metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.23D4, (C2×Q8)⋊6D9, (Q8×C18)⋊4C2, D18⋊C4⋊16C2, (C4×Dic9)⋊7C2, (C2×D36).9C2, (C2×C4).57D18, (C2×C12).66D6, C18.58(C2×D4), C9⋊4(C4.4D4), (C6×Q8).17S3, C4.11(C9⋊D4), C18.37(C4○D4), C12.20(C3⋊D4), (C2×C36).64C22, (C2×C18).59C23, C3.(C12.23D4), C2.9(Q8⋊3D9), C6.45(Q8⋊3S3), C22.65(C22×D9), (C2×Dic9).43C22, (C22×D9).13C22, C2.22(C2×C9⋊D4), C6.106(C2×C3⋊D4), (C2×C6).216(C22×S3), SmallGroup(288,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.23D4
G = < a,b,c | a36=b4=c2=1, bab-1=a17, cac=a-1, cbc=a18b-1 >
Subgroups: 564 in 114 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C9, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, D9, C18, C18, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4.4D4, Dic9, C36, C36, D18, C2×C18, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, D36, C2×Dic9, C2×C36, C2×C36, Q8×C9, C22×D9, C12.23D4, C4×Dic9, D18⋊C4, C2×D36, Q8×C18, C36.23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C3⋊D4, C22×S3, C4.4D4, D18, Q8⋊3S3, C2×C3⋊D4, C9⋊D4, C22×D9, C12.23D4, Q8⋊3D9, C2×C9⋊D4, C36.23D4
►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 92 54 120)(2 73 55 137)(3 90 56 118)(4 107 57 135)(5 88 58 116)(6 105 59 133)(7 86 60 114)(8 103 61 131)(9 84 62 112)(10 101 63 129)(11 82 64 110)(12 99 65 127)(13 80 66 144)(14 97 67 125)(15 78 68 142)(16 95 69 123)(17 76 70 140)(18 93 71 121)(19 74 72 138)(20 91 37 119)(21 108 38 136)(22 89 39 117)(23 106 40 134)(24 87 41 115)(25 104 42 132)(26 85 43 113)(27 102 44 130)(28 83 45 111)(29 100 46 128)(30 81 47 109)(31 98 48 126)(32 79 49 143)(33 96 50 124)(34 77 51 141)(35 94 52 122)(36 75 53 139)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(73 121)(74 120)(75 119)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 144)(87 143)(88 142)(89 141)(90 140)(91 139)(92 138)(93 137)(94 136)(95 135)(96 134)(97 133)(98 132)(99 131)(100 130)(101 129)(102 128)(103 127)(104 126)(105 125)(106 124)(107 123)(108 122)
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), (1,92,54,120)(2,73,55,137)(3,90,56,118)(4,107,57,135)(5,88,58,116)(6,105,59,133)(7,86,60,114)(8,103,61,131)(9,84,62,112)(10,101,63,129)(11,82,64,110)(12,99,65,127)(13,80,66,144)(14,97,67,125)(15,78,68,142)(16,95,69,123)(17,76,70,140)(18,93,71,121)(19,74,72,138)(20,91,37,119)(21,108,38,136)(22,89,39,117)(23,106,40,134)(24,87,41,115)(25,104,42,132)(26,85,43,113)(27,102,44,130)(28,83,45,111)(29,100,46,128)(30,81,47,109)(31,98,48,126)(32,79,49,143)(33,96,50,124)(34,77,51,141)(35,94,52,122)(36,75,53,139), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,144)(87,143)(88,142)(89,141)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,131)(100,130)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122)>;
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), (1,92,54,120)(2,73,55,137)(3,90,56,118)(4,107,57,135)(5,88,58,116)(6,105,59,133)(7,86,60,114)(8,103,61,131)(9,84,62,112)(10,101,63,129)(11,82,64,110)(12,99,65,127)(13,80,66,144)(14,97,67,125)(15,78,68,142)(16,95,69,123)(17,76,70,140)(18,93,71,121)(19,74,72,138)(20,91,37,119)(21,108,38,136)(22,89,39,117)(23,106,40,134)(24,87,41,115)(25,104,42,132)(26,85,43,113)(27,102,44,130)(28,83,45,111)(29,100,46,128)(30,81,47,109)(31,98,48,126)(32,79,49,143)(33,96,50,124)(34,77,51,141)(35,94,52,122)(36,75,53,139), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,144)(87,143)(88,142)(89,141)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,131)(100,130)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122) );
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)], [(1,92,54,120),(2,73,55,137),(3,90,56,118),(4,107,57,135),(5,88,58,116),(6,105,59,133),(7,86,60,114),(8,103,61,131),(9,84,62,112),(10,101,63,129),(11,82,64,110),(12,99,65,127),(13,80,66,144),(14,97,67,125),(15,78,68,142),(16,95,69,123),(17,76,70,140),(18,93,71,121),(19,74,72,138),(20,91,37,119),(21,108,38,136),(22,89,39,117),(23,106,40,134),(24,87,41,115),(25,104,42,132),(26,85,43,113),(27,102,44,130),(28,83,45,111),(29,100,46,128),(30,81,47,109),(31,98,48,126),(32,79,49,143),(33,96,50,124),(34,77,51,141),(35,94,52,122),(36,75,53,139)], [(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(73,121),(74,120),(75,119),(76,118),(77,117),(78,116),(79,115),(80,114),(81,113),(82,112),(83,111),(84,110),(85,109),(86,144),(87,143),(88,142),(89,141),(90,140),(91,139),(92,138),(93,137),(94,136),(95,135),(96,134),(97,133),(98,132),(99,131),(100,130),(101,129),(102,128),(103,127),(104,126),(105,125),(106,124),(107,123),(108,122)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C4○D4 | D9 | C3⋊D4 | D18 | C9⋊D4 | Q8⋊3S3 | Q8⋊3D9 |
| kernel | C36.23D4 | C4×Dic9 | D18⋊C4 | C2×D36 | Q8×C18 | C6×Q8 | C36 | C2×C12 | C18 | C2×Q8 | C12 | C2×C4 | C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 4 | 9 | 12 | 2 | 6 |
Matrix representation of C36.23D4 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 0 | 0 |
| 0 | 0 | 26 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 18 | 36 |
| 21 | 6 | 0 | 0 | 0 | 0 |
| 25 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 13 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 30 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 18 | 36 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,6,26,0,0,0,0,11,17,0,0,0,0,0,0,1,18,0,0,0,0,4,36],[21,25,0,0,0,0,6,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,3,0,0,0,0,13,6],[1,30,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0,0,36] >;
C36.23D4 in GAP, Magma, Sage, TeX
C_{36}._{23}D_4 % in TeX
G:=Group("C36.23D4"); // GroupNames label
G:=SmallGroup(288,157);
// by ID
G=gap.SmallGroup(288,157);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=a^18*b^-1>;
// generators/relations