metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.26D18, (C2×C36)⋊6C4, (C2×C4)⋊4Dic9, C36.39(C2×C4), C4⋊Dic9⋊17C2, (C22×C4).9D9, C9⋊4(C42⋊C2), (C4×Dic9)⋊15C2, (C2×C12).374D6, (C2×C4).102D18, C4.15(C2×Dic9), C18.16(C4○D4), C6.86(C4○D12), C18.24(C22×C4), (C22×C12).27S3, (C22×C36).10C2, (C2×C18).44C23, C12.45(C2×Dic3), (C2×C12).18Dic3, (C22×C6).138D6, C2.4(D36⋊5C2), C2.5(C22×Dic9), C22.5(C2×Dic9), (C2×C36).112C22, C18.D4.5C2, C3.(C23.26D6), C6.25(C22×Dic3), C22.22(C22×D9), (C22×C18).36C22, (C2×Dic9).38C22, (C2×C18).35(C2×C4), (C2×C6).40(C2×Dic3), (C2×C6).201(C22×S3), SmallGroup(288,136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.26D18
G = < a,b,c,d,e | a2=b2=c2=1, d18=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d17 >
Subgroups: 328 in 114 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C18, C18, C18, C2×Dic3, C2×C12, C2×C12, C22×C6, C42⋊C2, Dic9, C36, C2×C18, C2×C18, C2×C18, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C2×Dic9, C2×C36, C2×C36, C22×C18, C23.26D6, C4×Dic9, C4⋊Dic9, C18.D4, C22×C36, C23.26D18
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, D9, C2×Dic3, C22×S3, C42⋊C2, Dic9, D18, C4○D12, C22×Dic3, C2×Dic9, C22×D9, C23.26D6, D36⋊5C2, C22×Dic9, C23.26D18
►On 144 points
Generators in S144
(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(25 144)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 107)(38 108)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(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 60 138 76)(2 41 139 93)(3 58 140 74)(4 39 141 91)(5 56 142 108)(6 37 143 89)(7 54 144 106)(8 71 109 87)(9 52 110 104)(10 69 111 85)(11 50 112 102)(12 67 113 83)(13 48 114 100)(14 65 115 81)(15 46 116 98)(16 63 117 79)(17 44 118 96)(18 61 119 77)(19 42 120 94)(20 59 121 75)(21 40 122 92)(22 57 123 73)(23 38 124 90)(24 55 125 107)(25 72 126 88)(26 53 127 105)(27 70 128 86)(28 51 129 103)(29 68 130 84)(30 49 131 101)(31 66 132 82)(32 47 133 99)(33 64 134 80)(34 45 135 97)(35 62 136 78)(36 43 137 95)
G:=sub<Sym(144)| (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,144)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,107)(38,108)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (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), (1,60,138,76)(2,41,139,93)(3,58,140,74)(4,39,141,91)(5,56,142,108)(6,37,143,89)(7,54,144,106)(8,71,109,87)(9,52,110,104)(10,69,111,85)(11,50,112,102)(12,67,113,83)(13,48,114,100)(14,65,115,81)(15,46,116,98)(16,63,117,79)(17,44,118,96)(18,61,119,77)(19,42,120,94)(20,59,121,75)(21,40,122,92)(22,57,123,73)(23,38,124,90)(24,55,125,107)(25,72,126,88)(26,53,127,105)(27,70,128,86)(28,51,129,103)(29,68,130,84)(30,49,131,101)(31,66,132,82)(32,47,133,99)(33,64,134,80)(34,45,135,97)(35,62,136,78)(36,43,137,95)>;
G:=Group( (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,144)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,107)(38,108)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,106), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (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), (1,60,138,76)(2,41,139,93)(3,58,140,74)(4,39,141,91)(5,56,142,108)(6,37,143,89)(7,54,144,106)(8,71,109,87)(9,52,110,104)(10,69,111,85)(11,50,112,102)(12,67,113,83)(13,48,114,100)(14,65,115,81)(15,46,116,98)(16,63,117,79)(17,44,118,96)(18,61,119,77)(19,42,120,94)(20,59,121,75)(21,40,122,92)(22,57,123,73)(23,38,124,90)(24,55,125,107)(25,72,126,88)(26,53,127,105)(27,70,128,86)(28,51,129,103)(29,68,130,84)(30,49,131,101)(31,66,132,82)(32,47,133,99)(33,64,134,80)(34,45,135,97)(35,62,136,78)(36,43,137,95) );
G=PermutationGroup([[(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108)], [(1,120),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,141),(23,142),(24,143),(25,144),(26,109),(27,110),(28,111),(29,112),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(37,107),(38,108),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,105),(72,106)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144)], [(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)], [(1,60,138,76),(2,41,139,93),(3,58,140,74),(4,39,141,91),(5,56,142,108),(6,37,143,89),(7,54,144,106),(8,71,109,87),(9,52,110,104),(10,69,111,85),(11,50,112,102),(12,67,113,83),(13,48,114,100),(14,65,115,81),(15,46,116,98),(16,63,117,79),(17,44,118,96),(18,61,119,77),(19,42,120,94),(20,59,121,75),(21,40,122,92),(22,57,123,73),(23,38,124,90),(24,55,125,107),(25,72,126,88),(26,53,127,105),(27,70,128,86),(28,51,129,103),(29,68,130,84),(30,49,131,101),(31,66,132,82),(32,47,133,99),(33,64,134,80),(34,45,135,97),(35,62,136,78),(36,43,137,95)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | Dic3 | D6 | D6 | C4○D4 | D9 | Dic9 | D18 | D18 | C4○D12 | D36⋊5C2 |
| kernel | C23.26D18 | C4×Dic9 | C4⋊Dic9 | C18.D4 | C22×C36 | C2×C36 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C18 | C22×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 4 | 2 | 1 | 4 | 3 | 12 | 6 | 3 | 8 | 24 |
Matrix representation of C23.26D18 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 33 | 0 | 0 | 0 |
| 30 | 9 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 20 |
| 28 | 22 | 0 | 0 |
| 3 | 9 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 30 | 0 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[33,30,0,0,0,9,0,0,0,0,24,0,0,0,0,20],[28,3,0,0,22,9,0,0,0,0,0,30,0,0,16,0] >;
C23.26D18 in GAP, Magma, Sage, TeX
C_2^3._{26}D_{18} % in TeX
G:=Group("C2^3.26D18"); // GroupNames label
G:=SmallGroup(288,136);
// by ID
G=gap.SmallGroup(288,136);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,422,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^18=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^17>;
// generators/relations