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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4⋊D4, Dic9, C36, C36, D18, C2×C18, C2×C18, C2×C18, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, D36, C2×Dic9, C9⋊D4, C2×C36, C2×C36, C22×D9, C22×C18, C12⋊7D4, C4⋊Dic9, D18⋊C4, C2×D36, C2×C9⋊D4, C22×C36, C36⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, D12, C3⋊D4, C22×S3, C4⋊D4, D18, C2×D12, C4○D12, C2×C3⋊D4, D36, C9⋊D4, C22×D9, C12⋊7D4, C2×D36, D36⋊5C2, C2×C9⋊D4, C36⋊7D4
►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)
(1 109 92 59)(2 144 93 58)(3 143 94 57)(4 142 95 56)(5 141 96 55)(6 140 97 54)(7 139 98 53)(8 138 99 52)(9 137 100 51)(10 136 101 50)(11 135 102 49)(12 134 103 48)(13 133 104 47)(14 132 105 46)(15 131 106 45)(16 130 107 44)(17 129 108 43)(18 128 73 42)(19 127 74 41)(20 126 75 40)(21 125 76 39)(22 124 77 38)(23 123 78 37)(24 122 79 72)(25 121 80 71)(26 120 81 70)(27 119 82 69)(28 118 83 68)(29 117 84 67)(30 116 85 66)(31 115 86 65)(32 114 87 64)(33 113 88 63)(34 112 89 62)(35 111 90 61)(36 110 91 60)
(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)(37 131)(38 130)(39 129)(40 128)(41 127)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 144)(61 143)(62 142)(63 141)(64 140)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(71 133)(72 132)(73 75)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)
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), (1,109,92,59)(2,144,93,58)(3,143,94,57)(4,142,95,56)(5,141,96,55)(6,140,97,54)(7,139,98,53)(8,138,99,52)(9,137,100,51)(10,136,101,50)(11,135,102,49)(12,134,103,48)(13,133,104,47)(14,132,105,46)(15,131,106,45)(16,130,107,44)(17,129,108,43)(18,128,73,42)(19,127,74,41)(20,126,75,40)(21,125,76,39)(22,124,77,38)(23,123,78,37)(24,122,79,72)(25,121,80,71)(26,120,81,70)(27,119,82,69)(28,118,83,68)(29,117,84,67)(30,116,85,66)(31,115,86,65)(32,114,87,64)(33,113,88,63)(34,112,89,62)(35,111,90,61)(36,110,91,60), (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)(37,131)(38,130)(39,129)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,144)(61,143)(62,142)(63,141)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,75)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)>;
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), (1,109,92,59)(2,144,93,58)(3,143,94,57)(4,142,95,56)(5,141,96,55)(6,140,97,54)(7,139,98,53)(8,138,99,52)(9,137,100,51)(10,136,101,50)(11,135,102,49)(12,134,103,48)(13,133,104,47)(14,132,105,46)(15,131,106,45)(16,130,107,44)(17,129,108,43)(18,128,73,42)(19,127,74,41)(20,126,75,40)(21,125,76,39)(22,124,77,38)(23,123,78,37)(24,122,79,72)(25,121,80,71)(26,120,81,70)(27,119,82,69)(28,118,83,68)(29,117,84,67)(30,116,85,66)(31,115,86,65)(32,114,87,64)(33,113,88,63)(34,112,89,62)(35,111,90,61)(36,110,91,60), (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)(37,131)(38,130)(39,129)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,144)(61,143)(62,142)(63,141)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,75)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93) );
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)], [(1,109,92,59),(2,144,93,58),(3,143,94,57),(4,142,95,56),(5,141,96,55),(6,140,97,54),(7,139,98,53),(8,138,99,52),(9,137,100,51),(10,136,101,50),(11,135,102,49),(12,134,103,48),(13,133,104,47),(14,132,105,46),(15,131,106,45),(16,130,107,44),(17,129,108,43),(18,128,73,42),(19,127,74,41),(20,126,75,40),(21,125,76,39),(22,124,77,38),(23,123,78,37),(24,122,79,72),(25,121,80,71),(26,120,81,70),(27,119,82,69),(28,118,83,68),(29,117,84,67),(30,116,85,66),(31,115,86,65),(32,114,87,64),(33,113,88,63),(34,112,89,62),(35,111,90,61),(36,110,91,60)], [(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),(37,131),(38,130),(39,129),(40,128),(41,127),(42,126),(43,125),(44,124),(45,123),(46,122),(47,121),(48,120),(49,119),(50,118),(51,117),(52,116),(53,115),(54,114),(55,113),(56,112),(57,111),(58,110),(59,109),(60,144),(61,143),(62,142),(63,141),(64,140),(65,139),(66,138),(67,137),(68,136),(69,135),(70,134),(71,133),(72,132),(73,75),(76,108),(77,107),(78,106),(79,105),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93)]])
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