metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊7D4, C22⋊2D36, C23.29D18, (C2×C18)⋊5D4, D18⋊C4⋊3C2, (C2×D36)⋊6C2, C9⋊3(C4⋊D4), C4⋊3(C9⋊D4), C4⋊Dic9⋊9C2, (C22×C4)⋊6D9, (C22×C36)⋊6C2, C3.(C12⋊7D4), (C2×C6).31D12, (C2×C4).70D18, C2.17(C2×D36), C18.44(C2×D4), C6.46(C2×D12), (C2×C12).346D6, C6.89(C4○D12), C18.19(C4○D4), (C22×C12).20S3, (C2×C18).48C23, (C2×C36).78C22, (C22×C6).142D6, C12.111(C3⋊D4), C2.19(D36⋊5C2), C22.56(C22×D9), (C22×C18).40C22, (C2×Dic9).14C22, (C22×D9).10C22, (C2×C9⋊D4)⋊3C2, C2.7(C2×C9⋊D4), C6.91(C2×C3⋊D4), (C2×C6).205(C22×S3), SmallGroup(288,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36⋊7D4
G = < a,b,c | a36=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 692 in 141 conjugacy classes, 50 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C9, Dic3 [×2], C12 [×2], C12, D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], D9 [×2], C18 [×3], C18 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, Dic9 [×2], C36 [×2], C36, D18 [×6], C2×C18, C2×C18 [×2], C2×C18 [×2], C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, D36 [×2], C2×Dic9 [×2], C9⋊D4 [×4], C2×C36 [×2], C2×C36 [×2], C22×D9 [×2], C22×C18, C12⋊7D4, C4⋊Dic9, D18⋊C4 [×2], C2×D36, C2×C9⋊D4 [×2], C22×C36, C36⋊7D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D9, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, D18 [×3], C2×D12, C4○D12, C2×C3⋊D4, D36 [×2], C9⋊D4 [×2], C22×D9, C12⋊7D4, C2×D36, D36⋊5C2, C2×C9⋊D4, C36⋊7D4
►On 144 points
(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 99 68 140)(2 98 69 139)(3 97 70 138)(4 96 71 137)(5 95 72 136)(6 94 37 135)(7 93 38 134)(8 92 39 133)(9 91 40 132)(10 90 41 131)(11 89 42 130)(12 88 43 129)(13 87 44 128)(14 86 45 127)(15 85 46 126)(16 84 47 125)(17 83 48 124)(18 82 49 123)(19 81 50 122)(20 80 51 121)(21 79 52 120)(22 78 53 119)(23 77 54 118)(24 76 55 117)(25 75 56 116)(26 74 57 115)(27 73 58 114)(28 108 59 113)(29 107 60 112)(30 106 61 111)(31 105 62 110)(32 104 63 109)(33 103 64 144)(34 102 65 143)(35 101 66 142)(36 100 67 141)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(64 72)(65 71)(66 70)(67 69)(73 130)(74 129)(75 128)(76 127)(77 126)(78 125)(79 124)(80 123)(81 122)(82 121)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)
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), (1,99,68,140)(2,98,69,139)(3,97,70,138)(4,96,71,137)(5,95,72,136)(6,94,37,135)(7,93,38,134)(8,92,39,133)(9,91,40,132)(10,90,41,131)(11,89,42,130)(12,88,43,129)(13,87,44,128)(14,86,45,127)(15,85,46,126)(16,84,47,125)(17,83,48,124)(18,82,49,123)(19,81,50,122)(20,80,51,121)(21,79,52,120)(22,78,53,119)(23,77,54,118)(24,76,55,117)(25,75,56,116)(26,74,57,115)(27,73,58,114)(28,108,59,113)(29,107,60,112)(30,106,61,111)(31,105,62,110)(32,104,63,109)(33,103,64,144)(34,102,65,143)(35,101,66,142)(36,100,67,141), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(64,72)(65,71)(66,70)(67,69)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)>;
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), (1,99,68,140)(2,98,69,139)(3,97,70,138)(4,96,71,137)(5,95,72,136)(6,94,37,135)(7,93,38,134)(8,92,39,133)(9,91,40,132)(10,90,41,131)(11,89,42,130)(12,88,43,129)(13,87,44,128)(14,86,45,127)(15,85,46,126)(16,84,47,125)(17,83,48,124)(18,82,49,123)(19,81,50,122)(20,80,51,121)(21,79,52,120)(22,78,53,119)(23,77,54,118)(24,76,55,117)(25,75,56,116)(26,74,57,115)(27,73,58,114)(28,108,59,113)(29,107,60,112)(30,106,61,111)(31,105,62,110)(32,104,63,109)(33,103,64,144)(34,102,65,143)(35,101,66,142)(36,100,67,141), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(64,72)(65,71)(66,70)(67,69)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131) );
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)], [(1,99,68,140),(2,98,69,139),(3,97,70,138),(4,96,71,137),(5,95,72,136),(6,94,37,135),(7,93,38,134),(8,92,39,133),(9,91,40,132),(10,90,41,131),(11,89,42,130),(12,88,43,129),(13,87,44,128),(14,86,45,127),(15,85,46,126),(16,84,47,125),(17,83,48,124),(18,82,49,123),(19,81,50,122),(20,80,51,121),(21,79,52,120),(22,78,53,119),(23,77,54,118),(24,76,55,117),(25,75,56,116),(26,74,57,115),(27,73,58,114),(28,108,59,113),(29,107,60,112),(30,106,61,111),(31,105,62,110),(32,104,63,109),(33,103,64,144),(34,102,65,143),(35,101,66,142),(36,100,67,141)], [(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(64,72),(65,71),(66,70),(67,69),(73,130),(74,129),(75,128),(76,127),(77,126),(78,125),(79,124),(80,123),(81,122),(82,121),(83,120),(84,119),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,112),(92,111),(93,110),(94,109),(95,144),(96,143),(97,142),(98,141),(99,140),(100,139),(101,138),(102,137),(103,136),(104,135),(105,134),(106,133),(107,132),(108,131)])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | D12 | D18 | D18 | C4○D12 | C9⋊D4 | D36 | D36⋊5C2 |
| kernel | C36⋊7D4 | C4⋊Dic9 | D18⋊C4 | C2×D36 | C2×C9⋊D4 | C22×C36 | C22×C12 | C36 | C2×C18 | C2×C12 | C22×C6 | C18 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 3 | 4 | 4 | 6 | 3 | 4 | 12 | 12 | 12 |
Matrix representation of C36⋊7D4 ►in GL4(𝔽37) generated by
| 0 | 36 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 29 | 25 |
| 0 | 0 | 12 | 4 |
| 30 | 23 | 0 | 0 |
| 30 | 7 | 0 | 0 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 7 | 30 |
| 1 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(37))| [0,1,0,0,36,1,0,0,0,0,29,12,0,0,25,4],[30,30,0,0,23,7,0,0,0,0,7,7,0,0,14,30],[1,36,0,0,0,36,0,0,0,0,36,1,0,0,0,1] >;
C36⋊7D4 in GAP, Magma, Sage, TeX
C_{36}\rtimes_7D_4 % in TeX
G:=Group("C36:7D4"); // GroupNames label
G:=SmallGroup(288,140);
// by ID
G=gap.SmallGroup(288,140);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations