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## G = C32×C3⋊C16order 432 = 24·33

### Direct product of C32 and C3⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32×C3⋊C16
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C32×C24 — C32×C3⋊C16
 Lower central C3 — C32×C3⋊C16
 Upper central C1 — C3×C24

Generators and relations for C32×C3⋊C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 136 in 92 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C8, C32, C32, C32, C12, C12, C12, C16, C3×C6, C3×C6, C3×C6, C24, C24, C24, C33, C3×C12, C3×C12, C3×C12, C3⋊C16, C48, C32×C6, C3×C24, C3×C24, C3×C24, C32×C12, C3×C3⋊C16, C3×C48, C32×C24, C32×C3⋊C16
Quotients: C1, C2, C3, C4, S3, C6, C8, C32, Dic3, C12, C16, C3×S3, C3×C6, C3⋊C8, C24, C3×Dic3, C3×C12, C3⋊C16, C48, S3×C32, C3×C3⋊C8, C3×C24, C32×Dic3, C3×C3⋊C16, C3×C48, C32×C3⋊C8, C32×C3⋊C16

Smallest permutation representation of C32×C3⋊C16
On 144 points
Generators in S144
(1 118 141)(2 119 142)(3 120 143)(4 121 144)(5 122 129)(6 123 130)(7 124 131)(8 125 132)(9 126 133)(10 127 134)(11 128 135)(12 113 136)(13 114 137)(14 115 138)(15 116 139)(16 117 140)(17 108 59)(18 109 60)(19 110 61)(20 111 62)(21 112 63)(22 97 64)(23 98 49)(24 99 50)(25 100 51)(26 101 52)(27 102 53)(28 103 54)(29 104 55)(30 105 56)(31 106 57)(32 107 58)(33 77 83)(34 78 84)(35 79 85)(36 80 86)(37 65 87)(38 66 88)(39 67 89)(40 68 90)(41 69 91)(42 70 92)(43 71 93)(44 72 94)(45 73 95)(46 74 96)(47 75 81)(48 76 82)
(1 84 27)(2 85 28)(3 86 29)(4 87 30)(5 88 31)(6 89 32)(7 90 17)(8 91 18)(9 92 19)(10 93 20)(11 94 21)(12 95 22)(13 96 23)(14 81 24)(15 82 25)(16 83 26)(33 101 117)(34 102 118)(35 103 119)(36 104 120)(37 105 121)(38 106 122)(39 107 123)(40 108 124)(41 109 125)(42 110 126)(43 111 127)(44 112 128)(45 97 113)(46 98 114)(47 99 115)(48 100 116)(49 137 74)(50 138 75)(51 139 76)(52 140 77)(53 141 78)(54 142 79)(55 143 80)(56 144 65)(57 129 66)(58 130 67)(59 131 68)(60 132 69)(61 133 70)(62 134 71)(63 135 72)(64 136 73)
(1 34 53)(2 54 35)(3 36 55)(4 56 37)(5 38 57)(6 58 39)(7 40 59)(8 60 41)(9 42 61)(10 62 43)(11 44 63)(12 64 45)(13 46 49)(14 50 47)(15 48 51)(16 52 33)(17 124 68)(18 69 125)(19 126 70)(20 71 127)(21 128 72)(22 73 113)(23 114 74)(24 75 115)(25 116 76)(26 77 117)(27 118 78)(28 79 119)(29 120 80)(30 65 121)(31 122 66)(32 67 123)(81 138 99)(82 100 139)(83 140 101)(84 102 141)(85 142 103)(86 104 143)(87 144 105)(88 106 129)(89 130 107)(90 108 131)(91 132 109)(92 110 133)(93 134 111)(94 112 135)(95 136 97)(96 98 137)
(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)

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

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

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

216 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6Q 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12AH 16A ··· 16H 24A ··· 24AF 24AG ··· 24BP 48A ··· 48BL order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 16 ··· 16 24 ··· 24 24 ··· 24 48 ··· 48 size 1 1 1 ··· 1 2 ··· 2 1 1 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C48 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3⋊C16 C3×C3⋊C8 C3×C3⋊C16 kernel C32×C3⋊C16 C32×C24 C3×C3⋊C16 C32×C12 C3×C24 C32×C6 C3×C12 C33 C3×C6 C32 C3×C24 C3×C12 C24 C3×C6 C12 C32 C6 C3 # reps 1 1 8 2 8 4 16 8 32 64 1 1 8 2 8 4 16 32

Matrix representation of C32×C3⋊C16 in GL3(𝔽97) generated by

 1 0 0 0 35 0 0 0 35
,
 61 0 0 0 1 0 0 0 1
,
 1 0 0 0 35 0 0 0 61
,
 70 0 0 0 0 1 0 75 0
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,35],[61,0,0,0,1,0,0,0,1],[1,0,0,0,35,0,0,0,61],[70,0,0,0,0,75,0,1,0] >;

C32×C3⋊C16 in GAP, Magma, Sage, TeX

C_3^2\times C_3\rtimes C_{16}
% in TeX

G:=Group("C3^2xC3:C16");
// GroupNames label

G:=SmallGroup(432,229);
// by ID

G=gap.SmallGroup(432,229);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,-2,-3,126,80,102,14118]);
// Polycyclic

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

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