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G = C27⋊D4order 216 = 23·33

The semidirect product of C27 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C272D4, D542C2, Dic27⋊C2, C2.5D54, C18.12D6, C6.12D18, C222D27, 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

C1C54 — C27⋊D4
C1C3C9C27C54D54 — C27⋊D4
C27C54 — C27⋊D4
C1C2C22

Generators and relations for C27⋊D4
 G = < a,b,c | a27=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
54C2
27C4
27C22
2C6
18S3
27D4
9Dic3
9D6
2C18
6D9
9C3⋊D4
3Dic9
3D18
2D27
2C54
3C9⋊D4

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 61 38 100)(2 60 39 99)(3 59 40 98)(4 58 41 97)(5 57 42 96)(6 56 43 95)(7 55 44 94)(8 81 45 93)(9 80 46 92)(10 79 47 91)(11 78 48 90)(12 77 49 89)(13 76 50 88)(14 75 51 87)(15 74 52 86)(16 73 53 85)(17 72 54 84)(18 71 28 83)(19 70 29 82)(20 69 30 108)(21 68 31 107)(22 67 32 106)(23 66 33 105)(24 65 34 104)(25 64 35 103)(26 63 36 102)(27 62 37 101)
(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 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(49 54)(50 53)(51 52)(55 106)(56 105)(57 104)(58 103)(59 102)(60 101)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 108)(81 107)

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), (1,61,38,100)(2,60,39,99)(3,59,40,98)(4,58,41,97)(5,57,42,96)(6,56,43,95)(7,55,44,94)(8,81,45,93)(9,80,46,92)(10,79,47,91)(11,78,48,90)(12,77,49,89)(13,76,50,88)(14,75,51,87)(15,74,52,86)(16,73,53,85)(17,72,54,84)(18,71,28,83)(19,70,29,82)(20,69,30,108)(21,68,31,107)(22,67,32,106)(23,66,33,105)(24,65,34,104)(25,64,35,103)(26,63,36,102)(27,62,37,101), (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)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(49,54)(50,53)(51,52)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,108)(81,107)>;

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), (1,61,38,100)(2,60,39,99)(3,59,40,98)(4,58,41,97)(5,57,42,96)(6,56,43,95)(7,55,44,94)(8,81,45,93)(9,80,46,92)(10,79,47,91)(11,78,48,90)(12,77,49,89)(13,76,50,88)(14,75,51,87)(15,74,52,86)(16,73,53,85)(17,72,54,84)(18,71,28,83)(19,70,29,82)(20,69,30,108)(21,68,31,107)(22,67,32,106)(23,66,33,105)(24,65,34,104)(25,64,35,103)(26,63,36,102)(27,62,37,101), (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)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(49,54)(50,53)(51,52)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,108)(81,107) );

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)], [(1,61,38,100),(2,60,39,99),(3,59,40,98),(4,58,41,97),(5,57,42,96),(6,56,43,95),(7,55,44,94),(8,81,45,93),(9,80,46,92),(10,79,47,91),(11,78,48,90),(12,77,49,89),(13,76,50,88),(14,75,51,87),(15,74,52,86),(16,73,53,85),(17,72,54,84),(18,71,28,83),(19,70,29,82),(20,69,30,108),(21,68,31,107),(22,67,32,106),(23,66,33,105),(24,65,34,104),(25,64,35,103),(26,63,36,102),(27,62,37,101)], [(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),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(49,54),(50,53),(51,52),(55,106),(56,105),(57,104),(58,103),(59,102),(60,101),(61,100),(62,99),(63,98),(64,97),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,108),(81,107)])

C27⋊D4 is a maximal subgroup of   D1085C2  D4×D27  D42D27
C27⋊D4 is a maximal quotient of   Dic27⋊C4  D54⋊C4  D4.D27  D4⋊D27  C27⋊Q16  Q82D27  C54.D4

57 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C9A9B9C18A···18I27A···27I54A···54AA
order12223466699918···1827···2754···54
size112542542222222···22···22···2

57 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D6D9C3⋊D4D18D27C9⋊D4D54C27⋊D4
kernelC27⋊D4Dic27D54C2×C54C2×C18C27C18C2×C6C9C6C22C3C2C1
# reps111111132396918

Matrix representation of C27⋊D4 in GL2(𝔽109) generated by

3093
1646
,
612
6103
,
10
108108
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

Export

Subgroup lattice of C27⋊D4 in TeX

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