,116),(98,117),(99,118)], [(1,12,23),(2,13,24),(3,14,25),(4,15,26),(5,16,27),(6,17,28),(7,18,29),(8,19,30),(9,20,31),(10,21,32),(11,22,33),(34,112,71),(35,113,72),(36,114,73),(37,115,74),(38,116,75),(39,117,76),(40,118,77),(41,119,78),(42,120,79),(43,121,80),(44,122,81),(45,123,82),(46,124,83),(47,125,84),(48,126,85),(49,127,86),(50,128,87),(51,129,88),(52,130,89),(53,131,90),(54,132,91),(55,100,92),(56,101,93),(57,102,94),(58,103,95),(59,104,96),(60,105,97),(61,106,98),(62,107,99),(63,108,67),(64,109,68),(65,110,69),(66,111,70)]])
132 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33T | 33U | ··· | 33CB | 66A | ··· | 66T |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 3 | 1 | 1 | 4 | ··· | 4 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 4 | ··· | 4 | 3 | ··· | 3 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C3 | C11 | C33 | C33 | A4 | C3×A4 | C11×A4 | A4×C33 |
| kernel | A4×C33 | C11×A4 | C2×C66 | C3×A4 | A4 | C2×C6 | C33 | C11 | C3 | C1 |
| # reps | 1 | 6 | 2 | 10 | 60 | 20 | 1 | 2 | 10 | 20 |
Matrix representation of A4×C33 ►in GL4(𝔽67) generated by
| 29 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 14 |
| 1 | 0 | 0 | 0 |
| 0 | 66 | 0 | 0 |
| 0 | 66 | 0 | 1 |
| 0 | 66 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 66 |
| 0 | 1 | 0 | 66 |
| 0 | 0 | 0 | 66 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(67))| [29,0,0,0,0,14,0,0,0,0,14,0,0,0,0,14],[1,0,0,0,0,66,66,66,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,1,0,0,1,0,0,0,66,66,66],[29,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
A4×C33 in GAP, Magma, Sage, TeX
A_4\times C_{33} % in TeX
G:=Group("A4xC33"); // GroupNames label
G:=SmallGroup(396,24);
// by ID
G=gap.SmallGroup(396,24);
# by ID
G:=PCGroup([5,-3,-3,-11,-2,2,3963,7429]);
// Polycyclic
G:=Group<a,b,c,d|a^33=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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