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## G = C2×C18.A4order 432 = 24·33

### Direct product of C2 and C18.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C2×C18.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C9 — C18.A4 — C2×C18.A4
 Lower central Q8 — C3×Q8 — C2×C18.A4
 Upper central C1 — C2×C6 — C2×C18

Generators and relations for C2×C18.A4
G = < a,b,c,d,e | a2=b18=e3=1, c2=d2=b9, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b7, dcd-1=b9c, ece-1=b9cd, ede-1=c >

Subgroups: 211 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, Q8, C9, C9, C32, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, SL2(𝔽3), C2×C12, C3×Q8, C3×Q8, 3- 1+2, C36, C2×C18, C2×C18, C62, C2×SL2(𝔽3), C6×Q8, C2×3- 1+2, Q8⋊C9, C2×C36, Q8×C9, Q8×C9, C3×SL2(𝔽3), C22×3- 1+2, C2×Q8⋊C9, Q8×C18, C6×SL2(𝔽3), C18.A4, C2×C18.A4
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, SL2(𝔽3), C2×A4, 3- 1+2, C3×A4, C2×SL2(𝔽3), C2×3- 1+2, C3×SL2(𝔽3), C6×A4, C9⋊A4, C6×SL2(𝔽3), C18.A4, C2×C9⋊A4, C2×C18.A4

Smallest permutation representation of C2×C18.A4
On 144 points
Generators in S144
(1 139)(2 140)(3 141)(4 142)(5 143)(6 144)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 117 10 126)(2 118 11 109)(3 119 12 110)(4 120 13 111)(5 121 14 112)(6 122 15 113)(7 123 16 114)(8 124 17 115)(9 125 18 116)(19 87 28 78)(20 88 29 79)(21 89 30 80)(22 90 31 81)(23 73 32 82)(24 74 33 83)(25 75 34 84)(26 76 35 85)(27 77 36 86)(37 131 46 140)(38 132 47 141)(39 133 48 142)(40 134 49 143)(41 135 50 144)(42 136 51 127)(43 137 52 128)(44 138 53 129)(45 139 54 130)(55 97 64 106)(56 98 65 107)(57 99 66 108)(58 100 67 91)(59 101 68 92)(60 102 69 93)(61 103 70 94)(62 104 71 95)(63 105 72 96)
(1 35 10 26)(2 36 11 27)(3 19 12 28)(4 20 13 29)(5 21 14 30)(6 22 15 31)(7 23 16 32)(8 24 17 33)(9 25 18 34)(37 59 46 68)(38 60 47 69)(39 61 48 70)(40 62 49 71)(41 63 50 72)(42 64 51 55)(43 65 52 56)(44 66 53 57)(45 67 54 58)(73 123 82 114)(74 124 83 115)(75 125 84 116)(76 126 85 117)(77 109 86 118)(78 110 87 119)(79 111 88 120)(80 112 89 121)(81 113 90 122)(91 139 100 130)(92 140 101 131)(93 141 102 132)(94 142 103 133)(95 143 104 134)(96 144 105 135)(97 127 106 136)(98 128 107 137)(99 129 108 138)
(2 14 8)(3 9 15)(5 17 11)(6 12 18)(19 75 113)(20 88 120)(21 83 109)(22 78 116)(23 73 123)(24 86 112)(25 81 119)(26 76 126)(27 89 115)(28 84 122)(29 79 111)(30 74 118)(31 87 125)(32 82 114)(33 77 121)(34 90 110)(35 85 117)(36 80 124)(37 95 56)(38 108 63)(39 103 70)(40 98 59)(41 93 66)(42 106 55)(43 101 62)(44 96 69)(45 91 58)(46 104 65)(47 99 72)(48 94 61)(49 107 68)(50 102 57)(51 97 64)(52 92 71)(53 105 60)(54 100 67)(128 140 134)(129 135 141)(131 143 137)(132 138 144)

G:=sub<Sym(144)| (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,117,10,126)(2,118,11,109)(3,119,12,110)(4,120,13,111)(5,121,14,112)(6,122,15,113)(7,123,16,114)(8,124,17,115)(9,125,18,116)(19,87,28,78)(20,88,29,79)(21,89,30,80)(22,90,31,81)(23,73,32,82)(24,74,33,83)(25,75,34,84)(26,76,35,85)(27,77,36,86)(37,131,46,140)(38,132,47,141)(39,133,48,142)(40,134,49,143)(41,135,50,144)(42,136,51,127)(43,137,52,128)(44,138,53,129)(45,139,54,130)(55,97,64,106)(56,98,65,107)(57,99,66,108)(58,100,67,91)(59,101,68,92)(60,102,69,93)(61,103,70,94)(62,104,71,95)(63,105,72,96), (1,35,10,26)(2,36,11,27)(3,19,12,28)(4,20,13,29)(5,21,14,30)(6,22,15,31)(7,23,16,32)(8,24,17,33)(9,25,18,34)(37,59,46,68)(38,60,47,69)(39,61,48,70)(40,62,49,71)(41,63,50,72)(42,64,51,55)(43,65,52,56)(44,66,53,57)(45,67,54,58)(73,123,82,114)(74,124,83,115)(75,125,84,116)(76,126,85,117)(77,109,86,118)(78,110,87,119)(79,111,88,120)(80,112,89,121)(81,113,90,122)(91,139,100,130)(92,140,101,131)(93,141,102,132)(94,142,103,133)(95,143,104,134)(96,144,105,135)(97,127,106,136)(98,128,107,137)(99,129,108,138), (2,14,8)(3,9,15)(5,17,11)(6,12,18)(19,75,113)(20,88,120)(21,83,109)(22,78,116)(23,73,123)(24,86,112)(25,81,119)(26,76,126)(27,89,115)(28,84,122)(29,79,111)(30,74,118)(31,87,125)(32,82,114)(33,77,121)(34,90,110)(35,85,117)(36,80,124)(37,95,56)(38,108,63)(39,103,70)(40,98,59)(41,93,66)(42,106,55)(43,101,62)(44,96,69)(45,91,58)(46,104,65)(47,99,72)(48,94,61)(49,107,68)(50,102,57)(51,97,64)(52,92,71)(53,105,60)(54,100,67)(128,140,134)(129,135,141)(131,143,137)(132,138,144)>;

G:=Group( (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,117,10,126)(2,118,11,109)(3,119,12,110)(4,120,13,111)(5,121,14,112)(6,122,15,113)(7,123,16,114)(8,124,17,115)(9,125,18,116)(19,87,28,78)(20,88,29,79)(21,89,30,80)(22,90,31,81)(23,73,32,82)(24,74,33,83)(25,75,34,84)(26,76,35,85)(27,77,36,86)(37,131,46,140)(38,132,47,141)(39,133,48,142)(40,134,49,143)(41,135,50,144)(42,136,51,127)(43,137,52,128)(44,138,53,129)(45,139,54,130)(55,97,64,106)(56,98,65,107)(57,99,66,108)(58,100,67,91)(59,101,68,92)(60,102,69,93)(61,103,70,94)(62,104,71,95)(63,105,72,96), (1,35,10,26)(2,36,11,27)(3,19,12,28)(4,20,13,29)(5,21,14,30)(6,22,15,31)(7,23,16,32)(8,24,17,33)(9,25,18,34)(37,59,46,68)(38,60,47,69)(39,61,48,70)(40,62,49,71)(41,63,50,72)(42,64,51,55)(43,65,52,56)(44,66,53,57)(45,67,54,58)(73,123,82,114)(74,124,83,115)(75,125,84,116)(76,126,85,117)(77,109,86,118)(78,110,87,119)(79,111,88,120)(80,112,89,121)(81,113,90,122)(91,139,100,130)(92,140,101,131)(93,141,102,132)(94,142,103,133)(95,143,104,134)(96,144,105,135)(97,127,106,136)(98,128,107,137)(99,129,108,138), (2,14,8)(3,9,15)(5,17,11)(6,12,18)(19,75,113)(20,88,120)(21,83,109)(22,78,116)(23,73,123)(24,86,112)(25,81,119)(26,76,126)(27,89,115)(28,84,122)(29,79,111)(30,74,118)(31,87,125)(32,82,114)(33,77,121)(34,90,110)(35,85,117)(36,80,124)(37,95,56)(38,108,63)(39,103,70)(40,98,59)(41,93,66)(42,106,55)(43,101,62)(44,96,69)(45,91,58)(46,104,65)(47,99,72)(48,94,61)(49,107,68)(50,102,57)(51,97,64)(52,92,71)(53,105,60)(54,100,67)(128,140,134)(129,135,141)(131,143,137)(132,138,144) );

G=PermutationGroup([[(1,139),(2,140),(3,141),(4,142),(5,143),(6,144),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90)], [(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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,117,10,126),(2,118,11,109),(3,119,12,110),(4,120,13,111),(5,121,14,112),(6,122,15,113),(7,123,16,114),(8,124,17,115),(9,125,18,116),(19,87,28,78),(20,88,29,79),(21,89,30,80),(22,90,31,81),(23,73,32,82),(24,74,33,83),(25,75,34,84),(26,76,35,85),(27,77,36,86),(37,131,46,140),(38,132,47,141),(39,133,48,142),(40,134,49,143),(41,135,50,144),(42,136,51,127),(43,137,52,128),(44,138,53,129),(45,139,54,130),(55,97,64,106),(56,98,65,107),(57,99,66,108),(58,100,67,91),(59,101,68,92),(60,102,69,93),(61,103,70,94),(62,104,71,95),(63,105,72,96)], [(1,35,10,26),(2,36,11,27),(3,19,12,28),(4,20,13,29),(5,21,14,30),(6,22,15,31),(7,23,16,32),(8,24,17,33),(9,25,18,34),(37,59,46,68),(38,60,47,69),(39,61,48,70),(40,62,49,71),(41,63,50,72),(42,64,51,55),(43,65,52,56),(44,66,53,57),(45,67,54,58),(73,123,82,114),(74,124,83,115),(75,125,84,116),(76,126,85,117),(77,109,86,118),(78,110,87,119),(79,111,88,120),(80,112,89,121),(81,113,90,122),(91,139,100,130),(92,140,101,131),(93,141,102,132),(94,142,103,133),(95,143,104,134),(96,144,105,135),(97,127,106,136),(98,128,107,137),(99,129,108,138)], [(2,14,8),(3,9,15),(5,17,11),(6,12,18),(19,75,113),(20,88,120),(21,83,109),(22,78,116),(23,73,123),(24,86,112),(25,81,119),(26,76,126),(27,89,115),(28,84,122),(29,79,111),(30,74,118),(31,87,125),(32,82,114),(33,77,121),(34,90,110),(35,85,117),(36,80,124),(37,95,56),(38,108,63),(39,103,70),(40,98,59),(41,93,66),(42,106,55),(43,101,62),(44,96,69),(45,91,58),(46,104,65),(47,99,72),(48,94,61),(49,107,68),(50,102,57),(51,97,64),(52,92,71),(53,105,60),(54,100,67),(128,140,134),(129,135,141),(131,143,137),(132,138,144)]])

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 6A ··· 6F 6G ··· 6L 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A ··· 18F 18G ··· 18R 36A ··· 36L order 1 2 2 2 3 3 3 3 4 4 6 ··· 6 6 ··· 6 9 9 9 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 12 12 6 6 1 ··· 1 12 ··· 12 3 3 12 12 12 12 6 6 6 6 3 ··· 3 12 ··· 12 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 3 6 type + + - + + image C1 C2 C3 C3 C3 C6 C6 C6 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 C2×A4 3- 1+2 C3×A4 C2×3- 1+2 C6×A4 C9⋊A4 C2×C9⋊A4 C18.A4 kernel C2×C18.A4 C18.A4 C2×Q8⋊C9 Q8×C18 C6×SL2(𝔽3) Q8⋊C9 Q8×C9 C3×SL2(𝔽3) C18 C18 C6 C2×C18 C18 C2×Q8 C2×C6 Q8 C6 C22 C2 C2 # reps 1 1 4 2 2 4 2 2 2 4 12 1 1 2 2 2 2 6 6 4

Matrix representation of C2×C18.A4 in GL5(𝔽37)

 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 36 0 0 0 0 0 36 0 0 0 0 0 31 22 15 0 0 2 14 35 0 0 17 20 29
,
 0 36 0 0 0 1 0 0 0 0 0 0 0 1 36 0 0 1 0 36 0 0 0 0 36
,
 11 27 0 0 0 27 26 0 0 0 0 0 36 0 0 0 0 36 0 1 0 0 36 1 0
,
 1 0 0 0 0 27 26 0 0 0 0 0 1 36 0 0 0 0 36 1 0 0 0 36 0

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,36,0,0,0,0,0,31,2,17,0,0,22,14,20,0,0,15,35,29],[0,1,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,36,36,36],[11,27,0,0,0,27,26,0,0,0,0,0,36,36,36,0,0,0,0,1,0,0,0,1,0],[1,27,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,36,36,36,0,0,0,1,0] >;

C2×C18.A4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC18.A4");
// GroupNames label

G:=SmallGroup(432,328);
// by ID

G=gap.SmallGroup(432,328);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,261,79,1901,172,3414,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^18=e^3=1,c^2=d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^7,d*c*d^-1=b^9*c,e*c*e^-1=b^9*c*d,e*d*e^-1=c>;
// generators/relations

׿
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