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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

 Derived series C1 — C2 — D4×C27
 Chief series C1 — C3 — C9 — C18 — C54 — C2×C54 — D4×C27
 Lower central C1 — C2 — D4×C27
 Upper central C1 — C54 — D4×C27

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

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 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 9A ··· 9F 12A 12B 18A ··· 18F 18G ··· 18R 27A ··· 27R 36A ··· 36F 54A ··· 54R 54S ··· 54BB 108A ··· 108R order 1 2 2 2 3 3 4 6 6 6 6 6 6 9 ··· 9 12 12 18 ··· 18 18 ··· 18 27 ··· 27 36 ··· 36 54 ··· 54 54 ··· 54 108 ··· 108 size 1 1 2 2 1 1 2 1 1 2 2 2 2 1 ··· 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 C27 C54 C54 D4 C3×D4 D4×C9 D4×C27 kernel D4×C27 C108 C2×C54 D4×C9 C36 C2×C18 C3×D4 C12 C2×C6 D4 C4 C22 C27 C9 C3 C1 # reps 1 1 2 2 2 4 6 6 12 18 18 36 1 2 6 18

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

 21 0 0 21
,
 1 62 7 108
,
 1 62 0 108
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

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