metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.55D4, C18.6M4(2), C23.3Dic9, (C2×C18)⋊2C8, C22⋊2(C9⋊C8), C9⋊2(C22⋊C8), (C2×C36).5C4, C18.10(C2×C8), (C2×C4).94D18, (C2×C4).4Dic9, (C22×C4).3D9, (C2×C12).408D6, C4.30(C9⋊D4), (C22×C18).5C4, (C2×C12).7Dic3, (C22×C36).11C2, (C22×C12).31S3, C3.(C12.55D4), C6.6(C4.Dic3), C2.3(C4.Dic9), C12.125(C3⋊D4), C22.9(C2×Dic9), C18.12(C22⋊C4), (C2×C36).106C22, (C22×C6).11Dic3, C6.12(C6.D4), C2.1(C18.D4), C2.5(C2×C9⋊C8), (C2×C9⋊C8)⋊10C2, C6.10(C2×C3⋊C8), (C2×C6).5(C3⋊C8), (C2×C18).27(C2×C4), (C2×C6).31(C2×Dic3), SmallGroup(288,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.55D4
G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=cac-1=a17, cbc-1=a27b3 >
Subgroups: 172 in 75 conjugacy classes, 42 normal (28 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C22×C4, C18, C18, C3⋊C8, C2×C12, C2×C12, C22×C6, C22⋊C8, C36, C36, C2×C18, C2×C18, C2×C18, C2×C3⋊C8, C22×C12, C9⋊C8, C2×C36, C2×C36, C22×C18, C12.55D4, C2×C9⋊C8, C22×C36, C36.55D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), D9, C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, Dic9, D18, C2×C3⋊C8, C4.Dic3, C6.D4, C9⋊C8, C2×Dic9, C9⋊D4, C12.55D4, C2×C9⋊C8, C4.Dic9, C18.D4, C36.55D4
►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 65 84 125 19 47 102 143)(2 46 85 142 20 64 103 124)(3 63 86 123 21 45 104 141)(4 44 87 140 22 62 105 122)(5 61 88 121 23 43 106 139)(6 42 89 138 24 60 107 120)(7 59 90 119 25 41 108 137)(8 40 91 136 26 58 73 118)(9 57 92 117 27 39 74 135)(10 38 93 134 28 56 75 116)(11 55 94 115 29 37 76 133)(12 72 95 132 30 54 77 114)(13 53 96 113 31 71 78 131)(14 70 97 130 32 52 79 112)(15 51 98 111 33 69 80 129)(16 68 99 128 34 50 81 110)(17 49 100 109 35 67 82 127)(18 66 101 126 36 48 83 144)
(1 116 10 125 19 134 28 143)(2 133 11 142 20 115 29 124)(3 114 12 123 21 132 30 141)(4 131 13 140 22 113 31 122)(5 112 14 121 23 130 32 139)(6 129 15 138 24 111 33 120)(7 110 16 119 25 128 34 137)(8 127 17 136 26 109 35 118)(9 144 18 117 27 126 36 135)(37 76 46 85 55 94 64 103)(38 93 47 102 56 75 65 84)(39 74 48 83 57 92 66 101)(40 91 49 100 58 73 67 82)(41 108 50 81 59 90 68 99)(42 89 51 98 60 107 69 80)(43 106 52 79 61 88 70 97)(44 87 53 96 62 105 71 78)(45 104 54 77 63 86 72 95)
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), (1,65,84,125,19,47,102,143)(2,46,85,142,20,64,103,124)(3,63,86,123,21,45,104,141)(4,44,87,140,22,62,105,122)(5,61,88,121,23,43,106,139)(6,42,89,138,24,60,107,120)(7,59,90,119,25,41,108,137)(8,40,91,136,26,58,73,118)(9,57,92,117,27,39,74,135)(10,38,93,134,28,56,75,116)(11,55,94,115,29,37,76,133)(12,72,95,132,30,54,77,114)(13,53,96,113,31,71,78,131)(14,70,97,130,32,52,79,112)(15,51,98,111,33,69,80,129)(16,68,99,128,34,50,81,110)(17,49,100,109,35,67,82,127)(18,66,101,126,36,48,83,144), (1,116,10,125,19,134,28,143)(2,133,11,142,20,115,29,124)(3,114,12,123,21,132,30,141)(4,131,13,140,22,113,31,122)(5,112,14,121,23,130,32,139)(6,129,15,138,24,111,33,120)(7,110,16,119,25,128,34,137)(8,127,17,136,26,109,35,118)(9,144,18,117,27,126,36,135)(37,76,46,85,55,94,64,103)(38,93,47,102,56,75,65,84)(39,74,48,83,57,92,66,101)(40,91,49,100,58,73,67,82)(41,108,50,81,59,90,68,99)(42,89,51,98,60,107,69,80)(43,106,52,79,61,88,70,97)(44,87,53,96,62,105,71,78)(45,104,54,77,63,86,72,95)>;
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), (1,65,84,125,19,47,102,143)(2,46,85,142,20,64,103,124)(3,63,86,123,21,45,104,141)(4,44,87,140,22,62,105,122)(5,61,88,121,23,43,106,139)(6,42,89,138,24,60,107,120)(7,59,90,119,25,41,108,137)(8,40,91,136,26,58,73,118)(9,57,92,117,27,39,74,135)(10,38,93,134,28,56,75,116)(11,55,94,115,29,37,76,133)(12,72,95,132,30,54,77,114)(13,53,96,113,31,71,78,131)(14,70,97,130,32,52,79,112)(15,51,98,111,33,69,80,129)(16,68,99,128,34,50,81,110)(17,49,100,109,35,67,82,127)(18,66,101,126,36,48,83,144), (1,116,10,125,19,134,28,143)(2,133,11,142,20,115,29,124)(3,114,12,123,21,132,30,141)(4,131,13,140,22,113,31,122)(5,112,14,121,23,130,32,139)(6,129,15,138,24,111,33,120)(7,110,16,119,25,128,34,137)(8,127,17,136,26,109,35,118)(9,144,18,117,27,126,36,135)(37,76,46,85,55,94,64,103)(38,93,47,102,56,75,65,84)(39,74,48,83,57,92,66,101)(40,91,49,100,58,73,67,82)(41,108,50,81,59,90,68,99)(42,89,51,98,60,107,69,80)(43,106,52,79,61,88,70,97)(44,87,53,96,62,105,71,78)(45,104,54,77,63,86,72,95) );
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)], [(1,65,84,125,19,47,102,143),(2,46,85,142,20,64,103,124),(3,63,86,123,21,45,104,141),(4,44,87,140,22,62,105,122),(5,61,88,121,23,43,106,139),(6,42,89,138,24,60,107,120),(7,59,90,119,25,41,108,137),(8,40,91,136,26,58,73,118),(9,57,92,117,27,39,74,135),(10,38,93,134,28,56,75,116),(11,55,94,115,29,37,76,133),(12,72,95,132,30,54,77,114),(13,53,96,113,31,71,78,131),(14,70,97,130,32,52,79,112),(15,51,98,111,33,69,80,129),(16,68,99,128,34,50,81,110),(17,49,100,109,35,67,82,127),(18,66,101,126,36,48,83,144)], [(1,116,10,125,19,134,28,143),(2,133,11,142,20,115,29,124),(3,114,12,123,21,132,30,141),(4,131,13,140,22,113,31,122),(5,112,14,121,23,130,32,139),(6,129,15,138,24,111,33,120),(7,110,16,119,25,128,34,137),(8,127,17,136,26,109,35,118),(9,144,18,117,27,126,36,135),(37,76,46,85,55,94,64,103),(38,93,47,102,56,75,65,84),(39,74,48,83,57,92,66,101),(40,91,49,100,58,73,67,82),(41,108,50,81,59,90,68,99),(42,89,51,98,60,107,69,80),(43,106,52,79,61,88,70,97),(44,87,53,96,62,105,71,78),(45,104,54,77,63,86,72,95)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Dic3 | D6 | Dic3 | M4(2) | D9 | C3⋊D4 | C3⋊C8 | Dic9 | D18 | Dic9 | C4.Dic3 | C9⋊D4 | C9⋊C8 | C4.Dic9 |
| kernel | C36.55D4 | C2×C9⋊C8 | C22×C36 | C2×C36 | C22×C18 | C2×C18 | C22×C12 | C36 | C2×C12 | C2×C12 | C22×C6 | C18 | C22×C4 | C12 | C2×C6 | C2×C4 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 3 | 4 | 4 | 3 | 3 | 3 | 4 | 12 | 12 | 12 |
Matrix representation of C36.55D4 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 37 |
| 22 | 48 | 0 | 0 |
| 65 | 51 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 22 | 48 | 0 | 0 |
| 0 | 51 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,2,0,0,0,0,37],[22,65,0,0,48,51,0,0,0,0,0,72,0,0,1,0],[22,0,0,0,48,51,0,0,0,0,0,1,0,0,1,0] >;
C36.55D4 in GAP, Magma, Sage, TeX
C_{36}._{55}D_4 % in TeX
G:=Group("C36.55D4"); // GroupNames label
G:=SmallGroup(288,37);
// by ID
G=gap.SmallGroup(288,37);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^27*b^3>;
// generators/relations