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 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C9, C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C22×C4, C18 [×3], C18 [×2], C3⋊C8 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C22⋊C8, C36 [×2], C36, C2×C18, C2×C18 [×2], C2×C18 [×2], C2×C3⋊C8 [×2], C22×C12, C9⋊C8 [×2], C2×C36 [×2], C2×C36 [×2], C22×C18, C12.55D4, C2×C9⋊C8 [×2], C22×C36, C36.55D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×C8, M4(2), D9, C3⋊C8 [×2], C2×Dic3, C3⋊D4 [×2], C22⋊C8, Dic9 [×2], D18, C2×C3⋊C8, C4.Dic3, C6.D4, C9⋊C8 [×2], C2×Dic9, C9⋊D4 [×2], 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 61 103 122 19 43 85 140)(2 42 104 139 20 60 86 121)(3 59 105 120 21 41 87 138)(4 40 106 137 22 58 88 119)(5 57 107 118 23 39 89 136)(6 38 108 135 24 56 90 117)(7 55 73 116 25 37 91 134)(8 72 74 133 26 54 92 115)(9 53 75 114 27 71 93 132)(10 70 76 131 28 52 94 113)(11 51 77 112 29 69 95 130)(12 68 78 129 30 50 96 111)(13 49 79 110 31 67 97 128)(14 66 80 127 32 48 98 109)(15 47 81 144 33 65 99 126)(16 64 82 125 34 46 100 143)(17 45 83 142 35 63 101 124)(18 62 84 123 36 44 102 141)
(1 113 10 122 19 131 28 140)(2 130 11 139 20 112 29 121)(3 111 12 120 21 129 30 138)(4 128 13 137 22 110 31 119)(5 109 14 118 23 127 32 136)(6 126 15 135 24 144 33 117)(7 143 16 116 25 125 34 134)(8 124 17 133 26 142 35 115)(9 141 18 114 27 123 36 132)(37 91 46 100 55 73 64 82)(38 108 47 81 56 90 65 99)(39 89 48 98 57 107 66 80)(40 106 49 79 58 88 67 97)(41 87 50 96 59 105 68 78)(42 104 51 77 60 86 69 95)(43 85 52 94 61 103 70 76)(44 102 53 75 62 84 71 93)(45 83 54 92 63 101 72 74)
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), (1,61,103,122,19,43,85,140)(2,42,104,139,20,60,86,121)(3,59,105,120,21,41,87,138)(4,40,106,137,22,58,88,119)(5,57,107,118,23,39,89,136)(6,38,108,135,24,56,90,117)(7,55,73,116,25,37,91,134)(8,72,74,133,26,54,92,115)(9,53,75,114,27,71,93,132)(10,70,76,131,28,52,94,113)(11,51,77,112,29,69,95,130)(12,68,78,129,30,50,96,111)(13,49,79,110,31,67,97,128)(14,66,80,127,32,48,98,109)(15,47,81,144,33,65,99,126)(16,64,82,125,34,46,100,143)(17,45,83,142,35,63,101,124)(18,62,84,123,36,44,102,141), (1,113,10,122,19,131,28,140)(2,130,11,139,20,112,29,121)(3,111,12,120,21,129,30,138)(4,128,13,137,22,110,31,119)(5,109,14,118,23,127,32,136)(6,126,15,135,24,144,33,117)(7,143,16,116,25,125,34,134)(8,124,17,133,26,142,35,115)(9,141,18,114,27,123,36,132)(37,91,46,100,55,73,64,82)(38,108,47,81,56,90,65,99)(39,89,48,98,57,107,66,80)(40,106,49,79,58,88,67,97)(41,87,50,96,59,105,68,78)(42,104,51,77,60,86,69,95)(43,85,52,94,61,103,70,76)(44,102,53,75,62,84,71,93)(45,83,54,92,63,101,72,74)>;
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), (1,61,103,122,19,43,85,140)(2,42,104,139,20,60,86,121)(3,59,105,120,21,41,87,138)(4,40,106,137,22,58,88,119)(5,57,107,118,23,39,89,136)(6,38,108,135,24,56,90,117)(7,55,73,116,25,37,91,134)(8,72,74,133,26,54,92,115)(9,53,75,114,27,71,93,132)(10,70,76,131,28,52,94,113)(11,51,77,112,29,69,95,130)(12,68,78,129,30,50,96,111)(13,49,79,110,31,67,97,128)(14,66,80,127,32,48,98,109)(15,47,81,144,33,65,99,126)(16,64,82,125,34,46,100,143)(17,45,83,142,35,63,101,124)(18,62,84,123,36,44,102,141), (1,113,10,122,19,131,28,140)(2,130,11,139,20,112,29,121)(3,111,12,120,21,129,30,138)(4,128,13,137,22,110,31,119)(5,109,14,118,23,127,32,136)(6,126,15,135,24,144,33,117)(7,143,16,116,25,125,34,134)(8,124,17,133,26,142,35,115)(9,141,18,114,27,123,36,132)(37,91,46,100,55,73,64,82)(38,108,47,81,56,90,65,99)(39,89,48,98,57,107,66,80)(40,106,49,79,58,88,67,97)(41,87,50,96,59,105,68,78)(42,104,51,77,60,86,69,95)(43,85,52,94,61,103,70,76)(44,102,53,75,62,84,71,93)(45,83,54,92,63,101,72,74) );
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)], [(1,61,103,122,19,43,85,140),(2,42,104,139,20,60,86,121),(3,59,105,120,21,41,87,138),(4,40,106,137,22,58,88,119),(5,57,107,118,23,39,89,136),(6,38,108,135,24,56,90,117),(7,55,73,116,25,37,91,134),(8,72,74,133,26,54,92,115),(9,53,75,114,27,71,93,132),(10,70,76,131,28,52,94,113),(11,51,77,112,29,69,95,130),(12,68,78,129,30,50,96,111),(13,49,79,110,31,67,97,128),(14,66,80,127,32,48,98,109),(15,47,81,144,33,65,99,126),(16,64,82,125,34,46,100,143),(17,45,83,142,35,63,101,124),(18,62,84,123,36,44,102,141)], [(1,113,10,122,19,131,28,140),(2,130,11,139,20,112,29,121),(3,111,12,120,21,129,30,138),(4,128,13,137,22,110,31,119),(5,109,14,118,23,127,32,136),(6,126,15,135,24,144,33,117),(7,143,16,116,25,125,34,134),(8,124,17,133,26,142,35,115),(9,141,18,114,27,123,36,132),(37,91,46,100,55,73,64,82),(38,108,47,81,56,90,65,99),(39,89,48,98,57,107,66,80),(40,106,49,79,58,88,67,97),(41,87,50,96,59,105,68,78),(42,104,51,77,60,86,69,95),(43,85,52,94,61,103,70,76),(44,102,53,75,62,84,71,93),(45,83,54,92,63,101,72,74)])
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