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## G = A4×C2×C18order 432 = 24·33

### Direct product of C2×C18 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C2×C18
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9×A4 — A4×C18 — A4×C2×C18
 Lower central C22 — A4×C2×C18
 Upper central C1 — C2×C18

Generators and relations for A4×C2×C18
G = < a,b,c,d,e | a2=b18=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 376 in 157 conjugacy classes, 65 normal (16 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C23, C9, C9, C32, A4, C2×C6, C2×C6, C24, C18, C18, C3×C6, C2×A4, C22×C6, C22×C6, C3×C9, C3.A4, C2×C18, C2×C18, C3×A4, C62, C22×A4, C23×C6, C3×C18, C2×C3.A4, C22×C18, C22×C18, C6×A4, C9×A4, C6×C18, C22×C3.A4, C23×C18, A4×C2×C6, A4×C18, A4×C2×C18
Quotients: C1, C2, C3, C22, C6, C9, C32, A4, C2×C6, C18, C3×C6, C2×A4, C3×C9, C2×C18, C3×A4, C62, C22×A4, C3×C18, C6×A4, C9×A4, C6×C18, A4×C2×C6, A4×C18, A4×C2×C18

Smallest permutation representation of A4×C2×C18
On 108 points
Generators in S108
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 19)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 73)(69 74)(70 75)(71 76)(72 77)
(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)
(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 73)(69 74)(70 75)(71 76)(72 77)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 19)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 73)(69 74)(70 75)(71 76)(72 77)
(1 45 73)(2 46 74)(3 47 75)(4 48 76)(5 49 77)(6 50 78)(7 51 79)(8 52 80)(9 53 81)(10 54 82)(11 37 83)(12 38 84)(13 39 85)(14 40 86)(15 41 87)(16 42 88)(17 43 89)(18 44 90)(19 107 67)(20 108 68)(21 91 69)(22 92 70)(23 93 71)(24 94 72)(25 95 55)(26 96 56)(27 97 57)(28 98 58)(29 99 59)(30 100 60)(31 101 61)(32 102 62)(33 103 63)(34 104 64)(35 105 65)(36 106 66)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 6A ··· 6F 6G ··· 6N 6O ··· 6AF 9A ··· 9F 9G ··· 9R 18A ··· 18R 18S ··· 18AP 18AQ ··· 18BZ order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 1 1 3 3 3 3 1 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 1 ··· 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 A4 C2×A4 C3×A4 C6×A4 C9×A4 A4×C18 kernel A4×C2×C18 A4×C18 C22×C3.A4 C23×C18 A4×C2×C6 C2×C3.A4 C22×C18 C6×A4 C22×A4 C2×A4 C2×C18 C18 C2×C6 C6 C22 C2 # reps 1 3 4 2 2 12 6 6 18 54 1 3 2 6 6 18

Matrix representation of A4×C2×C18 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 6 0 0 0 0 0 12 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 18 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 1 1 1
,
 11 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 18 18 17 0 0 0 0 1

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[6,0,0,0,0,0,12,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,18,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,1,0,0,0,18,1,0,0,0,0,1],[11,0,0,0,0,0,11,0,0,0,0,0,0,18,0,0,0,1,18,0,0,0,0,17,1] >;

A4×C2×C18 in GAP, Magma, Sage, TeX

A_4\times C_2\times C_{18}
% in TeX

G:=Group("A4xC2xC18");
// GroupNames label

G:=SmallGroup(432,546);
// by ID

G=gap.SmallGroup(432,546);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,108,2287,3989]);
// Polycyclic

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

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