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 [×2], C4 [×2], C22, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C9, Dic3 [×2], C12 [×2], C2×C6, M4(2), M4(2), C2×Q8, C18, C18, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12, C4.10D4, Dic9 [×2], C36 [×2], C2×C18, C4.Dic3, C3×M4(2), C2×Dic6, C9⋊C8, C72, Dic18 [×2], C2×Dic9 [×2], C2×C36, C12.47D4, C4.Dic9, C9×M4(2), C2×Dic18, C4.D36
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], 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)
(1 127 10 118 19 109 28 136)(2 126 11 117 20 144 29 135)(3 125 12 116 21 143 30 134)(4 124 13 115 22 142 31 133)(5 123 14 114 23 141 32 132)(6 122 15 113 24 140 33 131)(7 121 16 112 25 139 34 130)(8 120 17 111 26 138 35 129)(9 119 18 110 27 137 36 128)(37 90 64 99 55 108 46 81)(38 89 65 98 56 107 47 80)(39 88 66 97 57 106 48 79)(40 87 67 96 58 105 49 78)(41 86 68 95 59 104 50 77)(42 85 69 94 60 103 51 76)(43 84 70 93 61 102 52 75)(44 83 71 92 62 101 53 74)(45 82 72 91 63 100 54 73)
(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)(37 118 55 136)(38 117 56 135)(39 116 57 134)(40 115 58 133)(41 114 59 132)(42 113 60 131)(43 112 61 130)(44 111 62 129)(45 110 63 128)(46 109 64 127)(47 144 65 126)(48 143 66 125)(49 142 67 124)(50 141 68 123)(51 140 69 122)(52 139 70 121)(53 138 71 120)(54 137 72 119)
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), (1,127,10,118,19,109,28,136)(2,126,11,117,20,144,29,135)(3,125,12,116,21,143,30,134)(4,124,13,115,22,142,31,133)(5,123,14,114,23,141,32,132)(6,122,15,113,24,140,33,131)(7,121,16,112,25,139,34,130)(8,120,17,111,26,138,35,129)(9,119,18,110,27,137,36,128)(37,90,64,99,55,108,46,81)(38,89,65,98,56,107,47,80)(39,88,66,97,57,106,48,79)(40,87,67,96,58,105,49,78)(41,86,68,95,59,104,50,77)(42,85,69,94,60,103,51,76)(43,84,70,93,61,102,52,75)(44,83,71,92,62,101,53,74)(45,82,72,91,63,100,54,73), (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)(37,118,55,136)(38,117,56,135)(39,116,57,134)(40,115,58,133)(41,114,59,132)(42,113,60,131)(43,112,61,130)(44,111,62,129)(45,110,63,128)(46,109,64,127)(47,144,65,126)(48,143,66,125)(49,142,67,124)(50,141,68,123)(51,140,69,122)(52,139,70,121)(53,138,71,120)(54,137,72,119)>;
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), (1,127,10,118,19,109,28,136)(2,126,11,117,20,144,29,135)(3,125,12,116,21,143,30,134)(4,124,13,115,22,142,31,133)(5,123,14,114,23,141,32,132)(6,122,15,113,24,140,33,131)(7,121,16,112,25,139,34,130)(8,120,17,111,26,138,35,129)(9,119,18,110,27,137,36,128)(37,90,64,99,55,108,46,81)(38,89,65,98,56,107,47,80)(39,88,66,97,57,106,48,79)(40,87,67,96,58,105,49,78)(41,86,68,95,59,104,50,77)(42,85,69,94,60,103,51,76)(43,84,70,93,61,102,52,75)(44,83,71,92,62,101,53,74)(45,82,72,91,63,100,54,73), (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)(37,118,55,136)(38,117,56,135)(39,116,57,134)(40,115,58,133)(41,114,59,132)(42,113,60,131)(43,112,61,130)(44,111,62,129)(45,110,63,128)(46,109,64,127)(47,144,65,126)(48,143,66,125)(49,142,67,124)(50,141,68,123)(51,140,69,122)(52,139,70,121)(53,138,71,120)(54,137,72,119) );
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)], [(1,127,10,118,19,109,28,136),(2,126,11,117,20,144,29,135),(3,125,12,116,21,143,30,134),(4,124,13,115,22,142,31,133),(5,123,14,114,23,141,32,132),(6,122,15,113,24,140,33,131),(7,121,16,112,25,139,34,130),(8,120,17,111,26,138,35,129),(9,119,18,110,27,137,36,128),(37,90,64,99,55,108,46,81),(38,89,65,98,56,107,47,80),(39,88,66,97,57,106,48,79),(40,87,67,96,58,105,49,78),(41,86,68,95,59,104,50,77),(42,85,69,94,60,103,51,76),(43,84,70,93,61,102,52,75),(44,83,71,92,62,101,53,74),(45,82,72,91,63,100,54,73)], [(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),(37,118,55,136),(38,117,56,135),(39,116,57,134),(40,115,58,133),(41,114,59,132),(42,113,60,131),(43,112,61,130),(44,111,62,129),(45,110,63,128),(46,109,64,127),(47,144,65,126),(48,143,66,125),(49,142,67,124),(50,141,68,123),(51,140,69,122),(52,139,70,121),(53,138,71,120),(54,137,72,119)])
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