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## G = C3×C32⋊7Q16order 432 = 24·33

### Direct product of C3 and C32⋊7Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊7Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C32⋊4Q8 — C3×C32⋊7Q16
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊7Q16
 Upper central C1 — C6 — C12 — C3×Q8

Generators and relations for C3×C327Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 388 in 156 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C32×C6, C3×C3⋊C8, C324C8, C3×Dic6, C324Q8, Q8×C32, Q8×C32, Q8×C32, C3×C3⋊Dic3, C32×C12, C32×C12, C3×C3⋊Q16, C327Q16, C3×C324C8, C3×C324Q8, Q8×C33, C3×C327Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊S3, C3⋊D4, C3×D4, S3×C6, C2×C3⋊S3, C3⋊Q16, C3×Q16, C3×C3⋊S3, C3×C3⋊D4, C327D4, C6×C3⋊S3, C3×C3⋊Q16, C327Q16, C3×C327D4, C3×C327Q16

Smallest permutation representation of C3×C327Q16
On 144 points
Generators in S144
(1 31 106)(2 32 107)(3 25 108)(4 26 109)(5 27 110)(6 28 111)(7 29 112)(8 30 105)(9 95 22)(10 96 23)(11 89 24)(12 90 17)(13 91 18)(14 92 19)(15 93 20)(16 94 21)(33 102 137)(34 103 138)(35 104 139)(36 97 140)(37 98 141)(38 99 142)(39 100 143)(40 101 144)(41 121 71)(42 122 72)(43 123 65)(44 124 66)(45 125 67)(46 126 68)(47 127 69)(48 128 70)(49 120 134)(50 113 135)(51 114 136)(52 115 129)(53 116 130)(54 117 131)(55 118 132)(56 119 133)(57 79 82)(58 80 83)(59 73 84)(60 74 85)(61 75 86)(62 76 87)(63 77 88)(64 78 81)
(1 72 11)(2 12 65)(3 66 13)(4 14 67)(5 68 15)(6 16 69)(7 70 9)(8 10 71)(17 123 107)(18 108 124)(19 125 109)(20 110 126)(21 127 111)(22 112 128)(23 121 105)(24 106 122)(25 44 91)(26 92 45)(27 46 93)(28 94 47)(29 48 95)(30 96 41)(31 42 89)(32 90 43)(33 113 79)(34 80 114)(35 115 73)(36 74 116)(37 117 75)(38 76 118)(39 119 77)(40 78 120)(49 144 64)(50 57 137)(51 138 58)(52 59 139)(53 140 60)(54 61 141)(55 142 62)(56 63 143)(81 134 101)(82 102 135)(83 136 103)(84 104 129)(85 130 97)(86 98 131)(87 132 99)(88 100 133)
(1 106 31)(2 32 107)(3 108 25)(4 26 109)(5 110 27)(6 28 111)(7 112 29)(8 30 105)(9 22 95)(10 96 23)(11 24 89)(12 90 17)(13 18 91)(14 92 19)(15 20 93)(16 94 21)(33 137 102)(34 103 138)(35 139 104)(36 97 140)(37 141 98)(38 99 142)(39 143 100)(40 101 144)(41 121 71)(42 72 122)(43 123 65)(44 66 124)(45 125 67)(46 68 126)(47 127 69)(48 70 128)(49 120 134)(50 135 113)(51 114 136)(52 129 115)(53 116 130)(54 131 117)(55 118 132)(56 133 119)(57 82 79)(58 80 83)(59 84 73)(60 74 85)(61 86 75)(62 76 87)(63 88 77)(64 78 81)
(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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 56 13 52)(10 55 14 51)(11 54 15 50)(12 53 16 49)(17 130 21 134)(18 129 22 133)(19 136 23 132)(20 135 24 131)(25 73 29 77)(26 80 30 76)(27 79 31 75)(28 78 32 74)(33 42 37 46)(34 41 38 45)(35 48 39 44)(36 47 40 43)(65 140 69 144)(66 139 70 143)(67 138 71 142)(68 137 72 141)(81 107 85 111)(82 106 86 110)(83 105 87 109)(84 112 88 108)(89 117 93 113)(90 116 94 120)(91 115 95 119)(92 114 96 118)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)

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

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

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

81 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 4A 4B 4C 6A 6B 6C ··· 6N 8A 8B 12A 12B 12C ··· 12AN 12AO 12AP 24A 24B 24C 24D order 1 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 8 8 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 1 1 2 ··· 2 2 4 36 1 1 2 ··· 2 18 18 2 2 4 ··· 4 36 36 18 18 18 18

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - - image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 Q16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×Q16 C3×C3⋊D4 C3⋊Q16 C3×C3⋊Q16 kernel C3×C32⋊7Q16 C3×C32⋊4C8 C3×C32⋊4Q8 Q8×C33 C32⋊7Q16 C32⋊4C8 C32⋊4Q8 Q8×C32 Q8×C32 C32×C6 C3×C12 C33 C3×Q8 C3×C6 C3×C6 C12 C32 C6 C32 C3 # reps 1 1 1 1 2 2 2 2 4 1 4 2 8 8 2 8 4 16 4 8

Matrix representation of C3×C327Q16 in GL6(𝔽73)

 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 36 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 64 0 0 0 0 0 37 8 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 63 0 0 0 0 9 69 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 59 0 0 0 0 47 32
,
 1 0 0 0 0 0 30 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 17 17 0 0 0 0 13 56

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,36,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,37,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,9,0,0,0,0,63,69,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,47,0,0,0,0,59,32],[1,30,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,13,0,0,0,0,17,56] >;

C3×C327Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_7Q_{16}
% in TeX

G:=Group("C3xC3^2:7Q16");
// GroupNames label

G:=SmallGroup(432,494);
// by ID

G=gap.SmallGroup(432,494);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,4037,14118]);
// 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,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
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