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## G = C42×3- 1+2order 432 = 24·33

### Direct product of C42 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C42×3- 1+2
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C22×3- 1+2 — C2×C4×3- 1+2 — C42×3- 1+2
 Lower central C1 — C3 — C42×3- 1+2
 Upper central C1 — C4×C12 — C42×3- 1+2

Generators and relations for C42×3- 1+2
G = < a,b,c,d | a4=b4=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 150 in 120 conjugacy classes, 105 normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C42, C18, C3×C6, C2×C12, C2×C12, 3- 1+2, C36, C2×C18, C3×C12, C62, C4×C12, C4×C12, C2×3- 1+2, C2×C36, C6×C12, C4×3- 1+2, C22×3- 1+2, C4×C36, C122, C2×C4×3- 1+2, C42×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, C3×C6, C2×C12, 3- 1+2, C3×C12, C62, C4×C12, C2×3- 1+2, C6×C12, C4×3- 1+2, C22×3- 1+2, C122, C2×C4×3- 1+2, C42×3- 1+2

Smallest permutation representation of C42×3- 1+2
On 144 points
Generators in S144
(1 128 52 142)(2 129 53 143)(3 130 54 144)(4 131 46 136)(5 132 47 137)(6 133 48 138)(7 134 49 139)(8 135 50 140)(9 127 51 141)(10 39 20 34)(11 40 21 35)(12 41 22 36)(13 42 23 28)(14 43 24 29)(15 44 25 30)(16 45 26 31)(17 37 27 32)(18 38 19 33)(55 105 69 91)(56 106 70 92)(57 107 71 93)(58 108 72 94)(59 100 64 95)(60 101 65 96)(61 102 66 97)(62 103 67 98)(63 104 68 99)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)
(1 56 40 88)(2 57 41 89)(3 58 42 90)(4 59 43 82)(5 60 44 83)(6 61 45 84)(7 62 37 85)(8 63 38 86)(9 55 39 87)(10 123 141 91)(11 124 142 92)(12 125 143 93)(13 126 144 94)(14 118 136 95)(15 119 137 96)(16 120 138 97)(17 121 139 98)(18 122 140 99)(19 113 135 104)(20 114 127 105)(21 115 128 106)(22 116 129 107)(23 117 130 108)(24 109 131 100)(25 110 132 101)(26 111 133 102)(27 112 134 103)(28 81 54 72)(29 73 46 64)(30 74 47 65)(31 75 48 66)(32 76 49 67)(33 77 50 68)(34 78 51 69)(35 79 52 70)(36 80 53 71)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(28 31 34)(30 36 33)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(55 58 61)(57 63 60)(65 71 68)(66 69 72)(74 80 77)(75 78 81)(83 89 86)(84 87 90)(91 94 97)(93 99 96)(101 107 104)(102 105 108)(110 116 113)(111 114 117)(119 125 122)(120 123 126)(127 130 133)(129 135 132)(137 143 140)(138 141 144)

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

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

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

176 conjugacy classes

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

176 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 3- 1+2 C2×3- 1+2 C4×3- 1+2 kernel C42×3- 1+2 C2×C4×3- 1+2 C4×C36 C122 C4×3- 1+2 C2×C36 C6×C12 C36 C3×C12 C42 C2×C4 C4 # reps 1 3 6 2 12 18 6 72 24 2 6 24

Matrix representation of C42×3- 1+2 in GL5(𝔽37)

 31 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 6 0 0 0 0 0 6 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 26 9 0 0 0 10 11 10 0 0 0 27 0
,
 26 0 0 0 0 0 26 0 0 0 0 0 1 0 0 0 0 26 10 0 0 0 36 0 26

G:=sub<GL(5,GF(37))| [31,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,26,10,0,0,0,9,11,27,0,0,0,10,0],[26,0,0,0,0,0,26,0,0,0,0,0,1,26,36,0,0,0,10,0,0,0,0,0,26] >;

C42×3- 1+2 in GAP, Magma, Sage, TeX

C_4^2\times 3_-^{1+2}
% in TeX

G:=Group("C4^2xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,202);
// by ID

G=gap.SmallGroup(432,202);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,512,772,1109]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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