metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D36, C23.21D18, D18⋊C4⋊6C2, C4⋊Dic9⋊5C2, C22⋊C4⋊6D9, C18.6(C2×D4), (C2×C12).5D6, C2.8(C2×D36), (C2×C18).4D4, (C2×C4).7D18, (C2×C6).5D12, C6.35(C2×D12), (C2×C36).6C22, (C22×C6).46D6, C18.23(C4○D4), (C2×C18).27C23, (C22×Dic9)⋊2C2, C9⋊2(C22.D4), C6.78(D4⋊2S3), C2.10(D4⋊2D9), C3.(C23.21D6), (C22×D9).6C22, C22.45(C22×D9), (C22×C18).16C22, (C2×Dic9).27C22, (C9×C22⋊C4)⋊4C2, (C2×C9⋊D4).5C2, (C3×C22⋊C4).4S3, (C2×C6).184(C22×S3), SmallGroup(288,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.4D36
G = < a,b,c,d | a2=b2=c36=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 524 in 117 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, C18, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22.D4, Dic9, C36, D18, C2×C18, C2×C18, C2×C18, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C2×Dic9, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, C22×D9, C22×C18, C23.21D6, C4⋊Dic9, D18⋊C4, C9×C22⋊C4, C22×Dic9, C2×C9⋊D4, C22.4D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, D12, C22×S3, C22.D4, D18, C2×D12, D4⋊2S3, D36, C22×D9, C23.21D6, C2×D36, D4⋊2D9, C22.4D36
►On 144 points
Generators in S144
(1 89)(2 136)(3 91)(4 138)(5 93)(6 140)(7 95)(8 142)(9 97)(10 144)(11 99)(12 110)(13 101)(14 112)(15 103)(16 114)(17 105)(18 116)(19 107)(20 118)(21 73)(22 120)(23 75)(24 122)(25 77)(26 124)(27 79)(28 126)(29 81)(30 128)(31 83)(32 130)(33 85)(34 132)(35 87)(36 134)(37 88)(38 135)(39 90)(40 137)(41 92)(42 139)(43 94)(44 141)(45 96)(46 143)(47 98)(48 109)(49 100)(50 111)(51 102)(52 113)(53 104)(54 115)(55 106)(56 117)(57 108)(58 119)(59 74)(60 121)(61 76)(62 123)(63 78)(64 125)(65 80)(66 127)(67 82)(68 129)(69 84)(70 131)(71 86)(72 133)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 37)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)
(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 37 38 36)(2 35 39 72)(3 71 40 34)(4 33 41 70)(5 69 42 32)(6 31 43 68)(7 67 44 30)(8 29 45 66)(9 65 46 28)(10 27 47 64)(11 63 48 26)(12 25 49 62)(13 61 50 24)(14 23 51 60)(15 59 52 22)(16 21 53 58)(17 57 54 20)(18 19 55 56)(73 104 119 114)(74 113 120 103)(75 102 121 112)(76 111 122 101)(77 100 123 110)(78 109 124 99)(79 98 125 144)(80 143 126 97)(81 96 127 142)(82 141 128 95)(83 94 129 140)(84 139 130 93)(85 92 131 138)(86 137 132 91)(87 90 133 136)(88 135 134 89)(105 108 115 118)(106 117 116 107)
G:=sub<Sym(144)| (1,89)(2,136)(3,91)(4,138)(5,93)(6,140)(7,95)(8,142)(9,97)(10,144)(11,99)(12,110)(13,101)(14,112)(15,103)(16,114)(17,105)(18,116)(19,107)(20,118)(21,73)(22,120)(23,75)(24,122)(25,77)(26,124)(27,79)(28,126)(29,81)(30,128)(31,83)(32,130)(33,85)(34,132)(35,87)(36,134)(37,88)(38,135)(39,90)(40,137)(41,92)(42,139)(43,94)(44,141)(45,96)(46,143)(47,98)(48,109)(49,100)(50,111)(51,102)(52,113)(53,104)(54,115)(55,106)(56,117)(57,108)(58,119)(59,74)(60,121)(61,76)(62,123)(63,78)(64,125)(65,80)(66,127)(67,82)(68,129)(69,84)(70,131)(71,86)(72,133), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,37)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118), (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), (1,37,38,36)(2,35,39,72)(3,71,40,34)(4,33,41,70)(5,69,42,32)(6,31,43,68)(7,67,44,30)(8,29,45,66)(9,65,46,28)(10,27,47,64)(11,63,48,26)(12,25,49,62)(13,61,50,24)(14,23,51,60)(15,59,52,22)(16,21,53,58)(17,57,54,20)(18,19,55,56)(73,104,119,114)(74,113,120,103)(75,102,121,112)(76,111,122,101)(77,100,123,110)(78,109,124,99)(79,98,125,144)(80,143,126,97)(81,96,127,142)(82,141,128,95)(83,94,129,140)(84,139,130,93)(85,92,131,138)(86,137,132,91)(87,90,133,136)(88,135,134,89)(105,108,115,118)(106,117,116,107)>;
G:=Group( (1,89)(2,136)(3,91)(4,138)(5,93)(6,140)(7,95)(8,142)(9,97)(10,144)(11,99)(12,110)(13,101)(14,112)(15,103)(16,114)(17,105)(18,116)(19,107)(20,118)(21,73)(22,120)(23,75)(24,122)(25,77)(26,124)(27,79)(28,126)(29,81)(30,128)(31,83)(32,130)(33,85)(34,132)(35,87)(36,134)(37,88)(38,135)(39,90)(40,137)(41,92)(42,139)(43,94)(44,141)(45,96)(46,143)(47,98)(48,109)(49,100)(50,111)(51,102)(52,113)(53,104)(54,115)(55,106)(56,117)(57,108)(58,119)(59,74)(60,121)(61,76)(62,123)(63,78)(64,125)(65,80)(66,127)(67,82)(68,129)(69,84)(70,131)(71,86)(72,133), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,37)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118), (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), (1,37,38,36)(2,35,39,72)(3,71,40,34)(4,33,41,70)(5,69,42,32)(6,31,43,68)(7,67,44,30)(8,29,45,66)(9,65,46,28)(10,27,47,64)(11,63,48,26)(12,25,49,62)(13,61,50,24)(14,23,51,60)(15,59,52,22)(16,21,53,58)(17,57,54,20)(18,19,55,56)(73,104,119,114)(74,113,120,103)(75,102,121,112)(76,111,122,101)(77,100,123,110)(78,109,124,99)(79,98,125,144)(80,143,126,97)(81,96,127,142)(82,141,128,95)(83,94,129,140)(84,139,130,93)(85,92,131,138)(86,137,132,91)(87,90,133,136)(88,135,134,89)(105,108,115,118)(106,117,116,107) );
G=PermutationGroup([[(1,89),(2,136),(3,91),(4,138),(5,93),(6,140),(7,95),(8,142),(9,97),(10,144),(11,99),(12,110),(13,101),(14,112),(15,103),(16,114),(17,105),(18,116),(19,107),(20,118),(21,73),(22,120),(23,75),(24,122),(25,77),(26,124),(27,79),(28,126),(29,81),(30,128),(31,83),(32,130),(33,85),(34,132),(35,87),(36,134),(37,88),(38,135),(39,90),(40,137),(41,92),(42,139),(43,94),(44,141),(45,96),(46,143),(47,98),(48,109),(49,100),(50,111),(51,102),(52,113),(53,104),(54,115),(55,106),(56,117),(57,108),(58,119),(59,74),(60,121),(61,76),(62,123),(63,78),(64,125),(65,80),(66,127),(67,82),(68,129),(69,84),(70,131),(71,86),(72,133)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,37),(73,119),(74,120),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118)], [(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)], [(1,37,38,36),(2,35,39,72),(3,71,40,34),(4,33,41,70),(5,69,42,32),(6,31,43,68),(7,67,44,30),(8,29,45,66),(9,65,46,28),(10,27,47,64),(11,63,48,26),(12,25,49,62),(13,61,50,24),(14,23,51,60),(15,59,52,22),(16,21,53,58),(17,57,54,20),(18,19,55,56),(73,104,119,114),(74,113,120,103),(75,102,121,112),(76,111,122,101),(77,100,123,110),(78,109,124,99),(79,98,125,144),(80,143,126,97),(81,96,127,142),(82,141,128,95),(83,94,129,140),(84,139,130,93),(85,92,131,138),(86,137,132,91),(87,90,133,136),(88,135,134,89),(105,108,115,118),(106,117,116,107)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 36 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D9 | D12 | D18 | D18 | D36 | D4⋊2S3 | D4⋊2D9 |
| kernel | C22.4D36 | C4⋊Dic9 | D18⋊C4 | C9×C22⋊C4 | C22×Dic9 | C2×C9⋊D4 | C3×C22⋊C4 | C2×C18 | C2×C12 | C22×C6 | C18 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 3 | 4 | 6 | 3 | 12 | 2 | 6 |
Matrix representation of C22.4D36 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 31 | 25 |
| 0 | 0 | 6 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 25 | 29 | 0 | 0 |
| 8 | 33 | 0 | 0 |
| 0 | 0 | 36 | 35 |
| 0 | 0 | 0 | 1 |
| 27 | 32 | 0 | 0 |
| 5 | 10 | 0 | 0 |
| 0 | 0 | 36 | 35 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,31,6,0,0,25,6],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[25,8,0,0,29,33,0,0,0,0,36,0,0,0,35,1],[27,5,0,0,32,10,0,0,0,0,36,1,0,0,35,1] >;
C22.4D36 in GAP, Magma, Sage, TeX
C_2^2._4D_{36} % in TeX
G:=Group("C2^2.4D36"); // GroupNames label
G:=SmallGroup(288,96);
// by ID
G=gap.SmallGroup(288,96);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,142,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations