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G = D4×C27order 216 = 23·33

Direct product of C27 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C27, C4⋊C54, C1083C2, C36.7C6, C12.3C18, C222C54, C54.6C22, C9.(C3×D4), C3.(D4×C9), C54(D4×C9), (D4×C9).C3, (C3×D4).C9, (C2×C54)⋊1C2, (C2×C6).2C18, C6.6(C2×C18), (C2×C18).4C6, C2.1(C2×C54), C18.14(C2×C6), SmallGroup(216,10)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C27
C1C3C9C18C54C2×C54 — D4×C27
C1C2 — D4×C27
C1C54 — D4×C27

Generators and relations for D4×C27
 G = < a,b,c | a27=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C6
2C6
2C18
2C18
2C54
2C54

Smallest permutation representation of D4×C27
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 74 96 31)(2 75 97 32)(3 76 98 33)(4 77 99 34)(5 78 100 35)(6 79 101 36)(7 80 102 37)(8 81 103 38)(9 55 104 39)(10 56 105 40)(11 57 106 41)(12 58 107 42)(13 59 108 43)(14 60 82 44)(15 61 83 45)(16 62 84 46)(17 63 85 47)(18 64 86 48)(19 65 87 49)(20 66 88 50)(21 67 89 51)(22 68 90 52)(23 69 91 53)(24 70 92 54)(25 71 93 28)(26 72 94 29)(27 73 95 30)
(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)

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

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

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

D4×C27 is a maximal subgroup of   D4.D27  D4⋊D27  D42D27

135 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F9A···9F12A12B18A···18F18G···18R27A···27R36A···36F54A···54R54S···54BB108A···108R
order12223346666669···9121218···1818···1827···2736···3654···5454···54108···108
size11221121122221···1221···12···21···12···21···12···22···2

135 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C3C6C6C9C18C18C27C54C54D4C3×D4D4×C9D4×C27
kernelD4×C27C108C2×C54D4×C9C36C2×C18C3×D4C12C2×C6D4C4C22C27C9C3C1
# reps112224661218183612618

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

210
021
,
162
7108
,
162
0108
G:=sub<GL(2,GF(109))| [21,0,0,21],[1,7,62,108],[1,0,62,108] >;

D4×C27 in GAP, Magma, Sage, TeX

D_4\times C_{27}
% in TeX

G:=Group("D4xC27");
// GroupNames label

G:=SmallGroup(216,10);
// by ID

G=gap.SmallGroup(216,10);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,122,118]);
// Polycyclic

G:=Group<a,b,c|a^27=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C27 in TeX

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