direct product, non-abelian, soluble
Aliases: C9×C4.A4, C36.4A4, SL2(𝔽3)⋊2C18, C4.(C9×A4), C6.7(C6×A4), Q8⋊C9.4C6, C2.3(A4×C18), C12.3(C3×A4), (Q8×C9).7C6, C18.16(C2×A4), Q8.C18⋊3C3, Q8.1(C3×C18), (C9×SL2(𝔽3))⋊5C2, (C3×SL2(𝔽3)).9C6, C4○D4⋊1(C3×C9), (C9×C4○D4)⋊1C3, C3.1(C3×C4.A4), (C3×C4.A4).2C3, (C3×Q8).4(C3×C6), (C3×C4○D4).1C32, SmallGroup(432,329)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C9×C4.A4 |
Generators and relations for C9×C4.A4
G = < a,b,c,d,e | a9=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >
Subgroups: 167 in 65 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C9, C9, C32, C12, C12, C2×C6, C4○D4, C18, C18, C3×C6, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C3×C9, C36, C36, C2×C18, C3×C12, C4.A4, C3×C4○D4, C3×C18, Q8⋊C9, C2×C36, D4×C9, Q8×C9, C3×SL2(𝔽3), C3×C36, Q8.C18, C9×C4○D4, C3×C4.A4, C9×SL2(𝔽3), C9×C4.A4
Quotients: C1, C2, C3, C6, C9, C32, A4, C18, C3×C6, C2×A4, C3×C9, C3×A4, C4.A4, C3×C18, C6×A4, C9×A4, C3×C4.A4, A4×C18, C9×C4.A4
►On 144 points
Generators in S144
(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 77 138)(2 23 78 139)(3 24 79 140)(4 25 80 141)(5 26 81 142)(6 27 73 143)(7 19 74 144)(8 20 75 136)(9 21 76 137)(10 114 55 86)(11 115 56 87)(12 116 57 88)(13 117 58 89)(14 109 59 90)(15 110 60 82)(16 111 61 83)(17 112 62 84)(18 113 63 85)(28 132 50 105)(29 133 51 106)(30 134 52 107)(31 135 53 108)(32 127 54 100)(33 128 46 101)(34 129 47 102)(35 130 48 103)(36 131 49 104)(37 92 64 119)(38 93 65 120)(39 94 66 121)(40 95 67 122)(41 96 68 123)(42 97 69 124)(43 98 70 125)(44 99 71 126)(45 91 72 118)
(1 119 77 92)(2 120 78 93)(3 121 79 94)(4 122 80 95)(5 123 81 96)(6 124 73 97)(7 125 74 98)(8 126 75 99)(9 118 76 91)(10 50 55 28)(11 51 56 29)(12 52 57 30)(13 53 58 31)(14 54 59 32)(15 46 60 33)(16 47 61 34)(17 48 62 35)(18 49 63 36)(19 43 144 70)(20 44 136 71)(21 45 137 72)(22 37 138 64)(23 38 139 65)(24 39 140 66)(25 40 141 67)(26 41 142 68)(27 42 143 69)(82 128 110 101)(83 129 111 102)(84 130 112 103)(85 131 113 104)(86 132 114 105)(87 133 115 106)(88 134 116 107)(89 135 117 108)(90 127 109 100)
(1 132 77 105)(2 133 78 106)(3 134 79 107)(4 135 80 108)(5 127 81 100)(6 128 73 101)(7 129 74 102)(8 130 75 103)(9 131 76 104)(10 64 55 37)(11 65 56 38)(12 66 57 39)(13 67 58 40)(14 68 59 41)(15 69 60 42)(16 70 61 43)(17 71 62 44)(18 72 63 45)(19 47 144 34)(20 48 136 35)(21 49 137 36)(22 50 138 28)(23 51 139 29)(24 52 140 30)(25 53 141 31)(26 54 142 32)(27 46 143 33)(82 97 110 124)(83 98 111 125)(84 99 112 126)(85 91 113 118)(86 92 114 119)(87 93 115 120)(88 94 116 121)(89 95 117 122)(90 96 109 123)
(1 4 7)(2 5 8)(3 6 9)(10 67 34)(11 68 35)(12 69 36)(13 70 28)(14 71 29)(15 72 30)(16 64 31)(17 65 32)(18 66 33)(19 22 25)(20 23 26)(21 24 27)(37 53 61)(38 54 62)(39 46 63)(40 47 55)(41 48 56)(42 49 57)(43 50 58)(44 51 59)(45 52 60)(73 76 79)(74 77 80)(75 78 81)(82 91 107)(83 92 108)(84 93 100)(85 94 101)(86 95 102)(87 96 103)(88 97 104)(89 98 105)(90 99 106)(109 126 133)(110 118 134)(111 119 135)(112 120 127)(113 121 128)(114 122 129)(115 123 130)(116 124 131)(117 125 132)(136 139 142)(137 140 143)(138 141 144)
G:=sub<Sym(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), (1,22,77,138)(2,23,78,139)(3,24,79,140)(4,25,80,141)(5,26,81,142)(6,27,73,143)(7,19,74,144)(8,20,75,136)(9,21,76,137)(10,114,55,86)(11,115,56,87)(12,116,57,88)(13,117,58,89)(14,109,59,90)(15,110,60,82)(16,111,61,83)(17,112,62,84)(18,113,63,85)(28,132,50,105)(29,133,51,106)(30,134,52,107)(31,135,53,108)(32,127,54,100)(33,128,46,101)(34,129,47,102)(35,130,48,103)(36,131,49,104)(37,92,64,119)(38,93,65,120)(39,94,66,121)(40,95,67,122)(41,96,68,123)(42,97,69,124)(43,98,70,125)(44,99,71,126)(45,91,72,118), (1,119,77,92)(2,120,78,93)(3,121,79,94)(4,122,80,95)(5,123,81,96)(6,124,73,97)(7,125,74,98)(8,126,75,99)(9,118,76,91)(10,50,55,28)(11,51,56,29)(12,52,57,30)(13,53,58,31)(14,54,59,32)(15,46,60,33)(16,47,61,34)(17,48,62,35)(18,49,63,36)(19,43,144,70)(20,44,136,71)(21,45,137,72)(22,37,138,64)(23,38,139,65)(24,39,140,66)(25,40,141,67)(26,41,142,68)(27,42,143,69)(82,128,110,101)(83,129,111,102)(84,130,112,103)(85,131,113,104)(86,132,114,105)(87,133,115,106)(88,134,116,107)(89,135,117,108)(90,127,109,100), (1,132,77,105)(2,133,78,106)(3,134,79,107)(4,135,80,108)(5,127,81,100)(6,128,73,101)(7,129,74,102)(8,130,75,103)(9,131,76,104)(10,64,55,37)(11,65,56,38)(12,66,57,39)(13,67,58,40)(14,68,59,41)(15,69,60,42)(16,70,61,43)(17,71,62,44)(18,72,63,45)(19,47,144,34)(20,48,136,35)(21,49,137,36)(22,50,138,28)(23,51,139,29)(24,52,140,30)(25,53,141,31)(26,54,142,32)(27,46,143,33)(82,97,110,124)(83,98,111,125)(84,99,112,126)(85,91,113,118)(86,92,114,119)(87,93,115,120)(88,94,116,121)(89,95,117,122)(90,96,109,123), (1,4,7)(2,5,8)(3,6,9)(10,67,34)(11,68,35)(12,69,36)(13,70,28)(14,71,29)(15,72,30)(16,64,31)(17,65,32)(18,66,33)(19,22,25)(20,23,26)(21,24,27)(37,53,61)(38,54,62)(39,46,63)(40,47,55)(41,48,56)(42,49,57)(43,50,58)(44,51,59)(45,52,60)(73,76,79)(74,77,80)(75,78,81)(82,91,107)(83,92,108)(84,93,100)(85,94,101)(86,95,102)(87,96,103)(88,97,104)(89,98,105)(90,99,106)(109,126,133)(110,118,134)(111,119,135)(112,120,127)(113,121,128)(114,122,129)(115,123,130)(116,124,131)(117,125,132)(136,139,142)(137,140,143)(138,141,144)>;
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)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,22,77,138)(2,23,78,139)(3,24,79,140)(4,25,80,141)(5,26,81,142)(6,27,73,143)(7,19,74,144)(8,20,75,136)(9,21,76,137)(10,114,55,86)(11,115,56,87)(12,116,57,88)(13,117,58,89)(14,109,59,90)(15,110,60,82)(16,111,61,83)(17,112,62,84)(18,113,63,85)(28,132,50,105)(29,133,51,106)(30,134,52,107)(31,135,53,108)(32,127,54,100)(33,128,46,101)(34,129,47,102)(35,130,48,103)(36,131,49,104)(37,92,64,119)(38,93,65,120)(39,94,66,121)(40,95,67,122)(41,96,68,123)(42,97,69,124)(43,98,70,125)(44,99,71,126)(45,91,72,118), (1,119,77,92)(2,120,78,93)(3,121,79,94)(4,122,80,95)(5,123,81,96)(6,124,73,97)(7,125,74,98)(8,126,75,99)(9,118,76,91)(10,50,55,28)(11,51,56,29)(12,52,57,30)(13,53,58,31)(14,54,59,32)(15,46,60,33)(16,47,61,34)(17,48,62,35)(18,49,63,36)(19,43,144,70)(20,44,136,71)(21,45,137,72)(22,37,138,64)(23,38,139,65)(24,39,140,66)(25,40,141,67)(26,41,142,68)(27,42,143,69)(82,128,110,101)(83,129,111,102)(84,130,112,103)(85,131,113,104)(86,132,114,105)(87,133,115,106)(88,134,116,107)(89,135,117,108)(90,127,109,100), (1,132,77,105)(2,133,78,106)(3,134,79,107)(4,135,80,108)(5,127,81,100)(6,128,73,101)(7,129,74,102)(8,130,75,103)(9,131,76,104)(10,64,55,37)(11,65,56,38)(12,66,57,39)(13,67,58,40)(14,68,59,41)(15,69,60,42)(16,70,61,43)(17,71,62,44)(18,72,63,45)(19,47,144,34)(20,48,136,35)(21,49,137,36)(22,50,138,28)(23,51,139,29)(24,52,140,30)(25,53,141,31)(26,54,142,32)(27,46,143,33)(82,97,110,124)(83,98,111,125)(84,99,112,126)(85,91,113,118)(86,92,114,119)(87,93,115,120)(88,94,116,121)(89,95,117,122)(90,96,109,123), (1,4,7)(2,5,8)(3,6,9)(10,67,34)(11,68,35)(12,69,36)(13,70,28)(14,71,29)(15,72,30)(16,64,31)(17,65,32)(18,66,33)(19,22,25)(20,23,26)(21,24,27)(37,53,61)(38,54,62)(39,46,63)(40,47,55)(41,48,56)(42,49,57)(43,50,58)(44,51,59)(45,52,60)(73,76,79)(74,77,80)(75,78,81)(82,91,107)(83,92,108)(84,93,100)(85,94,101)(86,95,102)(87,96,103)(88,97,104)(89,98,105)(90,99,106)(109,126,133)(110,118,134)(111,119,135)(112,120,127)(113,121,128)(114,122,129)(115,123,130)(116,124,131)(117,125,132)(136,139,142)(137,140,143)(138,141,144) );
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),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144)], [(1,22,77,138),(2,23,78,139),(3,24,79,140),(4,25,80,141),(5,26,81,142),(6,27,73,143),(7,19,74,144),(8,20,75,136),(9,21,76,137),(10,114,55,86),(11,115,56,87),(12,116,57,88),(13,117,58,89),(14,109,59,90),(15,110,60,82),(16,111,61,83),(17,112,62,84),(18,113,63,85),(28,132,50,105),(29,133,51,106),(30,134,52,107),(31,135,53,108),(32,127,54,100),(33,128,46,101),(34,129,47,102),(35,130,48,103),(36,131,49,104),(37,92,64,119),(38,93,65,120),(39,94,66,121),(40,95,67,122),(41,96,68,123),(42,97,69,124),(43,98,70,125),(44,99,71,126),(45,91,72,118)], [(1,119,77,92),(2,120,78,93),(3,121,79,94),(4,122,80,95),(5,123,81,96),(6,124,73,97),(7,125,74,98),(8,126,75,99),(9,118,76,91),(10,50,55,28),(11,51,56,29),(12,52,57,30),(13,53,58,31),(14,54,59,32),(15,46,60,33),(16,47,61,34),(17,48,62,35),(18,49,63,36),(19,43,144,70),(20,44,136,71),(21,45,137,72),(22,37,138,64),(23,38,139,65),(24,39,140,66),(25,40,141,67),(26,41,142,68),(27,42,143,69),(82,128,110,101),(83,129,111,102),(84,130,112,103),(85,131,113,104),(86,132,114,105),(87,133,115,106),(88,134,116,107),(89,135,117,108),(90,127,109,100)], [(1,132,77,105),(2,133,78,106),(3,134,79,107),(4,135,80,108),(5,127,81,100),(6,128,73,101),(7,129,74,102),(8,130,75,103),(9,131,76,104),(10,64,55,37),(11,65,56,38),(12,66,57,39),(13,67,58,40),(14,68,59,41),(15,69,60,42),(16,70,61,43),(17,71,62,44),(18,72,63,45),(19,47,144,34),(20,48,136,35),(21,49,137,36),(22,50,138,28),(23,51,139,29),(24,52,140,30),(25,53,141,31),(26,54,142,32),(27,46,143,33),(82,97,110,124),(83,98,111,125),(84,99,112,126),(85,91,113,118),(86,92,114,119),(87,93,115,120),(88,94,116,121),(89,95,117,122),(90,96,109,123)], [(1,4,7),(2,5,8),(3,6,9),(10,67,34),(11,68,35),(12,69,36),(13,70,28),(14,71,29),(15,72,30),(16,64,31),(17,65,32),(18,66,33),(19,22,25),(20,23,26),(21,24,27),(37,53,61),(38,54,62),(39,46,63),(40,47,55),(41,48,56),(42,49,57),(43,50,58),(44,51,59),(45,52,60),(73,76,79),(74,77,80),(75,78,81),(82,91,107),(83,92,108),(84,93,100),(85,94,101),(86,95,102),(87,96,103),(88,97,104),(89,98,105),(90,99,106),(109,126,133),(110,118,134),(111,119,135),(112,120,127),(113,121,128),(114,122,129),(115,123,130),(116,124,131),(117,125,132),(136,139,142),(137,140,143),(138,141,144)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 9A | ··· | 9F | 9G | ··· | 9R | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | 12R | 18A | ··· | 18F | 18G | ··· | 18R | 18S | ··· | 18X | 36A | ··· | 36L | 36M | ··· | 36AJ | 36AK | ··· | 36AP |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 6 | 1 | 1 | 4 | ··· | 4 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | C9 | C18 | C4.A4 | C3×C4.A4 | C9×C4.A4 | A4 | C2×A4 | C3×A4 | C6×A4 | C9×A4 | A4×C18 |
| kernel | C9×C4.A4 | C9×SL2(𝔽3) | Q8.C18 | C9×C4○D4 | C3×C4.A4 | Q8⋊C9 | Q8×C9 | C3×SL2(𝔽3) | C4.A4 | SL2(𝔽3) | C9 | C3 | C1 | C36 | C18 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 18 | 18 | 6 | 12 | 36 | 1 | 1 | 2 | 2 | 6 | 6 |
Matrix representation of C9×C4.A4 ►in GL2(𝔽37) generated by
| 9 | 0 |
| 0 | 9 |
| 6 | 0 |
| 0 | 6 |
| 26 | 10 |
| 10 | 11 |
| 0 | 36 |
| 1 | 0 |
| 0 | 27 |
| 26 | 36 |
G:=sub<GL(2,GF(37))| [9,0,0,9],[6,0,0,6],[26,10,10,11],[0,1,36,0],[0,26,27,36] >;
C9×C4.A4 in GAP, Magma, Sage, TeX
C_9\times C_4.A_4
% in TeX
G:=Group("C9xC4.A4"); // GroupNames label
G:=SmallGroup(432,329);
// by ID
G=gap.SmallGroup(432,329);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,79,1901,172,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^4=e^3=1,c^2=d^2=b^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=b^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations