metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊1D4, C4⋊2D36, D18⋊2D4, C12.5D12, C4⋊C4⋊3D9, D18⋊C4⋊7C2, (C2×D36)⋊4C2, C9⋊2(C4⋊D4), (C2×C12).9D6, (C2×C4).9D18, C2.13(D4×D9), C6.87(S3×D4), C18.7(C2×D4), C2.9(C2×D36), C3.(C12⋊D4), C6.36(C2×D12), C18.34(C4○D4), (C2×C18).36C23, (C2×C36).11C22, C2.6(Q8⋊3D9), C6.41(Q8⋊3S3), C22.50(C22×D9), (C2×Dic9).34C22, (C22×D9).20C22, (C2×C4×D9)⋊1C2, (C9×C4⋊C4)⋊6C2, (C3×C4⋊C4).13S3, (C2×C6).193(C22×S3), SmallGroup(288,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D36
G = < a,b,c | a4=b36=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 812 in 141 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4⋊D4, Dic9, C36, C36, D18, D18, C2×C18, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C4×D9, D36, C2×Dic9, C2×C36, C2×C36, C22×D9, C22×D9, C12⋊D4, D18⋊C4, C9×C4⋊C4, C2×C4×D9, C2×D36, C2×D36, C4⋊D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, D12, C22×S3, C4⋊D4, D18, C2×D12, S3×D4, Q8⋊3S3, D36, C22×D9, C12⋊D4, C2×D36, D4×D9, Q8⋊3D9, C4⋊D36
►On 144 points
Generators in S144
(1 75 59 144)(2 109 60 76)(3 77 61 110)(4 111 62 78)(5 79 63 112)(6 113 64 80)(7 81 65 114)(8 115 66 82)(9 83 67 116)(10 117 68 84)(11 85 69 118)(12 119 70 86)(13 87 71 120)(14 121 72 88)(15 89 37 122)(16 123 38 90)(17 91 39 124)(18 125 40 92)(19 93 41 126)(20 127 42 94)(21 95 43 128)(22 129 44 96)(23 97 45 130)(24 131 46 98)(25 99 47 132)(26 133 48 100)(27 101 49 134)(28 135 50 102)(29 103 51 136)(30 137 52 104)(31 105 53 138)(32 139 54 106)(33 107 55 140)(34 141 56 108)(35 73 57 142)(36 143 58 74)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(73 136)(74 135)(75 134)(76 133)(77 132)(78 131)(79 130)(80 129)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 120)(90 119)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 144)(102 143)(103 142)(104 141)(105 140)(106 139)(107 138)(108 137)
G:=sub<Sym(144)| (1,75,59,144)(2,109,60,76)(3,77,61,110)(4,111,62,78)(5,79,63,112)(6,113,64,80)(7,81,65,114)(8,115,66,82)(9,83,67,116)(10,117,68,84)(11,85,69,118)(12,119,70,86)(13,87,71,120)(14,121,72,88)(15,89,37,122)(16,123,38,90)(17,91,39,124)(18,125,40,92)(19,93,41,126)(20,127,42,94)(21,95,43,128)(22,129,44,96)(23,97,45,130)(24,131,46,98)(25,99,47,132)(26,133,48,100)(27,101,49,134)(28,135,50,102)(29,103,51,136)(30,137,52,104)(31,105,53,138)(32,139,54,106)(33,107,55,140)(34,141,56,108)(35,73,57,142)(36,143,58,74), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)>;
G:=Group( (1,75,59,144)(2,109,60,76)(3,77,61,110)(4,111,62,78)(5,79,63,112)(6,113,64,80)(7,81,65,114)(8,115,66,82)(9,83,67,116)(10,117,68,84)(11,85,69,118)(12,119,70,86)(13,87,71,120)(14,121,72,88)(15,89,37,122)(16,123,38,90)(17,91,39,124)(18,125,40,92)(19,93,41,126)(20,127,42,94)(21,95,43,128)(22,129,44,96)(23,97,45,130)(24,131,46,98)(25,99,47,132)(26,133,48,100)(27,101,49,134)(28,135,50,102)(29,103,51,136)(30,137,52,104)(31,105,53,138)(32,139,54,106)(33,107,55,140)(34,141,56,108)(35,73,57,142)(36,143,58,74), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137) );
G=PermutationGroup([[(1,75,59,144),(2,109,60,76),(3,77,61,110),(4,111,62,78),(5,79,63,112),(6,113,64,80),(7,81,65,114),(8,115,66,82),(9,83,67,116),(10,117,68,84),(11,85,69,118),(12,119,70,86),(13,87,71,120),(14,121,72,88),(15,89,37,122),(16,123,38,90),(17,91,39,124),(18,125,40,92),(19,93,41,126),(20,127,42,94),(21,95,43,128),(22,129,44,96),(23,97,45,130),(24,131,46,98),(25,99,47,132),(26,133,48,100),(27,101,49,134),(28,135,50,102),(29,103,51,136),(30,137,52,104),(31,105,53,138),(32,139,54,106),(33,107,55,140),(34,141,56,108),(35,73,57,142),(36,143,58,74)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(28,36),(29,35),(30,34),(31,33),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(73,136),(74,135),(75,134),(76,133),(77,132),(78,131),(79,130),(80,129),(81,128),(82,127),(83,126),(84,125),(85,124),(86,123),(87,122),(88,121),(89,120),(90,119),(91,118),(92,117),(93,116),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,144),(102,143),(103,142),(104,141),(105,140),(106,139),(107,138),(108,137)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | C4○D4 | D9 | D12 | D18 | D36 | S3×D4 | Q8⋊3S3 | D4×D9 | Q8⋊3D9 |
| kernel | C4⋊D36 | D18⋊C4 | C9×C4⋊C4 | C2×C4×D9 | C2×D36 | C3×C4⋊C4 | C36 | D18 | C2×C12 | C18 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 1 | 2 | 2 | 3 | 2 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of C4⋊D36 ►in GL4(𝔽37) generated by
| 0 | 6 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 33 | 29 |
| 0 | 0 | 8 | 25 |
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 11 | 6 |
G:=sub<GL(4,GF(37))| [0,6,0,0,6,0,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,1,0,0,0,0,0,33,8,0,0,29,25],[36,0,0,0,0,1,0,0,0,0,31,11,0,0,17,6] >;
C4⋊D36 in GAP, Magma, Sage, TeX
C_4\rtimes D_{36} % in TeX
G:=Group("C4:D36"); // GroupNames label
G:=SmallGroup(288,105);
// by ID
G=gap.SmallGroup(288,105);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^4=b^36=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations