direct product, abelian, monomial, 2-elementary
Aliases: C4×C36, SmallGroup(144,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C36 |
| C1 — C4×C36 |
| C1 — C4×C36 |
Generators and relations for C4×C36
G = < a,b | a4=b36=1, ab=ba >
Smallest permutation representation of C4×C36
►Regular action on 144 points
Generators in S144
(1 141 93 38)(2 142 94 39)(3 143 95 40)(4 144 96 41)(5 109 97 42)(6 110 98 43)(7 111 99 44)(8 112 100 45)(9 113 101 46)(10 114 102 47)(11 115 103 48)(12 116 104 49)(13 117 105 50)(14 118 106 51)(15 119 107 52)(16 120 108 53)(17 121 73 54)(18 122 74 55)(19 123 75 56)(20 124 76 57)(21 125 77 58)(22 126 78 59)(23 127 79 60)(24 128 80 61)(25 129 81 62)(26 130 82 63)(27 131 83 64)(28 132 84 65)(29 133 85 66)(30 134 86 67)(31 135 87 68)(32 136 88 69)(33 137 89 70)(34 138 90 71)(35 139 91 72)(36 140 92 37)
(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)
G:=sub<Sym(144)| (1,141,93,38)(2,142,94,39)(3,143,95,40)(4,144,96,41)(5,109,97,42)(6,110,98,43)(7,111,99,44)(8,112,100,45)(9,113,101,46)(10,114,102,47)(11,115,103,48)(12,116,104,49)(13,117,105,50)(14,118,106,51)(15,119,107,52)(16,120,108,53)(17,121,73,54)(18,122,74,55)(19,123,75,56)(20,124,76,57)(21,125,77,58)(22,126,78,59)(23,127,79,60)(24,128,80,61)(25,129,81,62)(26,130,82,63)(27,131,83,64)(28,132,84,65)(29,133,85,66)(30,134,86,67)(31,135,87,68)(32,136,88,69)(33,137,89,70)(34,138,90,71)(35,139,91,72)(36,140,92,37), (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)>;
G:=Group( (1,141,93,38)(2,142,94,39)(3,143,95,40)(4,144,96,41)(5,109,97,42)(6,110,98,43)(7,111,99,44)(8,112,100,45)(9,113,101,46)(10,114,102,47)(11,115,103,48)(12,116,104,49)(13,117,105,50)(14,118,106,51)(15,119,107,52)(16,120,108,53)(17,121,73,54)(18,122,74,55)(19,123,75,56)(20,124,76,57)(21,125,77,58)(22,126,78,59)(23,127,79,60)(24,128,80,61)(25,129,81,62)(26,130,82,63)(27,131,83,64)(28,132,84,65)(29,133,85,66)(30,134,86,67)(31,135,87,68)(32,136,88,69)(33,137,89,70)(34,138,90,71)(35,139,91,72)(36,140,92,37), (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) );
G=PermutationGroup([[(1,141,93,38),(2,142,94,39),(3,143,95,40),(4,144,96,41),(5,109,97,42),(6,110,98,43),(7,111,99,44),(8,112,100,45),(9,113,101,46),(10,114,102,47),(11,115,103,48),(12,116,104,49),(13,117,105,50),(14,118,106,51),(15,119,107,52),(16,120,108,53),(17,121,73,54),(18,122,74,55),(19,123,75,56),(20,124,76,57),(21,125,77,58),(22,126,78,59),(23,127,79,60),(24,128,80,61),(25,129,81,62),(26,130,82,63),(27,131,83,64),(28,132,84,65),(29,133,85,66),(30,134,86,67),(31,135,87,68),(32,136,88,69),(33,137,89,70),(34,138,90,71),(35,139,91,72),(36,140,92,37)], [(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)]])
C4×C36 is a maximal subgroup of
C42.D9 C36⋊C8 C42⋊4D9 C36⋊2Q8 C36.6Q8 C42⋊2D9 C42⋊6D9 C42⋊7D9 C42⋊3D9 C42⋊C27 C42⋊3- 1+2
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12X | 18A | ··· | 18R | 36A | ··· | 36BT |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 |
| kernel | C4×C36 | C2×C36 | C4×C12 | C36 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 2 | 12 | 6 | 6 | 24 | 18 | 72 |
Matrix representation of C4×C36 ►in GL2(𝔽37) generated by
| 6 | 0 |
| 0 | 1 |
| 29 | 0 |
| 0 | 2 |
G:=sub<GL(2,GF(37))| [6,0,0,1],[29,0,0,2] >;
C4×C36 in GAP, Magma, Sage, TeX
C_4\times C_{36} % in TeX
G:=Group("C4xC36"); // GroupNames label
G:=SmallGroup(144,20);
// by ID
G=gap.SmallGroup(144,20);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,151,230]);
// Polycyclic
G:=Group<a,b|a^4=b^36=1,a*b=b*a>;
// generators/relations
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