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## G = C3×C32⋊5Q16order 432 = 24·33

### Direct product of C3 and C32⋊5Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊5Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C32⋊4Q8 — C3×C32⋊5Q16
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊5Q16
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×C325Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 420 in 140 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C24, C24, C24, Dic6, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, Dic12, C3×Q16, C32×C6, C3×C24, C3×C24, C3×C24, C3×Dic6, C324Q8, C3×C3⋊Dic3, C32×C12, C3×Dic12, C325Q16, C32×C24, C3×C324Q8, C3×C325Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊S3, D12, C3×D4, S3×C6, C2×C3⋊S3, Dic12, C3×Q16, C3×C3⋊S3, C3×D12, C12⋊S3, C6×C3⋊S3, C3×Dic12, C325Q16, C3×C12⋊S3, C3×C325Q16

Smallest permutation representation of C3×C325Q16
On 144 points
Generators in S144
(1 79 121)(2 80 122)(3 73 123)(4 74 124)(5 75 125)(6 76 126)(7 77 127)(8 78 128)(9 52 20)(10 53 21)(11 54 22)(12 55 23)(13 56 24)(14 49 17)(15 50 18)(16 51 19)(25 58 90)(26 59 91)(27 60 92)(28 61 93)(29 62 94)(30 63 95)(31 64 96)(32 57 89)(33 45 141)(34 46 142)(35 47 143)(36 48 144)(37 41 137)(38 42 138)(39 43 139)(40 44 140)(65 129 88)(66 130 81)(67 131 82)(68 132 83)(69 133 84)(70 134 85)(71 135 86)(72 136 87)(97 117 112)(98 118 105)(99 119 106)(100 120 107)(101 113 108)(102 114 109)(103 115 110)(104 116 111)
(1 68 100)(2 69 101)(3 70 102)(4 71 103)(5 72 104)(6 65 97)(7 66 98)(8 67 99)(9 140 25)(10 141 26)(11 142 27)(12 143 28)(13 144 29)(14 137 30)(15 138 31)(16 139 32)(17 41 95)(18 42 96)(19 43 89)(20 44 90)(21 45 91)(22 46 92)(23 47 93)(24 48 94)(33 59 53)(34 60 54)(35 61 55)(36 62 56)(37 63 49)(38 64 50)(39 57 51)(40 58 52)(73 134 114)(74 135 115)(75 136 116)(76 129 117)(77 130 118)(78 131 119)(79 132 120)(80 133 113)(81 105 127)(82 106 128)(83 107 121)(84 108 122)(85 109 123)(86 110 124)(87 111 125)(88 112 126)
(1 132 107)(2 133 108)(3 134 109)(4 135 110)(5 136 111)(6 129 112)(7 130 105)(8 131 106)(9 44 58)(10 45 59)(11 46 60)(12 47 61)(13 48 62)(14 41 63)(15 42 64)(16 43 57)(17 37 30)(18 38 31)(19 39 32)(20 40 25)(21 33 26)(22 34 27)(23 35 28)(24 36 29)(49 137 95)(50 138 96)(51 139 89)(52 140 90)(53 141 91)(54 142 92)(55 143 93)(56 144 94)(65 117 126)(66 118 127)(67 119 128)(68 120 121)(69 113 122)(70 114 123)(71 115 124)(72 116 125)(73 85 102)(74 86 103)(75 87 104)(76 88 97)(77 81 98)(78 82 99)(79 83 100)(80 84 101)
(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 29 5 25)(2 28 6 32)(3 27 7 31)(4 26 8 30)(9 100 13 104)(10 99 14 103)(11 98 15 102)(12 97 16 101)(17 110 21 106)(18 109 22 105)(19 108 23 112)(20 107 24 111)(33 131 37 135)(34 130 38 134)(35 129 39 133)(36 136 40 132)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 115 53 119)(50 114 54 118)(51 113 55 117)(52 120 56 116)(57 80 61 76)(58 79 62 75)(59 78 63 74)(60 77 64 73)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)(89 122 93 126)(90 121 94 125)(91 128 95 124)(92 127 96 123)

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

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

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

117 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 4A 4B 4C 6A 6B 6C ··· 6N 8A 8B 12A ··· 12Z 12AA 12AB 12AC 12AD 24A ··· 24AZ order 1 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 8 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 ··· 2 2 36 36 1 1 2 ··· 2 2 2 2 ··· 2 36 36 36 36 2 ··· 2

117 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C3 C6 C6 S3 D4 D6 Q16 C3×S3 D12 C3×D4 S3×C6 Dic12 C3×Q16 C3×D12 C3×Dic12 kernel C3×C32⋊5Q16 C32×C24 C3×C32⋊4Q8 C32⋊5Q16 C3×C24 C32⋊4Q8 C3×C24 C32×C6 C3×C12 C33 C24 C3×C6 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 2 2 2 4 4 1 4 2 8 8 2 8 16 4 16 32

Matrix representation of C3×C325Q16 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 8 0 0 0 0 8
,
 64 0 0 0 0 8 0 0 0 0 64 0 0 0 0 8
,
 8 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 22 0 0 0 0 10 0 0 0 0 46 0 0 0 0 27
,
 0 1 0 0 72 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[64,0,0,0,0,8,0,0,0,0,64,0,0,0,0,8],[8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,10,0,0,0,0,46,0,0,0,0,27],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C325Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_5Q_{16}
% in TeX

G:=Group("C3xC3^2:5Q16");
// GroupNames label

G:=SmallGroup(432,484);
// by ID

G=gap.SmallGroup(432,484);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,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^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽