direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D27, D54.C2, C108⋊2C2, C36.5S3, C12.5D9, C18.9D6, C6.9D18, C2.1D54, Dic27⋊2C2, C54.2C22, C9.(C4×S3), C3.(C4×D9), C27⋊1(C2×C4), SmallGroup(216,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C4×D27 |
Generators and relations for C4×D27
G = < a,b,c | a4=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D27
►On 108 points
Generators in S108
(1 104 40 77)(2 105 41 78)(3 106 42 79)(4 107 43 80)(5 108 44 81)(6 82 45 55)(7 83 46 56)(8 84 47 57)(9 85 48 58)(10 86 49 59)(11 87 50 60)(12 88 51 61)(13 89 52 62)(14 90 53 63)(15 91 54 64)(16 92 28 65)(17 93 29 66)(18 94 30 67)(19 95 31 68)(20 96 32 69)(21 97 33 70)(22 98 34 71)(23 99 35 72)(24 100 36 73)(25 101 37 74)(26 102 38 75)(27 103 39 76)
(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)
(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(55 99)(56 98)(57 97)(58 96)(59 95)(60 94)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)
G:=sub<Sym(108)| (1,104,40,77)(2,105,41,78)(3,106,42,79)(4,107,43,80)(5,108,44,81)(6,82,45,55)(7,83,46,56)(8,84,47,57)(9,85,48,58)(10,86,49,59)(11,87,50,60)(12,88,51,61)(13,89,52,62)(14,90,53,63)(15,91,54,64)(16,92,28,65)(17,93,29,66)(18,94,30,67)(19,95,31,68)(20,96,32,69)(21,97,33,70)(22,98,34,71)(23,99,35,72)(24,100,36,73)(25,101,37,74)(26,102,38,75)(27,103,39,76), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)>;
G:=Group( (1,104,40,77)(2,105,41,78)(3,106,42,79)(4,107,43,80)(5,108,44,81)(6,82,45,55)(7,83,46,56)(8,84,47,57)(9,85,48,58)(10,86,49,59)(11,87,50,60)(12,88,51,61)(13,89,52,62)(14,90,53,63)(15,91,54,64)(16,92,28,65)(17,93,29,66)(18,94,30,67)(19,95,31,68)(20,96,32,69)(21,97,33,70)(22,98,34,71)(23,99,35,72)(24,100,36,73)(25,101,37,74)(26,102,38,75)(27,103,39,76), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100) );
G=PermutationGroup([[(1,104,40,77),(2,105,41,78),(3,106,42,79),(4,107,43,80),(5,108,44,81),(6,82,45,55),(7,83,46,56),(8,84,47,57),(9,85,48,58),(10,86,49,59),(11,87,50,60),(12,88,51,61),(13,89,52,62),(14,90,53,63),(15,91,54,64),(16,92,28,65),(17,93,29,66),(18,94,30,67),(19,95,31,68),(20,96,32,69),(21,97,33,70),(22,98,34,71),(23,99,35,72),(24,100,36,73),(25,101,37,74),(26,102,38,75),(27,103,39,76)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(55,99),(56,98),(57,97),(58,96),(59,95),(60,94),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,100)]])
C4×D27 is a maximal subgroup of
C8⋊D27 D108⋊5C2 D4⋊2D27 Q8⋊3D27
C4×D27 is a maximal quotient of C8⋊D27 Dic27⋊C4 D54⋊C4
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 27A | ··· | 27I | 36A | ··· | 36F | 54A | ··· | 54I | 108A | ··· | 108R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 27 | 27 | 2 | 1 | 1 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D9 | C4×S3 | D18 | D27 | C4×D9 | D54 | C4×D27 |
| kernel | C4×D27 | Dic27 | C108 | D54 | D27 | C36 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 3 | 9 | 6 | 9 | 18 |
Matrix representation of C4×D27 ►in GL2(𝔽109) generated by
| 76 | 0 |
| 0 | 76 |
| 80 | 87 |
| 22 | 58 |
| 27 | 32 |
| 59 | 82 |
G:=sub<GL(2,GF(109))| [76,0,0,76],[80,22,87,58],[27,59,32,82] >;
C4×D27 in GAP, Magma, Sage, TeX
C_4\times D_{27} % in TeX
G:=Group("C4xD27"); // GroupNames label
G:=SmallGroup(216,5);
// by ID
G=gap.SmallGroup(216,5);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,963,381,3604,208,5189]);
// Polycyclic
G:=Group<a,b,c|a^4=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export