direct product, non-abelian, soluble
Aliases: C3×C6×SL2(𝔽3), C62.12A4, (C2×Q8)⋊C33, Q8⋊(C32×C6), (C6×Q8)⋊C32, C6.19(C6×A4), (Q8×C32)⋊11C6, C22.2(C32×A4), C2.2(A4×C3×C6), (Q8×C3×C6)⋊3C3, (C3×Q8)⋊2(C3×C6), (C2×C6).27(C3×A4), (C3×C6).26(C2×A4), SmallGroup(432,698)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C3×C6×SL2(𝔽3) |
Generators and relations for C3×C6×SL2(𝔽3)
G = < a,b,c,d,e | a3=b6=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Subgroups: 598 in 220 conjugacy classes, 86 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, Q8, Q8, C32, C32, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, SL2(𝔽3), C2×C12, C3×Q8, C3×Q8, C33, C3×C12, C62, C62, C2×SL2(𝔽3), C6×Q8, C32×C6, C3×SL2(𝔽3), C6×C12, Q8×C32, Q8×C32, C3×C62, C6×SL2(𝔽3), Q8×C3×C6, C32×SL2(𝔽3), C3×C6×SL2(𝔽3)
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, SL2(𝔽3), C2×A4, C33, C3×A4, C2×SL2(𝔽3), C32×C6, C3×SL2(𝔽3), C6×A4, C32×A4, C6×SL2(𝔽3), C32×SL2(𝔽3), A4×C3×C6, C3×C6×SL2(𝔽3)
►On 144 points
Generators in S144
(1 77 143)(2 78 144)(3 73 139)(4 74 140)(5 75 141)(6 76 142)(7 71 51)(8 72 52)(9 67 53)(10 68 54)(11 69 49)(12 70 50)(13 59 63)(14 60 64)(15 55 65)(16 56 66)(17 57 61)(18 58 62)(19 131 39)(20 132 40)(21 127 41)(22 128 42)(23 129 37)(24 130 38)(25 96 45)(26 91 46)(27 92 47)(28 93 48)(29 94 43)(30 95 44)(31 125 111)(32 126 112)(33 121 113)(34 122 114)(35 123 109)(36 124 110)(79 89 137)(80 90 138)(81 85 133)(82 86 134)(83 87 135)(84 88 136)(97 120 106)(98 115 107)(99 116 108)(100 117 103)(101 118 104)(102 119 105)
(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)
(1 137 97 34)(2 138 98 35)(3 133 99 36)(4 134 100 31)(5 135 101 32)(6 136 102 33)(7 17 48 130)(8 18 43 131)(9 13 44 132)(10 14 45 127)(11 15 46 128)(12 16 47 129)(19 52 62 94)(20 53 63 95)(21 54 64 96)(22 49 65 91)(23 50 66 92)(24 51 61 93)(25 41 68 60)(26 42 69 55)(27 37 70 56)(28 38 71 57)(29 39 72 58)(30 40 67 59)(73 81 116 124)(74 82 117 125)(75 83 118 126)(76 84 119 121)(77 79 120 122)(78 80 115 123)(85 108 110 139)(86 103 111 140)(87 104 112 141)(88 105 113 142)(89 106 114 143)(90 107 109 144)
(1 62 97 19)(2 63 98 20)(3 64 99 21)(4 65 100 22)(5 66 101 23)(6 61 102 24)(7 121 48 84)(8 122 43 79)(9 123 44 80)(10 124 45 81)(11 125 46 82)(12 126 47 83)(13 115 132 78)(14 116 127 73)(15 117 128 74)(16 118 129 75)(17 119 130 76)(18 120 131 77)(25 85 68 110)(26 86 69 111)(27 87 70 112)(28 88 71 113)(29 89 72 114)(30 90 67 109)(31 91 134 49)(32 92 135 50)(33 93 136 51)(34 94 137 52)(35 95 138 53)(36 96 133 54)(37 141 56 104)(38 142 57 105)(39 143 58 106)(40 144 59 107)(41 139 60 108)(42 140 55 103)
(1 75 139)(2 76 140)(3 77 141)(4 78 142)(5 73 143)(6 74 144)(7 55 138)(8 56 133)(9 57 134)(10 58 135)(11 59 136)(12 60 137)(13 88 49)(14 89 50)(15 90 51)(16 85 52)(17 86 53)(18 87 54)(19 126 25)(20 121 26)(21 122 27)(22 123 28)(23 124 29)(24 125 30)(31 44 38)(32 45 39)(33 46 40)(34 47 41)(35 48 42)(36 43 37)(61 82 67)(62 83 68)(63 84 69)(64 79 70)(65 80 71)(66 81 72)(91 132 113)(92 127 114)(93 128 109)(94 129 110)(95 130 111)(96 131 112)(97 118 108)(98 119 103)(99 120 104)(100 115 105)(101 116 106)(102 117 107)
G:=sub<Sym(144)| (1,77,143)(2,78,144)(3,73,139)(4,74,140)(5,75,141)(6,76,142)(7,71,51)(8,72,52)(9,67,53)(10,68,54)(11,69,49)(12,70,50)(13,59,63)(14,60,64)(15,55,65)(16,56,66)(17,57,61)(18,58,62)(19,131,39)(20,132,40)(21,127,41)(22,128,42)(23,129,37)(24,130,38)(25,96,45)(26,91,46)(27,92,47)(28,93,48)(29,94,43)(30,95,44)(31,125,111)(32,126,112)(33,121,113)(34,122,114)(35,123,109)(36,124,110)(79,89,137)(80,90,138)(81,85,133)(82,86,134)(83,87,135)(84,88,136)(97,120,106)(98,115,107)(99,116,108)(100,117,103)(101,118,104)(102,119,105), (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), (1,137,97,34)(2,138,98,35)(3,133,99,36)(4,134,100,31)(5,135,101,32)(6,136,102,33)(7,17,48,130)(8,18,43,131)(9,13,44,132)(10,14,45,127)(11,15,46,128)(12,16,47,129)(19,52,62,94)(20,53,63,95)(21,54,64,96)(22,49,65,91)(23,50,66,92)(24,51,61,93)(25,41,68,60)(26,42,69,55)(27,37,70,56)(28,38,71,57)(29,39,72,58)(30,40,67,59)(73,81,116,124)(74,82,117,125)(75,83,118,126)(76,84,119,121)(77,79,120,122)(78,80,115,123)(85,108,110,139)(86,103,111,140)(87,104,112,141)(88,105,113,142)(89,106,114,143)(90,107,109,144), (1,62,97,19)(2,63,98,20)(3,64,99,21)(4,65,100,22)(5,66,101,23)(6,61,102,24)(7,121,48,84)(8,122,43,79)(9,123,44,80)(10,124,45,81)(11,125,46,82)(12,126,47,83)(13,115,132,78)(14,116,127,73)(15,117,128,74)(16,118,129,75)(17,119,130,76)(18,120,131,77)(25,85,68,110)(26,86,69,111)(27,87,70,112)(28,88,71,113)(29,89,72,114)(30,90,67,109)(31,91,134,49)(32,92,135,50)(33,93,136,51)(34,94,137,52)(35,95,138,53)(36,96,133,54)(37,141,56,104)(38,142,57,105)(39,143,58,106)(40,144,59,107)(41,139,60,108)(42,140,55,103), (1,75,139)(2,76,140)(3,77,141)(4,78,142)(5,73,143)(6,74,144)(7,55,138)(8,56,133)(9,57,134)(10,58,135)(11,59,136)(12,60,137)(13,88,49)(14,89,50)(15,90,51)(16,85,52)(17,86,53)(18,87,54)(19,126,25)(20,121,26)(21,122,27)(22,123,28)(23,124,29)(24,125,30)(31,44,38)(32,45,39)(33,46,40)(34,47,41)(35,48,42)(36,43,37)(61,82,67)(62,83,68)(63,84,69)(64,79,70)(65,80,71)(66,81,72)(91,132,113)(92,127,114)(93,128,109)(94,129,110)(95,130,111)(96,131,112)(97,118,108)(98,119,103)(99,120,104)(100,115,105)(101,116,106)(102,117,107)>;
G:=Group( (1,77,143)(2,78,144)(3,73,139)(4,74,140)(5,75,141)(6,76,142)(7,71,51)(8,72,52)(9,67,53)(10,68,54)(11,69,49)(12,70,50)(13,59,63)(14,60,64)(15,55,65)(16,56,66)(17,57,61)(18,58,62)(19,131,39)(20,132,40)(21,127,41)(22,128,42)(23,129,37)(24,130,38)(25,96,45)(26,91,46)(27,92,47)(28,93,48)(29,94,43)(30,95,44)(31,125,111)(32,126,112)(33,121,113)(34,122,114)(35,123,109)(36,124,110)(79,89,137)(80,90,138)(81,85,133)(82,86,134)(83,87,135)(84,88,136)(97,120,106)(98,115,107)(99,116,108)(100,117,103)(101,118,104)(102,119,105), (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), (1,137,97,34)(2,138,98,35)(3,133,99,36)(4,134,100,31)(5,135,101,32)(6,136,102,33)(7,17,48,130)(8,18,43,131)(9,13,44,132)(10,14,45,127)(11,15,46,128)(12,16,47,129)(19,52,62,94)(20,53,63,95)(21,54,64,96)(22,49,65,91)(23,50,66,92)(24,51,61,93)(25,41,68,60)(26,42,69,55)(27,37,70,56)(28,38,71,57)(29,39,72,58)(30,40,67,59)(73,81,116,124)(74,82,117,125)(75,83,118,126)(76,84,119,121)(77,79,120,122)(78,80,115,123)(85,108,110,139)(86,103,111,140)(87,104,112,141)(88,105,113,142)(89,106,114,143)(90,107,109,144), (1,62,97,19)(2,63,98,20)(3,64,99,21)(4,65,100,22)(5,66,101,23)(6,61,102,24)(7,121,48,84)(8,122,43,79)(9,123,44,80)(10,124,45,81)(11,125,46,82)(12,126,47,83)(13,115,132,78)(14,116,127,73)(15,117,128,74)(16,118,129,75)(17,119,130,76)(18,120,131,77)(25,85,68,110)(26,86,69,111)(27,87,70,112)(28,88,71,113)(29,89,72,114)(30,90,67,109)(31,91,134,49)(32,92,135,50)(33,93,136,51)(34,94,137,52)(35,95,138,53)(36,96,133,54)(37,141,56,104)(38,142,57,105)(39,143,58,106)(40,144,59,107)(41,139,60,108)(42,140,55,103), (1,75,139)(2,76,140)(3,77,141)(4,78,142)(5,73,143)(6,74,144)(7,55,138)(8,56,133)(9,57,134)(10,58,135)(11,59,136)(12,60,137)(13,88,49)(14,89,50)(15,90,51)(16,85,52)(17,86,53)(18,87,54)(19,126,25)(20,121,26)(21,122,27)(22,123,28)(23,124,29)(24,125,30)(31,44,38)(32,45,39)(33,46,40)(34,47,41)(35,48,42)(36,43,37)(61,82,67)(62,83,68)(63,84,69)(64,79,70)(65,80,71)(66,81,72)(91,132,113)(92,127,114)(93,128,109)(94,129,110)(95,130,111)(96,131,112)(97,118,108)(98,119,103)(99,120,104)(100,115,105)(101,116,106)(102,117,107) );
G=PermutationGroup([[(1,77,143),(2,78,144),(3,73,139),(4,74,140),(5,75,141),(6,76,142),(7,71,51),(8,72,52),(9,67,53),(10,68,54),(11,69,49),(12,70,50),(13,59,63),(14,60,64),(15,55,65),(16,56,66),(17,57,61),(18,58,62),(19,131,39),(20,132,40),(21,127,41),(22,128,42),(23,129,37),(24,130,38),(25,96,45),(26,91,46),(27,92,47),(28,93,48),(29,94,43),(30,95,44),(31,125,111),(32,126,112),(33,121,113),(34,122,114),(35,123,109),(36,124,110),(79,89,137),(80,90,138),(81,85,133),(82,86,134),(83,87,135),(84,88,136),(97,120,106),(98,115,107),(99,116,108),(100,117,103),(101,118,104),(102,119,105)], [(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)], [(1,137,97,34),(2,138,98,35),(3,133,99,36),(4,134,100,31),(5,135,101,32),(6,136,102,33),(7,17,48,130),(8,18,43,131),(9,13,44,132),(10,14,45,127),(11,15,46,128),(12,16,47,129),(19,52,62,94),(20,53,63,95),(21,54,64,96),(22,49,65,91),(23,50,66,92),(24,51,61,93),(25,41,68,60),(26,42,69,55),(27,37,70,56),(28,38,71,57),(29,39,72,58),(30,40,67,59),(73,81,116,124),(74,82,117,125),(75,83,118,126),(76,84,119,121),(77,79,120,122),(78,80,115,123),(85,108,110,139),(86,103,111,140),(87,104,112,141),(88,105,113,142),(89,106,114,143),(90,107,109,144)], [(1,62,97,19),(2,63,98,20),(3,64,99,21),(4,65,100,22),(5,66,101,23),(6,61,102,24),(7,121,48,84),(8,122,43,79),(9,123,44,80),(10,124,45,81),(11,125,46,82),(12,126,47,83),(13,115,132,78),(14,116,127,73),(15,117,128,74),(16,118,129,75),(17,119,130,76),(18,120,131,77),(25,85,68,110),(26,86,69,111),(27,87,70,112),(28,88,71,113),(29,89,72,114),(30,90,67,109),(31,91,134,49),(32,92,135,50),(33,93,136,51),(34,94,137,52),(35,95,138,53),(36,96,133,54),(37,141,56,104),(38,142,57,105),(39,143,58,106),(40,144,59,107),(41,139,60,108),(42,140,55,103)], [(1,75,139),(2,76,140),(3,77,141),(4,78,142),(5,73,143),(6,74,144),(7,55,138),(8,56,133),(9,57,134),(10,58,135),(11,59,136),(12,60,137),(13,88,49),(14,89,50),(15,90,51),(16,85,52),(17,86,53),(18,87,54),(19,126,25),(20,121,26),(21,122,27),(22,123,28),(23,124,29),(24,125,30),(31,44,38),(32,45,39),(33,46,40),(34,47,41),(35,48,42),(36,43,37),(61,82,67),(62,83,68),(63,84,69),(64,79,70),(65,80,71),(66,81,72),(91,132,113),(92,127,114),(93,128,109),(94,129,110),(95,130,111),(96,131,112),(97,118,108),(98,119,103),(99,120,104),(100,115,105),(101,116,106),(102,117,107)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Z | 4A | 4B | 6A | ··· | 6X | 6Y | ··· | 6BZ | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | - | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | SL2(𝔽3) | SL2(𝔽3) | C3×SL2(𝔽3) | A4 | C2×A4 | C3×A4 | C6×A4 |
| kernel | C3×C6×SL2(𝔽3) | C32×SL2(𝔽3) | C6×SL2(𝔽3) | Q8×C3×C6 | C3×SL2(𝔽3) | Q8×C32 | C3×C6 | C3×C6 | C6 | C62 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 1 | 24 | 2 | 24 | 2 | 2 | 4 | 48 | 1 | 1 | 8 | 8 |
Matrix representation of C3×C6×SL2(𝔽3) ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 4 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 4 | 10 |
| 0 | 10 | 9 |
| 1 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 0 |
| 3 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 12 | 3 |
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[4,0,0,0,3,0,0,0,3],[1,0,0,0,4,10,0,10,9],[1,0,0,0,0,1,0,12,0],[3,0,0,0,9,12,0,0,3] >;
C3×C6×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_3\times C_6\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C3xC6xSL(2,3)"); // GroupNames label
G:=SmallGroup(432,698);
// by ID
G=gap.SmallGroup(432,698);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1901,172,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^4=e^3=1,d^2=c^2,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,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations