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## G = C4×C27⋊C3order 324 = 22·34

### Direct product of C4 and C27⋊C3

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×C27⋊C3
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C2×C27⋊C3 — C4×C27⋊C3
 Lower central C1 — C3 — C4×C27⋊C3
 Upper central C1 — C36 — C4×C27⋊C3

Generators and relations for C4×C27⋊C3
G = < a,b,c | a4=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >

Smallest permutation representation of C4×C27⋊C3
On 108 points
Generators in S108
(1 88 53 73)(2 89 54 74)(3 90 28 75)(4 91 29 76)(5 92 30 77)(6 93 31 78)(7 94 32 79)(8 95 33 80)(9 96 34 81)(10 97 35 55)(11 98 36 56)(12 99 37 57)(13 100 38 58)(14 101 39 59)(15 102 40 60)(16 103 41 61)(17 104 42 62)(18 105 43 63)(19 106 44 64)(20 107 45 65)(21 108 46 66)(22 82 47 67)(23 83 48 68)(24 84 49 69)(25 85 50 70)(26 86 51 71)(27 87 52 72)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(28 37 46)(30 48 39)(31 40 49)(33 51 42)(34 43 52)(36 54 45)(56 74 65)(57 66 75)(59 77 68)(60 69 78)(62 80 71)(63 72 81)(83 101 92)(84 93 102)(86 104 95)(87 96 105)(89 107 98)(90 99 108)

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

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

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

132 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 9A ··· 9F 9G 9H 9I 9J 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 18G 18H 18I 18J 27A ··· 27R 36A ··· 36L 36M ··· 36T 54A ··· 54R 108A ··· 108AJ order 1 2 3 3 3 3 4 4 6 6 6 6 9 ··· 9 9 9 9 9 12 12 12 12 12 12 12 12 18 ··· 18 18 18 18 18 27 ··· 27 36 ··· 36 36 ··· 36 54 ··· 54 108 ··· 108 size 1 1 1 1 3 3 1 1 1 1 3 3 1 ··· 1 3 3 3 3 1 1 1 1 3 3 3 3 1 ··· 1 3 3 3 3 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C3 C4 C6 C6 C9 C9 C12 C12 C18 C18 C36 C36 C27⋊C3 C2×C27⋊C3 C4×C27⋊C3 kernel C4×C27⋊C3 C2×C27⋊C3 C108 C3×C36 C27⋊C3 C54 C3×C18 C36 C3×C12 C27 C3×C9 C18 C3×C6 C9 C32 C4 C2 C1 # reps 1 1 6 2 2 6 2 12 6 12 4 12 6 24 12 6 6 12

Matrix representation of C4×C27⋊C3 in GL4(𝔽109) generated by

 76 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 45 0 0 0 0 0 1 0 0 0 0 45 0 75 0 0
,
 63 0 0 0 0 1 0 0 0 0 45 0 0 0 0 63
G:=sub<GL(4,GF(109))| [76,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[45,0,0,0,0,0,0,75,0,1,0,0,0,0,45,0],[63,0,0,0,0,1,0,0,0,0,45,0,0,0,0,63] >;

C4×C27⋊C3 in GAP, Magma, Sage, TeX

C_4\times C_{27}\rtimes C_3
% in TeX

G:=Group("C4xC27:C3");
// GroupNames label

G:=SmallGroup(324,30);
// by ID

G=gap.SmallGroup(324,30);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,223,1034,118]);
// Polycyclic

G:=Group<a,b,c|a^4=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

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