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G = C42⋊C27order 432 = 24·33

The semidirect product of C42 and C27 acting via C27/C9=C3

metabelian, soluble, monomial, A-group

Aliases: C42⋊C27, (C4×C36).C3, (C4×C12).C9, C9.(C42⋊C3), C3.(C42⋊C9), (C2×C18).1A4, C22.(C9.A4), (C2×C6).1(C3.A4), SmallGroup(432,3)

Series: Derived Chief Lower central Upper central

C1C42 — C42⋊C27
C1C22C42C4×C12C4×C36 — C42⋊C27
C42 — C42⋊C27
C1C9

Generators and relations for C42⋊C27
 G = < a,b,c | a4=b4=c27=1, ab=ba, cac-1=ab-1, cbc-1=a-1b2 >

3C2
3C4
3C4
3C6
3C2×C4
3C12
3C12
3C18
16C27
3C2×C12
3C36
3C36
3C2×C36
4C9.A4

Smallest permutation representation of C42⋊C27
On 108 points
Generators in S108
(1 72)(2 50 73 85)(3 51 74 86)(4 75)(5 53 76 88)(6 54 77 89)(7 78)(8 29 79 91)(9 30 80 92)(10 81)(11 32 55 94)(12 33 56 95)(13 57)(14 35 58 97)(15 36 59 98)(16 60)(17 38 61 100)(18 39 62 101)(19 63)(20 41 64 103)(21 42 65 104)(22 66)(23 44 67 106)(24 45 68 107)(25 69)(26 47 70 82)(27 48 71 83)(28 90)(31 93)(34 96)(37 99)(40 102)(43 105)(46 108)(49 84)(52 87)
(1 49 72 84)(3 86 74 51)(4 52 75 87)(6 89 77 54)(7 28 78 90)(9 92 80 30)(10 31 81 93)(12 95 56 33)(13 34 57 96)(15 98 59 36)(16 37 60 99)(18 101 62 39)(19 40 63 102)(21 104 65 42)(22 43 66 105)(24 107 68 45)(25 46 69 108)(27 83 71 48)
(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)

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

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

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

72 conjugacy classes

class 1  2 3A3B4A4B4C4D6A6B9A···9F12A···12H18A···18F27A···27R36A···36X
order12334444669···912···1218···1827···2736···36
size13113333331···13···33···316···163···3

72 irreducible representations

dim1111333333
type++
imageC1C3C9C27A4C3.A4C42⋊C3C9.A4C42⋊C9C42⋊C27
kernelC42⋊C27C4×C36C4×C12C42C2×C18C2×C6C9C22C3C1
# reps126181246824

Matrix representation of C42⋊C27 in GL3(𝔽109) generated by

10800
0330
0033
,
3300
0760
001
,
010
001
6600
G:=sub<GL(3,GF(109))| [108,0,0,0,33,0,0,0,33],[33,0,0,0,76,0,0,0,1],[0,0,66,1,0,0,0,1,0] >;

C42⋊C27 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_{27}
% in TeX

G:=Group("C4^2:C27");
// GroupNames label

G:=SmallGroup(432,3);
// by ID

G=gap.SmallGroup(432,3);
# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,21,50,1515,360,10399,102,9077,15882]);
// Polycyclic

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

Export

Subgroup lattice of C42⋊C27 in TeX

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