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G = C18.S4order 432 = 24·33

3rd non-split extension by C18 of S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C18.3S4, C23.D27, C22:Dic27, C9.A4:C4, C9.(A4:C4), (C2xC6).Dic9, (C2xC18).Dic3, C2.1(C9.S4), C3.(C6.S4), C6.3(C3.S4), (C22xC6).2D9, (C22xC18).2S3, (C2xC9.A4).C2, SmallGroup(432,39)

Series: Derived Chief Lower central Upper central

C1C22C9.A4 — C18.S4
C1C22C2xC6C2xC18C9.A4C2xC9.A4 — C18.S4
C9.A4 — C18.S4
C1C2

Generators and relations for C18.S4
 G = < a,b,c,d,e | a18=b2=c2=1, d3=a2, e2=a9, ab=ba, ac=ca, ad=da, eae-1=a-1, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=a16d2 >

Subgroups: 336 in 45 conjugacy classes, 15 normal (all characteristic)
Quotients: C1, C2, C4, S3, Dic3, D9, S4, Dic9, A4:C4, D27, C3.S4, Dic27, C6.S4, C9.S4, C18.S4
3C2
3C2
3C22
3C22
54C4
54C4
3C6
3C6
27C2xC4
27C2xC4
3C2xC6
3C2xC6
18Dic3
18Dic3
3C18
3C18
4C27
27C22:C4
9C2xDic3
9C2xDic3
3C2xC18
3C2xC18
6Dic9
6Dic9
4C54
9C6.D4
3C2xDic9
3C2xDic9
4Dic27
3C18.D4

Smallest permutation representation of C18.S4
On 108 points
Generators in S108
(1 77 4 80 7 56 10 59 13 62 16 65 19 68 22 71 25 74)(2 78 5 81 8 57 11 60 14 63 17 66 20 69 23 72 26 75)(3 79 6 55 9 58 12 61 15 64 18 67 21 70 24 73 27 76)(28 92 31 95 34 98 37 101 40 104 43 107 46 83 49 86 52 89)(29 93 32 96 35 99 38 102 41 105 44 108 47 84 50 87 53 90)(30 94 33 97 36 100 39 103 42 106 45 82 48 85 51 88 54 91)
(2 63)(3 64)(5 66)(6 67)(8 69)(9 70)(11 72)(12 73)(14 75)(15 76)(17 78)(18 79)(20 81)(21 55)(23 57)(24 58)(26 60)(27 61)(28 104)(29 105)(31 107)(32 108)(34 83)(35 84)(37 86)(38 87)(40 89)(41 90)(43 92)(44 93)(46 95)(47 96)(49 98)(50 99)(52 101)(53 102)
(1 62)(3 64)(4 65)(6 67)(7 68)(9 70)(10 71)(12 73)(13 74)(15 76)(16 77)(18 79)(19 80)(21 55)(22 56)(24 58)(25 59)(27 61)(29 105)(30 106)(32 108)(33 82)(35 84)(36 85)(38 87)(39 88)(41 90)(42 91)(44 93)(45 94)(47 96)(48 97)(50 99)(51 100)(53 102)(54 103)
(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 90 62 41)(2 89 63 40)(3 88 64 39)(4 87 65 38)(5 86 66 37)(6 85 67 36)(7 84 68 35)(8 83 69 34)(9 82 70 33)(10 108 71 32)(11 107 72 31)(12 106 73 30)(13 105 74 29)(14 104 75 28)(15 103 76 54)(16 102 77 53)(17 101 78 52)(18 100 79 51)(19 99 80 50)(20 98 81 49)(21 97 55 48)(22 96 56 47)(23 95 57 46)(24 94 58 45)(25 93 59 44)(26 92 60 43)(27 91 61 42)

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

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

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

42 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D6A6B6C9A9B9C18A18B18C18D···18I27A···27I54A···54I
order12223444466699918181818···1827···2754···54
size11332545454542662222226···68···88···8

42 irreducible representations

dim111222222336666
type+++-+-+-++-+-
imageC1C2C4S3Dic3D9Dic9D27Dic27S4A4:C4C3.S4C6.S4C9.S4C18.S4
kernelC18.S4C2xC9.A4C9.A4C22xC18C2xC18C22xC6C2xC6C23C22C18C9C6C3C2C1
# reps112113399221133

Matrix representation of C18.S4 in GL5(F109)

8259000
5032000
00100
00010
00001
,
10000
01000
0010800
0001080
0011081
,
10000
01000
0010800
00010
0001108
,
3016000
9346000
0001080
0011082
00001
,
1519000
3494000
001081107
00010
00001

G:=sub<GL(5,GF(109))| [82,50,0,0,0,59,32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,108,0,1,0,0,0,108,108,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,108,0,0,0,0,0,1,1,0,0,0,0,108],[30,93,0,0,0,16,46,0,0,0,0,0,0,1,0,0,0,108,108,0,0,0,0,2,1],[15,34,0,0,0,19,94,0,0,0,0,0,108,0,0,0,0,1,1,0,0,0,107,0,1] >;

C18.S4 in GAP, Magma, Sage, TeX

C_{18}.S_4
% in TeX

G:=Group("C18.S4");
// GroupNames label

G:=SmallGroup(432,39);
// by ID

G=gap.SmallGroup(432,39);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,422,331,1683,192,2524,9077,2287,5298,3989]);
// Polycyclic

G:=Group<a,b,c,d,e|a^18=b^2=c^2=1,d^3=a^2,e^2=a^9,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=a^16*d^2>;
// generators/relations

Export

Subgroup lattice of C18.S4 in TeX

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