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## G = C32×CSU2(𝔽3)  order 432 = 24·33

### Direct product of C32 and CSU2(𝔽3)

Aliases: C32×CSU2(𝔽3), C6.20(C3×S4), (C3×C6).25S4, Q8.(S3×C32), C2.2(C32×S4), SL2(𝔽3).(C3×C6), (Q8×C32).17S3, (C3×SL2(𝔽3)).7C6, (C32×SL2(𝔽3)).3C2, (C3×Q8).10(C3×S3), SmallGroup(432,613)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C32×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C32×SL2(𝔽3) — C32×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C32×CSU2(𝔽3)
 Upper central C1 — C3×C6

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

Subgroups: 354 in 102 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C4, C6, C6, C8, Q8, Q8, C32, C32, Dic3, C12, Q16, C3×C6, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C3×Q8, C3×Q8, C33, C3×Dic3, C3×C12, C3×Q16, CSU2(𝔽3), C32×C6, C3×C24, C3×SL2(𝔽3), C3×SL2(𝔽3), Q8×C32, Q8×C32, C32×Dic3, C32×Q16, C3×CSU2(𝔽3), C32×SL2(𝔽3), C32×CSU2(𝔽3)
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, S4, CSU2(𝔽3), S3×C32, C3×S4, C3×CSU2(𝔽3), C32×S4, C32×CSU2(𝔽3)

Smallest permutation representation of C32×CSU2(𝔽3)
On 144 points
Generators in S144
(1 54 30)(2 55 31)(3 56 32)(4 53 29)(5 123 99)(6 124 100)(7 121 97)(8 122 98)(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)(77 125 101)(78 126 102)(79 127 103)(80 128 104)(81 129 105)(82 130 106)(83 131 107)(84 132 108)(85 133 109)(86 134 110)(87 135 111)(88 136 112)(89 137 113)(90 138 114)(91 139 115)(92 140 116)(93 141 117)(94 142 118)(95 143 119)(96 144 120)
(1 22 14)(2 23 15)(3 24 16)(4 21 13)(5 139 131)(6 140 132)(7 137 129)(8 138 130)(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 93 85)(78 94 86)(79 95 87)(80 96 88)(81 97 89)(82 98 90)(83 99 91)(84 100 92)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 121 113)(106 122 114)(107 123 115)(108 124 116)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(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)
(2 11 10)(4 9 12)(5 8 142)(6 144 7)(13 17 20)(15 19 18)(21 25 28)(23 27 26)(29 33 36)(31 35 34)(37 41 44)(39 43 42)(45 49 52)(47 51 50)(53 57 60)(55 59 58)(61 65 68)(63 67 66)(69 73 76)(71 75 74)(78 83 82)(80 81 84)(86 91 90)(88 89 92)(94 99 98)(96 97 100)(102 107 106)(104 105 108)(110 115 114)(112 113 116)(118 123 122)(120 121 124)(126 131 130)(128 129 132)(134 139 138)(136 137 140)
(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)(5,123,99)(6,124,100)(7,121,97)(8,122,98)(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)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,139,131)(6,140,132)(7,137,129)(8,138,130)(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)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,121,113)(106,122,114)(107,123,115)(108,124,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (2,11,10)(4,9,12)(5,8,142)(6,144,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(29,33,36)(31,35,34)(37,41,44)(39,43,42)(45,49,52)(47,51,50)(53,57,60)(55,59,58)(61,65,68)(63,67,66)(69,73,76)(71,75,74)(78,83,82)(80,81,84)(86,91,90)(88,89,92)(94,99,98)(96,97,100)(102,107,106)(104,105,108)(110,115,114)(112,113,116)(118,123,122)(120,121,124)(126,131,130)(128,129,132)(134,139,138)(136,137,140), (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)(5,123,99)(6,124,100)(7,121,97)(8,122,98)(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)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,139,131)(6,140,132)(7,137,129)(8,138,130)(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)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,121,113)(106,122,114)(107,123,115)(108,124,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (2,11,10)(4,9,12)(5,8,142)(6,144,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(29,33,36)(31,35,34)(37,41,44)(39,43,42)(45,49,52)(47,51,50)(53,57,60)(55,59,58)(61,65,68)(63,67,66)(69,73,76)(71,75,74)(78,83,82)(80,81,84)(86,91,90)(88,89,92)(94,99,98)(96,97,100)(102,107,106)(104,105,108)(110,115,114)(112,113,116)(118,123,122)(120,121,124)(126,131,130)(128,129,132)(134,139,138)(136,137,140), (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),(5,123,99),(6,124,100),(7,121,97),(8,122,98),(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),(77,125,101),(78,126,102),(79,127,103),(80,128,104),(81,129,105),(82,130,106),(83,131,107),(84,132,108),(85,133,109),(86,134,110),(87,135,111),(88,136,112),(89,137,113),(90,138,114),(91,139,115),(92,140,116),(93,141,117),(94,142,118),(95,143,119),(96,144,120)], [(1,22,14),(2,23,15),(3,24,16),(4,21,13),(5,139,131),(6,140,132),(7,137,129),(8,138,130),(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),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(81,97,89),(82,98,90),(83,99,91),(84,100,92),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(105,121,113),(106,122,114),(107,123,115),(108,124,116),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(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)], [(2,11,10),(4,9,12),(5,8,142),(6,144,7),(13,17,20),(15,19,18),(21,25,28),(23,27,26),(29,33,36),(31,35,34),(37,41,44),(39,43,42),(45,49,52),(47,51,50),(53,57,60),(55,59,58),(61,65,68),(63,67,66),(69,73,76),(71,75,74),(78,83,82),(80,81,84),(86,91,90),(88,89,92),(94,99,98),(96,97,100),(102,107,106),(104,105,108),(110,115,114),(112,113,116),(118,123,122),(120,121,124),(126,131,130),(128,129,132),(134,139,138),(136,137,140)], [(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)]])

72 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6Q 8A 8B 12A ··· 12H 12I ··· 12P 24A ··· 24P order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 ··· 1 8 ··· 8 6 12 1 ··· 1 8 ··· 8 6 6 6 ··· 6 12 ··· 12 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 4 4 type + + + - + - image C1 C2 C3 C6 S3 C3×S3 CSU2(𝔽3) C3×CSU2(𝔽3) S4 C3×S4 CSU2(𝔽3) C3×CSU2(𝔽3) kernel C32×CSU2(𝔽3) C32×SL2(𝔽3) C3×CSU2(𝔽3) C3×SL2(𝔽3) Q8×C32 C3×Q8 C32 C3 C3×C6 C6 C32 C3 # reps 1 1 8 8 1 8 2 16 2 16 1 8

Matrix representation of C32×CSU2(𝔽3) in GL3(𝔽73) generated by

 1 0 0 0 8 0 0 0 8
,
 64 0 0 0 1 0 0 0 1
,
 1 0 0 0 21 3 0 23 52
,
 1 0 0 0 2 50 0 51 71
,
 1 0 0 0 23 52 0 2 49
,
 1 0 0 0 12 51 0 63 61
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[64,0,0,0,1,0,0,0,1],[1,0,0,0,21,23,0,3,52],[1,0,0,0,2,51,0,50,71],[1,0,0,0,23,2,0,52,49],[1,0,0,0,12,63,0,51,61] >;

C32×CSU2(𝔽3) in GAP, Magma, Sage, TeX

C_3^2\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("C3^2xCSU(2,3)");
// GroupNames label

G:=SmallGroup(432,613);
// by ID

G=gap.SmallGroup(432,613);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,1011,3784,655,172,2273,404,285,124]);
// 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,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,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

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