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## G = C3⋊Dic3.2A4order 432 = 24·33

### The non-split extension by C3⋊Dic3 of A4 acting through Inn(C3⋊Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C32 — C3⋊Dic3.2A4
 Chief series C1 — C2 — C6 — C3×C6 — Q8×C32 — C32×SL2(𝔽3) — C3⋊Dic3.2A4
 Lower central Q8×C32 — C3⋊Dic3.2A4
 Upper central C1 — C2

Generators and relations for C3⋊Dic3.2A4
G = < a,b,c,d,e,f | a3=b6=f3=1, c2=d2=e2=b3, ab=ba, cac-1=a-1, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b3d, fdf-1=b3de, fef-1=d >

Subgroups: 718 in 114 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, D6, C4○D4, C3⋊S3, C3×C6, C3×C6, SL2(𝔽3), SL2(𝔽3), C4×S3, D12, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C4.A4, Q83S3, C32×C6, C3×SL2(𝔽3), C3×SL2(𝔽3), C4×C3⋊S3, C12⋊S3, Q8×C32, C3×C3⋊Dic3, Dic3.A4, C12.26D6, C32×SL2(𝔽3), C3⋊Dic3.2A4
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C3⋊S3, C2×A4, C4.A4, C3×C3⋊S3, S3×A4, Dic3.A4, A4×C3⋊S3, C3⋊Dic3.2A4

Smallest permutation representation of C3⋊Dic3.2A4
On 144 points
Generators in S144
(1 28 22)(2 29 23)(3 30 24)(4 25 19)(5 26 20)(6 27 21)(7 16 144)(8 17 139)(9 18 140)(10 13 141)(11 14 142)(12 15 143)(31 37 46)(32 38 47)(33 39 48)(34 40 43)(35 41 44)(36 42 45)(49 55 64)(50 56 65)(51 57 66)(52 58 61)(53 59 62)(54 60 63)(67 79 76)(68 80 77)(69 81 78)(70 82 73)(71 83 74)(72 84 75)(85 97 92)(86 98 93)(87 99 94)(88 100 95)(89 101 96)(90 102 91)(103 115 110)(104 116 111)(105 117 112)(106 118 113)(107 119 114)(108 120 109)(121 133 128)(122 134 129)(123 135 130)(124 136 131)(125 137 132)(126 138 127)
(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 86 4 89)(2 85 5 88)(3 90 6 87)(7 72 10 69)(8 71 11 68)(9 70 12 67)(13 78 16 75)(14 77 17 74)(15 76 18 73)(19 101 22 98)(20 100 23 97)(21 99 24 102)(25 96 28 93)(26 95 29 92)(27 94 30 91)(31 116 34 119)(32 115 35 118)(33 120 36 117)(37 104 40 107)(38 103 41 106)(39 108 42 105)(43 114 46 111)(44 113 47 110)(45 112 48 109)(49 134 52 137)(50 133 53 136)(51 138 54 135)(55 122 58 125)(56 121 59 124)(57 126 60 123)(61 132 64 129)(62 131 65 128)(63 130 66 127)(79 140 82 143)(80 139 83 142)(81 144 84 141)
(1 40 4 37)(2 41 5 38)(3 42 6 39)(7 135 10 138)(8 136 11 133)(9 137 12 134)(13 127 16 130)(14 128 17 131)(15 129 18 132)(19 31 22 34)(20 32 23 35)(21 33 24 36)(25 46 28 43)(26 47 29 44)(27 48 30 45)(49 67 52 70)(50 68 53 71)(51 69 54 72)(55 79 58 82)(56 80 59 83)(57 81 60 84)(61 73 64 76)(62 74 65 77)(63 75 66 78)(85 106 88 103)(86 107 89 104)(87 108 90 105)(91 112 94 109)(92 113 95 110)(93 114 96 111)(97 118 100 115)(98 119 101 116)(99 120 102 117)(121 139 124 142)(122 140 125 143)(123 141 126 144)
(1 58 4 55)(2 59 5 56)(3 60 6 57)(7 120 10 117)(8 115 11 118)(9 116 12 119)(13 112 16 109)(14 113 17 110)(15 114 18 111)(19 49 22 52)(20 50 23 53)(21 51 24 54)(25 64 28 61)(26 65 29 62)(27 66 30 63)(31 70 34 67)(32 71 35 68)(33 72 36 69)(37 82 40 79)(38 83 41 80)(39 84 42 81)(43 76 46 73)(44 77 47 74)(45 78 48 75)(85 124 88 121)(86 125 89 122)(87 126 90 123)(91 130 94 127)(92 131 95 128)(93 132 96 129)(97 136 100 133)(98 137 101 134)(99 138 102 135)(103 142 106 139)(104 143 107 140)(105 144 108 141)
(7 120 138)(8 115 133)(9 116 134)(10 117 135)(11 118 136)(12 119 137)(13 112 130)(14 113 131)(15 114 132)(16 109 127)(17 110 128)(18 111 129)(31 49 67)(32 50 68)(33 51 69)(34 52 70)(35 53 71)(36 54 72)(37 55 79)(38 56 80)(39 57 81)(40 58 82)(41 59 83)(42 60 84)(43 61 73)(44 62 74)(45 63 75)(46 64 76)(47 65 77)(48 66 78)(103 121 139)(104 122 140)(105 123 141)(106 124 142)(107 125 143)(108 126 144)

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

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

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

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 3G ··· 3N 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G ··· 6N 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 3 3 3 3 3 3 3 ··· 3 4 4 4 6 6 6 6 6 6 6 ··· 6 12 12 12 12 12 12 12 12 size 1 1 54 2 2 2 2 4 4 8 ··· 8 6 9 9 2 2 2 2 4 4 8 ··· 8 12 12 12 12 36 36 36 36

42 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 4 4 6 type + + + + + + + image C1 C2 C3 C6 S3 C3×S3 C4.A4 A4 C2×A4 Dic3.A4 Dic3.A4 S3×A4 kernel C3⋊Dic3.2A4 C32×SL2(𝔽3) C12.26D6 Q8×C32 C3×SL2(𝔽3) C3×Q8 C32 C3⋊Dic3 C3×C6 C3 C3 C6 # reps 1 1 2 2 4 8 6 1 1 4 8 4

Matrix representation of C3⋊Dic3.2A4 in GL6(𝔽13)

 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 5 11
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 3 0 0 0 0 6 10 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 2 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 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 10 9 0 0 0 0 9 3 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 9 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,2,11],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,6,0,0,0,0,3,10,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,2,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,9,0,0,0,0,9,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,9,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3⋊Dic3.2A4 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_3._2A_4
% in TeX

G:=Group("C3:Dic3.2A4");
// GroupNames label

G:=SmallGroup(432,625);
// by ID

G=gap.SmallGroup(432,625);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-3,-2,1512,198,772,94,1081,528,1684,6053]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^6=f^3=1,c^2=d^2=e^2=b^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=b^3*d,f*d*f^-1=b^3*d*e,f*e*f^-1=d>;
// generators/relations

׿
×
𝔽