Aliases: C36.A4, C4.(C9⋊A4), Q8⋊C9.2C6, C18.A4⋊3C2, C9⋊2(C4.A4), C6.22(C6×A4), C18.6(C2×A4), C12.4(C3×A4), (Q8×C9).8C6, Q8.C18⋊1C3, C4○D4⋊13- 1+2, (C3×SL2(𝔽3)).3C6, Q8.1(C2×3- 1+2), C2.3(C2×C9⋊A4), (C3×C4.A4).C3, (C9×C4○D4)⋊2C3, C3.3(C3×C4.A4), (C3×Q8).5(C3×C6), (C3×C4○D4).2C32, SmallGroup(432,330)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.A4
G = < a,b,c,d | a36=d3=1, b2=c2=a18, ab=ba, ac=ca, dad-1=a25, cbc-1=a18b, dbd-1=a18bc, dcd-1=b >
Smallest permutation representation of C36.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 67 19 49)(2 68 20 50)(3 69 21 51)(4 70 22 52)(5 71 23 53)(6 72 24 54)(7 37 25 55)(8 38 26 56)(9 39 27 57)(10 40 28 58)(11 41 29 59)(12 42 30 60)(13 43 31 61)(14 44 32 62)(15 45 33 63)(16 46 34 64)(17 47 35 65)(18 48 36 66)(73 142 91 124)(74 143 92 125)(75 144 93 126)(76 109 94 127)(77 110 95 128)(78 111 96 129)(79 112 97 130)(80 113 98 131)(81 114 99 132)(82 115 100 133)(83 116 101 134)(84 117 102 135)(85 118 103 136)(86 119 104 137)(87 120 105 138)(88 121 106 139)(89 122 107 140)(90 123 108 141)
(1 138 19 120)(2 139 20 121)(3 140 21 122)(4 141 22 123)(5 142 23 124)(6 143 24 125)(7 144 25 126)(8 109 26 127)(9 110 27 128)(10 111 28 129)(11 112 29 130)(12 113 30 131)(13 114 31 132)(14 115 32 133)(15 116 33 134)(16 117 34 135)(17 118 35 136)(18 119 36 137)(37 75 55 93)(38 76 56 94)(39 77 57 95)(40 78 58 96)(41 79 59 97)(42 80 60 98)(43 81 61 99)(44 82 62 100)(45 83 63 101)(46 84 64 102)(47 85 65 103)(48 86 66 104)(49 87 67 105)(50 88 68 106)(51 89 69 107)(52 90 70 108)(53 91 71 73)(54 92 72 74)
(2 14 26)(3 27 15)(5 17 29)(6 30 18)(8 20 32)(9 33 21)(11 23 35)(12 36 24)(37 144 93)(38 121 82)(39 134 107)(40 111 96)(41 124 85)(42 137 74)(43 114 99)(44 127 88)(45 140 77)(46 117 102)(47 130 91)(48 143 80)(49 120 105)(50 133 94)(51 110 83)(52 123 108)(53 136 97)(54 113 86)(55 126 75)(56 139 100)(57 116 89)(58 129 78)(59 142 103)(60 119 92)(61 132 81)(62 109 106)(63 122 95)(64 135 84)(65 112 73)(66 125 98)(67 138 87)(68 115 76)(69 128 101)(70 141 90)(71 118 79)(72 131 104)
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), (1,67,19,49)(2,68,20,50)(3,69,21,51)(4,70,22,52)(5,71,23,53)(6,72,24,54)(7,37,25,55)(8,38,26,56)(9,39,27,57)(10,40,28,58)(11,41,29,59)(12,42,30,60)(13,43,31,61)(14,44,32,62)(15,45,33,63)(16,46,34,64)(17,47,35,65)(18,48,36,66)(73,142,91,124)(74,143,92,125)(75,144,93,126)(76,109,94,127)(77,110,95,128)(78,111,96,129)(79,112,97,130)(80,113,98,131)(81,114,99,132)(82,115,100,133)(83,116,101,134)(84,117,102,135)(85,118,103,136)(86,119,104,137)(87,120,105,138)(88,121,106,139)(89,122,107,140)(90,123,108,141), (1,138,19,120)(2,139,20,121)(3,140,21,122)(4,141,22,123)(5,142,23,124)(6,143,24,125)(7,144,25,126)(8,109,26,127)(9,110,27,128)(10,111,28,129)(11,112,29,130)(12,113,30,131)(13,114,31,132)(14,115,32,133)(15,116,33,134)(16,117,34,135)(17,118,35,136)(18,119,36,137)(37,75,55,93)(38,76,56,94)(39,77,57,95)(40,78,58,96)(41,79,59,97)(42,80,60,98)(43,81,61,99)(44,82,62,100)(45,83,63,101)(46,84,64,102)(47,85,65,103)(48,86,66,104)(49,87,67,105)(50,88,68,106)(51,89,69,107)(52,90,70,108)(53,91,71,73)(54,92,72,74), (2,14,26)(3,27,15)(5,17,29)(6,30,18)(8,20,32)(9,33,21)(11,23,35)(12,36,24)(37,144,93)(38,121,82)(39,134,107)(40,111,96)(41,124,85)(42,137,74)(43,114,99)(44,127,88)(45,140,77)(46,117,102)(47,130,91)(48,143,80)(49,120,105)(50,133,94)(51,110,83)(52,123,108)(53,136,97)(54,113,86)(55,126,75)(56,139,100)(57,116,89)(58,129,78)(59,142,103)(60,119,92)(61,132,81)(62,109,106)(63,122,95)(64,135,84)(65,112,73)(66,125,98)(67,138,87)(68,115,76)(69,128,101)(70,141,90)(71,118,79)(72,131,104)>;
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), (1,67,19,49)(2,68,20,50)(3,69,21,51)(4,70,22,52)(5,71,23,53)(6,72,24,54)(7,37,25,55)(8,38,26,56)(9,39,27,57)(10,40,28,58)(11,41,29,59)(12,42,30,60)(13,43,31,61)(14,44,32,62)(15,45,33,63)(16,46,34,64)(17,47,35,65)(18,48,36,66)(73,142,91,124)(74,143,92,125)(75,144,93,126)(76,109,94,127)(77,110,95,128)(78,111,96,129)(79,112,97,130)(80,113,98,131)(81,114,99,132)(82,115,100,133)(83,116,101,134)(84,117,102,135)(85,118,103,136)(86,119,104,137)(87,120,105,138)(88,121,106,139)(89,122,107,140)(90,123,108,141), (1,138,19,120)(2,139,20,121)(3,140,21,122)(4,141,22,123)(5,142,23,124)(6,143,24,125)(7,144,25,126)(8,109,26,127)(9,110,27,128)(10,111,28,129)(11,112,29,130)(12,113,30,131)(13,114,31,132)(14,115,32,133)(15,116,33,134)(16,117,34,135)(17,118,35,136)(18,119,36,137)(37,75,55,93)(38,76,56,94)(39,77,57,95)(40,78,58,96)(41,79,59,97)(42,80,60,98)(43,81,61,99)(44,82,62,100)(45,83,63,101)(46,84,64,102)(47,85,65,103)(48,86,66,104)(49,87,67,105)(50,88,68,106)(51,89,69,107)(52,90,70,108)(53,91,71,73)(54,92,72,74), (2,14,26)(3,27,15)(5,17,29)(6,30,18)(8,20,32)(9,33,21)(11,23,35)(12,36,24)(37,144,93)(38,121,82)(39,134,107)(40,111,96)(41,124,85)(42,137,74)(43,114,99)(44,127,88)(45,140,77)(46,117,102)(47,130,91)(48,143,80)(49,120,105)(50,133,94)(51,110,83)(52,123,108)(53,136,97)(54,113,86)(55,126,75)(56,139,100)(57,116,89)(58,129,78)(59,142,103)(60,119,92)(61,132,81)(62,109,106)(63,122,95)(64,135,84)(65,112,73)(66,125,98)(67,138,87)(68,115,76)(69,128,101)(70,141,90)(71,118,79)(72,131,104) );
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)], [(1,67,19,49),(2,68,20,50),(3,69,21,51),(4,70,22,52),(5,71,23,53),(6,72,24,54),(7,37,25,55),(8,38,26,56),(9,39,27,57),(10,40,28,58),(11,41,29,59),(12,42,30,60),(13,43,31,61),(14,44,32,62),(15,45,33,63),(16,46,34,64),(17,47,35,65),(18,48,36,66),(73,142,91,124),(74,143,92,125),(75,144,93,126),(76,109,94,127),(77,110,95,128),(78,111,96,129),(79,112,97,130),(80,113,98,131),(81,114,99,132),(82,115,100,133),(83,116,101,134),(84,117,102,135),(85,118,103,136),(86,119,104,137),(87,120,105,138),(88,121,106,139),(89,122,107,140),(90,123,108,141)], [(1,138,19,120),(2,139,20,121),(3,140,21,122),(4,141,22,123),(5,142,23,124),(6,143,24,125),(7,144,25,126),(8,109,26,127),(9,110,27,128),(10,111,28,129),(11,112,29,130),(12,113,30,131),(13,114,31,132),(14,115,32,133),(15,116,33,134),(16,117,34,135),(17,118,35,136),(18,119,36,137),(37,75,55,93),(38,76,56,94),(39,77,57,95),(40,78,58,96),(41,79,59,97),(42,80,60,98),(43,81,61,99),(44,82,62,100),(45,83,63,101),(46,84,64,102),(47,85,65,103),(48,86,66,104),(49,87,67,105),(50,88,68,106),(51,89,69,107),(52,90,70,108),(53,91,71,73),(54,92,72,74)], [(2,14,26),(3,27,15),(5,17,29),(6,30,18),(8,20,32),(9,33,21),(11,23,35),(12,36,24),(37,144,93),(38,121,82),(39,134,107),(40,111,96),(41,124,85),(42,137,74),(43,114,99),(44,127,88),(45,140,77),(46,117,102),(47,130,91),(48,143,80),(49,120,105),(50,133,94),(51,110,83),(52,123,108),(53,136,97),(54,113,86),(55,126,75),(56,139,100),(57,116,89),(58,129,78),(59,142,103),(60,119,92),(61,132,81),(62,109,106),(63,122,95),(64,135,84),(65,112,73),(66,125,98),(67,138,87),(68,115,76),(69,128,101),(70,141,90),(71,118,79),(72,131,104)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 18A | 18B | 18C | ··· | 18H | 18I | 18J | 18K | 18L | 36A | 36B | 36C | 36D | 36E | ··· | 36J | 36K | ··· | 36R |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | ··· | 18 | 18 | 18 | 18 | 18 | 36 | 36 | 36 | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 1 | 1 | 12 | 12 | 1 | 1 | 6 | 1 | 1 | 6 | 6 | 12 | 12 | 3 | 3 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 12 | 12 | 3 | 3 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 12 | ··· | 12 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | C4.A4 | C3×C4.A4 | A4 | C2×A4 | 3- 1+2 | C3×A4 | C2×3- 1+2 | C6×A4 | C9⋊A4 | C2×C9⋊A4 | C36.A4 |
| kernel | C36.A4 | C18.A4 | Q8.C18 | C9×C4○D4 | C3×C4.A4 | Q8⋊C9 | Q8×C9 | C3×SL2(𝔽3) | C9 | C3 | C36 | C18 | C4○D4 | C12 | Q8 | C6 | C4 | C2 | C1 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 4 |
Matrix representation of C36.A4 ►in GL5(𝔽37)
| 6 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 20 | 24 | 13 |
| 0 | 0 | 5 | 2 | 32 |
| 0 | 0 | 18 | 19 | 15 |
| 10 | 11 | 0 | 0 | 0 |
| 11 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 1 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 1 | 36 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 1 | 0 | 36 |
| 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 |
| 27 | 26 | 0 | 0 | 0 |
| 0 | 0 | 26 | 11 | 0 |
| 0 | 0 | 0 | 11 | 26 |
| 0 | 0 | 0 | 11 | 0 |
G:=sub<GL(5,GF(37))| [6,0,0,0,0,0,6,0,0,0,0,0,20,5,18,0,0,24,2,19,0,0,13,32,15],[10,11,0,0,0,11,27,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[0,1,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,36,36,36],[1,27,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,11,11,11,0,0,0,26,0] >;
C36.A4 in GAP, Magma, Sage, TeX
C_{36}.A_4 % in TeX
G:=Group("C36.A4"); // GroupNames label
G:=SmallGroup(432,330);
// by ID
G=gap.SmallGroup(432,330);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,261,79,1901,172,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^36=d^3=1,b^2=c^2=a^18,a*b=b*a,a*c=c*a,d*a*d^-1=a^25,c*b*c^-1=a^18*b,d*b*d^-1=a^18*b*c,d*c*d^-1=b>;
// generators/relations
Export