Copied to
clipboard

## G = C32×C4.A4order 432 = 24·33

### Direct product of C32 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C32×C4.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — C32×SL2(𝔽3) — C32×C4.A4
 Lower central Q8 — C32×C4.A4
 Upper central C1 — C3×C12

Generators and relations for C32×C4.A4
G = < a,b,c,d,e,f | a3=b3=c4=f3=1, d2=e2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=c2d, fdf-1=c2de, fef-1=d >

Subgroups: 446 in 170 conjugacy classes, 74 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C32, C12, C12, C2×C6, C4○D4, C3×C6, C3×C6, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C33, C3×C12, C3×C12, C62, C4.A4, C3×C4○D4, C32×C6, C3×SL2(𝔽3), C6×C12, D4×C32, Q8×C32, C32×C12, C3×C4.A4, C32×C4○D4, C32×SL2(𝔽3), C32×C4.A4
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, C33, C3×A4, C4.A4, C32×C6, C6×A4, C32×A4, C3×C4.A4, A4×C3×C6, C32×C4.A4

Smallest permutation representation of C32×C4.A4
On 144 points
Generators in S144
(1 79 95)(2 80 96)(3 77 93)(4 78 94)(5 26 10)(6 27 11)(7 28 12)(8 25 9)(13 33 29)(14 34 30)(15 35 31)(16 36 32)(17 37 21)(18 38 22)(19 39 23)(20 40 24)(41 75 57)(42 76 58)(43 73 59)(44 74 60)(45 65 61)(46 66 62)(47 67 63)(48 68 64)(49 69 53)(50 70 54)(51 71 55)(52 72 56)(81 101 97)(82 102 98)(83 103 99)(84 104 100)(85 105 89)(86 106 90)(87 107 91)(88 108 92)(109 141 125)(110 142 126)(111 143 127)(112 144 128)(113 133 129)(114 134 130)(115 135 131)(116 136 132)(117 137 121)(118 138 122)(119 139 123)(120 140 124)
(1 87 83)(2 88 84)(3 85 81)(4 86 82)(5 38 34)(6 39 35)(7 40 36)(8 37 33)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 29 25)(22 30 26)(23 31 27)(24 32 28)(41 49 45)(42 50 46)(43 51 47)(44 52 48)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(65 75 69)(66 76 70)(67 73 71)(68 74 72)(77 105 101)(78 106 102)(79 107 103)(80 108 104)(89 97 93)(90 98 94)(91 99 95)(92 100 96)(109 117 113)(110 118 114)(111 119 115)(112 120 116)(121 129 125)(122 130 126)(123 131 127)(124 132 128)(133 141 137)(134 142 138)(135 143 139)(136 144 140)
(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)
(1 109 3 111)(2 110 4 112)(5 76 7 74)(6 73 8 75)(9 41 11 43)(10 42 12 44)(13 45 15 47)(14 46 16 48)(17 49 19 51)(18 50 20 52)(21 53 23 55)(22 54 24 56)(25 57 27 59)(26 58 28 60)(29 61 31 63)(30 62 32 64)(33 65 35 67)(34 66 36 68)(37 69 39 71)(38 70 40 72)(77 143 79 141)(78 144 80 142)(81 115 83 113)(82 116 84 114)(85 119 87 117)(86 120 88 118)(89 123 91 121)(90 124 92 122)(93 127 95 125)(94 128 96 126)(97 131 99 129)(98 132 100 130)(101 135 103 133)(102 136 104 134)(105 139 107 137)(106 140 108 138)
(1 9 3 11)(2 10 4 12)(5 78 7 80)(6 79 8 77)(13 81 15 83)(14 82 16 84)(17 85 19 87)(18 86 20 88)(21 89 23 91)(22 90 24 92)(25 93 27 95)(26 94 28 96)(29 97 31 99)(30 98 32 100)(33 101 35 103)(34 102 36 104)(37 105 39 107)(38 106 40 108)(41 109 43 111)(42 110 44 112)(45 113 47 115)(46 114 48 116)(49 117 51 119)(50 118 52 120)(53 121 55 123)(54 122 56 124)(57 125 59 127)(58 126 60 128)(61 129 63 131)(62 130 64 132)(65 133 67 135)(66 134 68 136)(69 137 71 139)(70 138 72 140)(73 143 75 141)(74 144 76 142)
(1 79 95)(2 80 96)(3 77 93)(4 78 94)(5 58 110)(6 59 111)(7 60 112)(8 57 109)(9 75 125)(10 76 126)(11 73 127)(12 74 128)(13 65 129)(14 66 130)(15 67 131)(16 68 132)(17 69 121)(18 70 122)(19 71 123)(20 72 124)(21 49 137)(22 50 138)(23 51 139)(24 52 140)(25 41 141)(26 42 142)(27 43 143)(28 44 144)(29 45 133)(30 46 134)(31 47 135)(32 48 136)(33 61 113)(34 62 114)(35 63 115)(36 64 116)(37 53 117)(38 54 118)(39 55 119)(40 56 120)(81 101 97)(82 102 98)(83 103 99)(84 104 100)(85 105 89)(86 106 90)(87 107 91)(88 108 92)

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

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

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

126 conjugacy classes

 class 1 2A 2B 3A ··· 3H 3I ··· 3Z 4A 4B 4C 6A ··· 6H 6I ··· 6Z 6AA ··· 6AH 12A ··· 12P 12Q ··· 12AZ 12BA ··· 12BH order 1 2 2 3 ··· 3 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 6 1 ··· 1 4 ··· 4 1 1 6 1 ··· 1 4 ··· 4 6 ··· 6 1 ··· 1 4 ··· 4 6 ··· 6

126 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 C4.A4 C3×C4.A4 A4 C2×A4 C3×A4 C6×A4 kernel C32×C4.A4 C32×SL2(𝔽3) C3×C4.A4 C32×C4○D4 C3×SL2(𝔽3) Q8×C32 C32 C3 C3×C12 C3×C6 C12 C6 # reps 1 1 24 2 24 2 6 48 1 1 8 8

Matrix representation of C32×C4.A4 in GL3(𝔽13) generated by

 3 0 0 0 3 0 0 0 3
,
 9 0 0 0 1 0 0 0 1
,
 12 0 0 0 5 0 0 0 5
,
 1 0 0 0 0 1 0 12 0
,
 1 0 0 0 4 3 0 3 9
,
 9 0 0 0 3 0 0 9 1
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[9,0,0,0,1,0,0,0,1],[12,0,0,0,5,0,0,0,5],[1,0,0,0,0,12,0,1,0],[1,0,0,0,4,3,0,3,9],[9,0,0,0,3,9,0,0,1] >;

C32×C4.A4 in GAP, Magma, Sage, TeX

C_3^2\times C_4.A_4
% in TeX

G:=Group("C3^2xC4.A4");
// GroupNames label

G:=SmallGroup(432,699);
// by ID

G=gap.SmallGroup(432,699);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,1901,172,3414,285,124]);
// Polycyclic

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

׿
×
𝔽