C3^2:4CSU(2,3) - GroupNames

G = C₃²⋊₄CSU₂(𝔽₃) order 432 = 2⁴·3³

2^nd semidirect product of C₃² and CSU₂(𝔽₃) acting via CSU₂(𝔽₃)/SL₂(𝔽₃)=C₂

Aliases: C₃²⋊₄CSU₂(𝔽₃), C₃⋊(C₆.₅S₄), C₆.₉(C₃⋊S₄), (C₃×C₆).₂₂S₄, Q₈.(C₃³⋊C₂), (Q₈×C₃²).₁₆S₃, SL₂(𝔽₃).(C₃⋊S₃), C₂.₂(C₃²⋊₄S₄), (C₃×SL₂(𝔽₃)).₈S₃, (C₃²×SL₂(𝔽₃)).₂C₂, (C₃×Q₈).₅(C₃⋊S₃), SmallGroup(432,619)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — Q₈ — C₃²×SL₂(𝔽₃) — C₃²⋊₄CSU₂(𝔽₃)

Chief series

C₁ — C₂ — Q₈ — C₃×Q₈ — Q₈×C₃² — C₃²×SL₂(𝔽₃) — C₃²⋊₄CSU₂(𝔽₃)

Lower central

C₃²×SL₂(𝔽₃) — C₃²⋊₄CSU₂(𝔽₃)

Upper central

C₁ — C₂

Generators and relations for C₃²⋊₄CSU₂(𝔽₃)
G = < a,b,c,d,e,f | a³=b³=c⁴=e³=1, d²=f²=c², ab=ba, ac=ca, ad=da, ae=ea, faf^-1=a^-1, bc=cb, bd=db, be=eb, fbf^-1=b^-1, dcd^-1=fdf^-1=c^-1, ece^-1=cd, fcf^-1=c²d, ede^-1=c, fef^-1=e^-1 >

Subgroups: 1154 in 158 conjugacy classes, 41 normal (8 characteristic)
C₁, C₂, C₃, C₃, C₄, C₆, C₆, C₈, Q₈, Q₈, C₃², C₃², Dic₃, C₁₂, Q₁₆, C₃×C₆, C₃×C₆, C₃⋊C₈, SL₂(𝔽₃), Dic₆, C₃×Q₈, C₃³, C₃⋊Dic₃, C₃×C₁₂, C₃⋊Q₁₆, CSU₂(𝔽₃), C₃²×C₆, C₃²⋊₄C₈, C₃×SL₂(𝔽₃), C₃²⋊₄Q₈, Q₈×C₃², C₃³⋊₅C₄, C₃²⋊₇Q₁₆, C₆.₅S₄, C₃²×SL₂(𝔽₃), C₃²⋊₄CSU₂(𝔽₃)
Quotients: C₁, C₂, S₃, C₃⋊S₃, S₄, CSU₂(𝔽₃), C₃³⋊C₂, C₃⋊S₄, C₆.₅S₄, C₃²⋊₄S₄, C₃²⋊₄CSU₂(𝔽₃)

Smallest permutation representation of C₃²⋊₄CSU₂(𝔽₃)
►On 144 points

Generators in S₁₄₄

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

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

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

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

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

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

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

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

36 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	···	3M	4A	4B	6A	6B	6C	6D	6E	···	6M	8A	8B	12A	12B	12C	12D
order	1	2	3	3	3	3	3	···	3	4	4	6	6	6	6	6	···	6	8	8	12	12	12	12
size	1	1	2	2	2	2	8	···	8	6	108	2	2	2	2	8	···	8	54	54	12	12	12	12

36 irreducible representations

dim	1	1	2	2	2	3	4	4	6
type	+	+	+	+	-	+	-	-	+
image	C₁	C₂	S₃	S₃	CSU₂(𝔽₃)	S₄	CSU₂(𝔽₃)	C₆.₅S₄	C₃⋊S₄
kernel	C₃²⋊₄CSU₂(𝔽₃)	C₃²×SL₂(𝔽₃)	C₃×SL₂(𝔽₃)	Q₈×C₃²	C₃²	C₃×C₆	C₃²	C₃	C₆
# reps	1	1	12	1	2	2	1	12	4

Matrix representation of C₃²⋊₄CSU₂(𝔽₃) ►in GL₆(𝔽₇₃)

72	72	0	0	0	0
1	0	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

72	72	0	0	0	0
1	0	0	0	0	0
0	0	72	1	0	0
0	0	72	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	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	6	4
0	0	0	0	9	67

1	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	63	7
0	0	0	0	69	10

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	72	0	0
0	0	1	72	0	0
0	0	0	0	72	1
0	0	0	0	72	0

1	0	0	0	0	0
72	72	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	36	48
0	0	0	0	11	37

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,9,0,0,0,0,4,67],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,63,69,0,0,0,0,7,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,11,0,0,0,0,48,37] >;

C₃²⋊₄CSU₂(𝔽₃) in GAP, Magma, Sage, TeX

C_3^2\rtimes_4{\rm CSU}_2({\mathbb F}_3)

% in TeX

G:=Group("C3^2:4CSU(2,3)");

// GroupNames label

G:=SmallGroup(432,619);

// by ID

G=gap.SmallGroup(432,619);

# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,57,254,1011,3784,5681,172,2273,3414,285,124]);

// Polycyclic

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

// generators/relations

G = C32⋊4CSU2(𝔽3) order 432 = 24·33

2nd semidirect product of C32 and CSU2(𝔽3) acting via CSU2(𝔽3)/SL2(𝔽3)=C2

G = C₃²⋊₄CSU₂(𝔽₃) order 432 = 2⁴·3³

2^nd semidirect product of C₃² and CSU₂(𝔽₃) acting via CSU₂(𝔽₃)/SL₂(𝔽₃)=C₂