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 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C9, Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C18, C18 [×2], C18 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C42⋊C2, Dic9 [×4], C36 [×4], C2×C18, C2×C18 [×2], C2×C18 [×2], C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×2], C22×C12, C2×Dic9 [×4], C2×C36 [×2], C2×C36 [×4], C22×C18, C23.26D6, C4×Dic9 [×2], C4⋊Dic9 [×2], C18.D4 [×2], C22×C36, C23.26D18
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, C4○D4 [×2], D9, C2×Dic3 [×6], C22×S3, C42⋊C2, Dic9 [×4], D18 [×3], C4○D12 [×2], C22×Dic3, C2×Dic9 [×6], C22×D9, C23.26D6, D36⋊5C2 [×2], 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)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(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)
(1 104 139 65)(2 85 140 46)(3 102 141 63)(4 83 142 44)(5 100 143 61)(6 81 144 42)(7 98 109 59)(8 79 110 40)(9 96 111 57)(10 77 112 38)(11 94 113 55)(12 75 114 72)(13 92 115 53)(14 73 116 70)(15 90 117 51)(16 107 118 68)(17 88 119 49)(18 105 120 66)(19 86 121 47)(20 103 122 64)(21 84 123 45)(22 101 124 62)(23 82 125 43)(24 99 126 60)(25 80 127 41)(26 97 128 58)(27 78 129 39)(28 95 130 56)(29 76 131 37)(30 93 132 54)(31 74 133 71)(32 91 134 52)(33 108 135 69)(34 89 136 50)(35 106 137 67)(36 87 138 48)
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), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (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), (1,104,139,65)(2,85,140,46)(3,102,141,63)(4,83,142,44)(5,100,143,61)(6,81,144,42)(7,98,109,59)(8,79,110,40)(9,96,111,57)(10,77,112,38)(11,94,113,55)(12,75,114,72)(13,92,115,53)(14,73,116,70)(15,90,117,51)(16,107,118,68)(17,88,119,49)(18,105,120,66)(19,86,121,47)(20,103,122,64)(21,84,123,45)(22,101,124,62)(23,82,125,43)(24,99,126,60)(25,80,127,41)(26,97,128,58)(27,78,129,39)(28,95,130,56)(29,76,131,37)(30,93,132,54)(31,74,133,71)(32,91,134,52)(33,108,135,69)(34,89,136,50)(35,106,137,67)(36,87,138,48)>;
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), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (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), (1,104,139,65)(2,85,140,46)(3,102,141,63)(4,83,142,44)(5,100,143,61)(6,81,144,42)(7,98,109,59)(8,79,110,40)(9,96,111,57)(10,77,112,38)(11,94,113,55)(12,75,114,72)(13,92,115,53)(14,73,116,70)(15,90,117,51)(16,107,118,68)(17,88,119,49)(18,105,120,66)(19,86,121,47)(20,103,122,64)(21,84,123,45)(22,101,124,62)(23,82,125,43)(24,99,126,60)(25,80,127,41)(26,97,128,58)(27,78,129,39)(28,95,130,56)(29,76,131,37)(30,93,132,54)(31,74,133,71)(32,91,134,52)(33,108,135,69)(34,89,136,50)(35,106,137,67)(36,87,138,48) );
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)], [(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,141),(22,142),(23,143),(24,144),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,94),(38,95),(39,96),(40,97),(41,98),(42,99),(43,100),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93)], [(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)], [(1,104,139,65),(2,85,140,46),(3,102,141,63),(4,83,142,44),(5,100,143,61),(6,81,144,42),(7,98,109,59),(8,79,110,40),(9,96,111,57),(10,77,112,38),(11,94,113,55),(12,75,114,72),(13,92,115,53),(14,73,116,70),(15,90,117,51),(16,107,118,68),(17,88,119,49),(18,105,120,66),(19,86,121,47),(20,103,122,64),(21,84,123,45),(22,101,124,62),(23,82,125,43),(24,99,126,60),(25,80,127,41),(26,97,128,58),(27,78,129,39),(28,95,130,56),(29,76,131,37),(30,93,132,54),(31,74,133,71),(32,91,134,52),(33,108,135,69),(34,89,136,50),(35,106,137,67),(36,87,138,48)])
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