metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.3D12, C36.47D4, C4.11D36, M4(2).2D9, (C2×C4).1D18, (C2×Dic9).C4, (C2×C12).42D6, C22.4(C4×D9), C6.15(D6⋊C4), C2.9(D18⋊C4), C4.21(C9⋊D4), C9⋊1(C4.10D4), C4.Dic9.3C2, C18.8(C22⋊C4), (C2×C36).20C22, C3.(C12.47D4), (C2×Dic18).6C2, (C9×M4(2)).2C2, (C3×M4(2)).6S3, C12.108(C3⋊D4), (C2×C6).3(C4×S3), (C2×C18).2(C2×C4), SmallGroup(288,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D36
G = < a,b,c | a36=1, b4=c2=a18, bab-1=cac-1=a-1, cbc-1=a9b3 >
Subgroups: 228 in 57 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, M4(2), M4(2), C2×Q8, C18, C18, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C4.10D4, Dic9, C36, C2×C18, C4.Dic3, C3×M4(2), C2×Dic6, C9⋊C8, C72, Dic18, C2×Dic9, C2×C36, C12.47D4, C4.Dic9, C9×M4(2), C2×Dic18, C4.D36
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4.10D4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C12.47D4, D18⋊C4, C4.D36
►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)
(1 114 10 141 19 132 28 123)(2 113 11 140 20 131 29 122)(3 112 12 139 21 130 30 121)(4 111 13 138 22 129 31 120)(5 110 14 137 23 128 32 119)(6 109 15 136 24 127 33 118)(7 144 16 135 25 126 34 117)(8 143 17 134 26 125 35 116)(9 142 18 133 27 124 36 115)(37 98 64 107 55 80 46 89)(38 97 65 106 56 79 47 88)(39 96 66 105 57 78 48 87)(40 95 67 104 58 77 49 86)(41 94 68 103 59 76 50 85)(42 93 69 102 60 75 51 84)(43 92 70 101 61 74 52 83)(44 91 71 100 62 73 53 82)(45 90 72 99 63 108 54 81)
(1 90 19 108)(2 89 20 107)(3 88 21 106)(4 87 22 105)(5 86 23 104)(6 85 24 103)(7 84 25 102)(8 83 26 101)(9 82 27 100)(10 81 28 99)(11 80 29 98)(12 79 30 97)(13 78 31 96)(14 77 32 95)(15 76 33 94)(16 75 34 93)(17 74 35 92)(18 73 36 91)(37 113 55 131)(38 112 56 130)(39 111 57 129)(40 110 58 128)(41 109 59 127)(42 144 60 126)(43 143 61 125)(44 142 62 124)(45 141 63 123)(46 140 64 122)(47 139 65 121)(48 138 66 120)(49 137 67 119)(50 136 68 118)(51 135 69 117)(52 134 70 116)(53 133 71 115)(54 132 72 114)
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), (1,114,10,141,19,132,28,123)(2,113,11,140,20,131,29,122)(3,112,12,139,21,130,30,121)(4,111,13,138,22,129,31,120)(5,110,14,137,23,128,32,119)(6,109,15,136,24,127,33,118)(7,144,16,135,25,126,34,117)(8,143,17,134,26,125,35,116)(9,142,18,133,27,124,36,115)(37,98,64,107,55,80,46,89)(38,97,65,106,56,79,47,88)(39,96,66,105,57,78,48,87)(40,95,67,104,58,77,49,86)(41,94,68,103,59,76,50,85)(42,93,69,102,60,75,51,84)(43,92,70,101,61,74,52,83)(44,91,71,100,62,73,53,82)(45,90,72,99,63,108,54,81), (1,90,19,108)(2,89,20,107)(3,88,21,106)(4,87,22,105)(5,86,23,104)(6,85,24,103)(7,84,25,102)(8,83,26,101)(9,82,27,100)(10,81,28,99)(11,80,29,98)(12,79,30,97)(13,78,31,96)(14,77,32,95)(15,76,33,94)(16,75,34,93)(17,74,35,92)(18,73,36,91)(37,113,55,131)(38,112,56,130)(39,111,57,129)(40,110,58,128)(41,109,59,127)(42,144,60,126)(43,143,61,125)(44,142,62,124)(45,141,63,123)(46,140,64,122)(47,139,65,121)(48,138,66,120)(49,137,67,119)(50,136,68,118)(51,135,69,117)(52,134,70,116)(53,133,71,115)(54,132,72,114)>;
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), (1,114,10,141,19,132,28,123)(2,113,11,140,20,131,29,122)(3,112,12,139,21,130,30,121)(4,111,13,138,22,129,31,120)(5,110,14,137,23,128,32,119)(6,109,15,136,24,127,33,118)(7,144,16,135,25,126,34,117)(8,143,17,134,26,125,35,116)(9,142,18,133,27,124,36,115)(37,98,64,107,55,80,46,89)(38,97,65,106,56,79,47,88)(39,96,66,105,57,78,48,87)(40,95,67,104,58,77,49,86)(41,94,68,103,59,76,50,85)(42,93,69,102,60,75,51,84)(43,92,70,101,61,74,52,83)(44,91,71,100,62,73,53,82)(45,90,72,99,63,108,54,81), (1,90,19,108)(2,89,20,107)(3,88,21,106)(4,87,22,105)(5,86,23,104)(6,85,24,103)(7,84,25,102)(8,83,26,101)(9,82,27,100)(10,81,28,99)(11,80,29,98)(12,79,30,97)(13,78,31,96)(14,77,32,95)(15,76,33,94)(16,75,34,93)(17,74,35,92)(18,73,36,91)(37,113,55,131)(38,112,56,130)(39,111,57,129)(40,110,58,128)(41,109,59,127)(42,144,60,126)(43,143,61,125)(44,142,62,124)(45,141,63,123)(46,140,64,122)(47,139,65,121)(48,138,66,120)(49,137,67,119)(50,136,68,118)(51,135,69,117)(52,134,70,116)(53,133,71,115)(54,132,72,114) );
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)], [(1,114,10,141,19,132,28,123),(2,113,11,140,20,131,29,122),(3,112,12,139,21,130,30,121),(4,111,13,138,22,129,31,120),(5,110,14,137,23,128,32,119),(6,109,15,136,24,127,33,118),(7,144,16,135,25,126,34,117),(8,143,17,134,26,125,35,116),(9,142,18,133,27,124,36,115),(37,98,64,107,55,80,46,89),(38,97,65,106,56,79,47,88),(39,96,66,105,57,78,48,87),(40,95,67,104,58,77,49,86),(41,94,68,103,59,76,50,85),(42,93,69,102,60,75,51,84),(43,92,70,101,61,74,52,83),(44,91,71,100,62,73,53,82),(45,90,72,99,63,108,54,81)], [(1,90,19,108),(2,89,20,107),(3,88,21,106),(4,87,22,105),(5,86,23,104),(6,85,24,103),(7,84,25,102),(8,83,26,101),(9,82,27,100),(10,81,28,99),(11,80,29,98),(12,79,30,97),(13,78,31,96),(14,77,32,95),(15,76,33,94),(16,75,34,93),(17,74,35,92),(18,73,36,91),(37,113,55,131),(38,112,56,130),(39,111,57,129),(40,110,58,128),(41,109,59,127),(42,144,60,126),(43,143,61,125),(44,142,62,124),(45,141,63,123),(46,140,64,122),(47,139,65,121),(48,138,66,120),(49,137,67,119),(50,136,68,118),(51,135,69,117),(52,134,70,116),(53,133,71,115),(54,132,72,114)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | 36H | 36I | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D9 | D12 | C3⋊D4 | C4×S3 | D18 | D36 | C9⋊D4 | C4×D9 | C4.10D4 | C12.47D4 | C4.D36 |
| kernel | C4.D36 | C4.Dic9 | C9×M4(2) | C2×Dic18 | C2×Dic9 | C3×M4(2) | C36 | C2×C12 | M4(2) | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of C4.D36 ►in GL4(𝔽73) generated by
| 25 | 19 | 0 | 0 |
| 54 | 44 | 0 | 0 |
| 0 | 0 | 25 | 19 |
| 0 | 0 | 54 | 44 |
| 0 | 0 | 14 | 5 |
| 0 | 0 | 19 | 59 |
| 51 | 12 | 0 | 0 |
| 63 | 22 | 0 | 0 |
| 14 | 5 | 0 | 0 |
| 19 | 59 | 0 | 0 |
| 0 | 0 | 14 | 5 |
| 0 | 0 | 19 | 59 |
G:=sub<GL(4,GF(73))| [25,54,0,0,19,44,0,0,0,0,25,54,0,0,19,44],[0,0,51,63,0,0,12,22,14,19,0,0,5,59,0,0],[14,19,0,0,5,59,0,0,0,0,14,19,0,0,5,59] >;
C4.D36 in GAP, Magma, Sage, TeX
C_4.D_{36} % in TeX
G:=Group("C4.D36"); // GroupNames label
G:=SmallGroup(288,30);
// by ID
G=gap.SmallGroup(288,30);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^4=c^2=a^18,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^3>;
// generators/relations