metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C27⋊2D4, D54⋊2C2, Dic27⋊C2, C2.5D54, C18.12D6, C6.12D18, C22⋊2D27, C54.5C22, (C2×C54)⋊2C2, C9.(C3⋊D4), C3.(C9⋊D4), (C2×C6).3D9, (C2×C18).3S3, SmallGroup(216,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C27⋊D4
G = < a,b,c | a27=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C27⋊D4
►On 108 points
Generators in S108
(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 66 42 88)(2 65 43 87)(3 64 44 86)(4 63 45 85)(5 62 46 84)(6 61 47 83)(7 60 48 82)(8 59 49 108)(9 58 50 107)(10 57 51 106)(11 56 52 105)(12 55 53 104)(13 81 54 103)(14 80 28 102)(15 79 29 101)(16 78 30 100)(17 77 31 99)(18 76 32 98)(19 75 33 97)(20 74 34 96)(21 73 35 95)(22 72 36 94)(23 71 37 93)(24 70 38 92)(25 69 39 91)(26 68 40 90)(27 67 41 89)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 29)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(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)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(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), (1,66,42,88)(2,65,43,87)(3,64,44,86)(4,63,45,85)(5,62,46,84)(6,61,47,83)(7,60,48,82)(8,59,49,108)(9,58,50,107)(10,57,51,106)(11,56,52,105)(12,55,53,104)(13,81,54,103)(14,80,28,102)(15,79,29,101)(16,78,30,100)(17,77,31,99)(18,76,32,98)(19,75,33,97)(20,74,34,96)(21,73,35,95)(22,72,36,94)(23,71,37,93)(24,70,38,92)(25,69,39,91)(26,68,40,90)(27,67,41,89), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,29)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(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( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(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), (1,66,42,88)(2,65,43,87)(3,64,44,86)(4,63,45,85)(5,62,46,84)(6,61,47,83)(7,60,48,82)(8,59,49,108)(9,58,50,107)(10,57,51,106)(11,56,52,105)(12,55,53,104)(13,81,54,103)(14,80,28,102)(15,79,29,101)(16,78,30,100)(17,77,31,99)(18,76,32,98)(19,75,33,97)(20,74,34,96)(21,73,35,95)(22,72,36,94)(23,71,37,93)(24,70,38,92)(25,69,39,91)(26,68,40,90)(27,67,41,89), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,29)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(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([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27),(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)], [(1,66,42,88),(2,65,43,87),(3,64,44,86),(4,63,45,85),(5,62,46,84),(6,61,47,83),(7,60,48,82),(8,59,49,108),(9,58,50,107),(10,57,51,106),(11,56,52,105),(12,55,53,104),(13,81,54,103),(14,80,28,102),(15,79,29,101),(16,78,30,100),(17,77,31,99),(18,76,32,98),(19,75,33,97),(20,74,34,96),(21,73,35,95),(22,72,36,94),(23,71,37,93),(24,70,38,92),(25,69,39,91),(26,68,40,90),(27,67,41,89)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,29),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(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)]])
C27⋊D4 is a maximal subgroup of
D108⋊5C2 D4×D27 D4⋊2D27
C27⋊D4 is a maximal quotient of Dic27⋊C4 D54⋊C4 D4.D27 D4⋊D27 C27⋊Q16 Q8⋊2D27 C54.D4
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 9A | 9B | 9C | 18A | ··· | 18I | 27A | ··· | 27I | 54A | ··· | 54AA |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 2 | 54 | 2 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D9 | C3⋊D4 | D18 | D27 | C9⋊D4 | D54 | C27⋊D4 |
| kernel | C27⋊D4 | Dic27 | D54 | C2×C54 | C2×C18 | C27 | C18 | C2×C6 | C9 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 9 | 6 | 9 | 18 |
Matrix representation of C27⋊D4 ►in GL2(𝔽109) generated by
| 30 | 93 |
| 16 | 46 |
| 6 | 12 |
| 6 | 103 |
| 1 | 0 |
| 108 | 108 |
G:=sub<GL(2,GF(109))| [30,16,93,46],[6,6,12,103],[1,108,0,108] >;
C27⋊D4 in GAP, Magma, Sage, TeX
C_{27}\rtimes D_4 % in TeX
G:=Group("C27:D4"); // GroupNames label
G:=SmallGroup(216,8);
// by ID
G=gap.SmallGroup(216,8);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,963,381,3604,208,5189]);
// Polycyclic
G:=Group<a,b,c|a^27=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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