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## G = C33⋊7M4(2)  order 432 = 24·33

### 3rd semidirect product of C33 and M4(2) acting via M4(2)/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊7M4(2)
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — S3×C3×C12 — C33⋊7M4(2)
 Lower central C33 — C32×C6 — C33⋊7M4(2)
 Upper central C1 — C4

Generators and relations for C337M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d5 >

Subgroups: 520 in 152 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C3, C3 [×4], C3 [×4], C4, C4, C22, S3, C6, C6 [×4], C6 [×8], C8 [×2], C2×C4, C32, C32 [×4], C32 [×4], Dic3, C12, C12 [×4], C12 [×8], D6, C2×C6 [×4], M4(2), C3×S3 [×4], C3×C6, C3×C6 [×4], C3×C6 [×5], C3⋊C8 [×13], C24, C4×S3, C2×C12 [×4], C33, C3×Dic3 [×4], C3×C12, C3×C12 [×4], C3×C12 [×5], S3×C6 [×4], C62, C8⋊S3, C4.Dic3 [×4], S3×C32, C32×C6, C3×C3⋊C8 [×4], C324C8, C324C8 [×9], S3×C12 [×4], C6×C12, C32×Dic3, C32×C12, S3×C3×C6, D6.Dic3 [×4], C12.58D6, C3×C324C8, C337C8, S3×C3×C12, C337M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, Dic3 [×8], D6 [×5], M4(2), C3⋊S3, C4×S3, C2×Dic3 [×4], C3⋊Dic3 [×2], S32 [×4], C2×C3⋊S3, C8⋊S3, C4.Dic3 [×4], S3×Dic3 [×4], C2×C3⋊Dic3, S3×C3⋊S3, D6.Dic3 [×4], C12.58D6, S3×C3⋊Dic3, C337M4(2)

Smallest permutation representation of C337M4(2)
On 144 points
Generators in S144
```(1 27 57)(2 58 28)(3 29 59)(4 60 30)(5 31 61)(6 62 32)(7 25 63)(8 64 26)(9 24 129)(10 130 17)(11 18 131)(12 132 19)(13 20 133)(14 134 21)(15 22 135)(16 136 23)(33 41 122)(34 123 42)(35 43 124)(36 125 44)(37 45 126)(38 127 46)(39 47 128)(40 121 48)(49 114 92)(50 93 115)(51 116 94)(52 95 117)(53 118 96)(54 89 119)(55 120 90)(56 91 113)(65 81 103)(66 104 82)(67 83 97)(68 98 84)(69 85 99)(70 100 86)(71 87 101)(72 102 88)(73 110 139)(74 140 111)(75 112 141)(76 142 105)(77 106 143)(78 144 107)(79 108 137)(80 138 109)
(1 54 15)(2 16 55)(3 56 9)(4 10 49)(5 50 11)(6 12 51)(7 52 13)(8 14 53)(17 92 30)(18 31 93)(19 94 32)(20 25 95)(21 96 26)(22 27 89)(23 90 28)(24 29 91)(33 82 108)(34 109 83)(35 84 110)(36 111 85)(37 86 112)(38 105 87)(39 88 106)(40 107 81)(41 66 137)(42 138 67)(43 68 139)(44 140 69)(45 70 141)(46 142 71)(47 72 143)(48 144 65)(57 119 135)(58 136 120)(59 113 129)(60 130 114)(61 115 131)(62 132 116)(63 117 133)(64 134 118)(73 124 98)(74 99 125)(75 126 100)(76 101 127)(77 128 102)(78 103 121)(79 122 104)(80 97 123)
(1 119 22)(2 23 120)(3 113 24)(4 17 114)(5 115 18)(6 19 116)(7 117 20)(8 21 118)(9 59 91)(10 92 60)(11 61 93)(12 94 62)(13 63 95)(14 96 64)(15 57 89)(16 90 58)(25 52 133)(26 134 53)(27 54 135)(28 136 55)(29 56 129)(30 130 49)(31 50 131)(32 132 51)(33 137 104)(34 97 138)(35 139 98)(36 99 140)(37 141 100)(38 101 142)(39 143 102)(40 103 144)(41 79 82)(42 83 80)(43 73 84)(44 85 74)(45 75 86)(46 87 76)(47 77 88)(48 81 78)(65 107 121)(66 122 108)(67 109 123)(68 124 110)(69 111 125)(70 126 112)(71 105 127)(72 128 106)
(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 44)(2 41)(3 46)(4 43)(5 48)(6 45)(7 42)(8 47)(9 71)(10 68)(11 65)(12 70)(13 67)(14 72)(15 69)(16 66)(17 84)(18 81)(19 86)(20 83)(21 88)(22 85)(23 82)(24 87)(25 34)(26 39)(27 36)(28 33)(29 38)(30 35)(31 40)(32 37)(49 139)(50 144)(51 141)(52 138)(53 143)(54 140)(55 137)(56 142)(57 125)(58 122)(59 127)(60 124)(61 121)(62 126)(63 123)(64 128)(73 114)(74 119)(75 116)(76 113)(77 118)(78 115)(79 120)(80 117)(89 111)(90 108)(91 105)(92 110)(93 107)(94 112)(95 109)(96 106)(97 133)(98 130)(99 135)(100 132)(101 129)(102 134)(103 131)(104 136)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 8A 8B 8C 8D 12A ··· 12J 12K ··· 12R 12S ··· 12Z 24A 24B 24C 24D order 1 2 2 3 ··· 3 3 3 3 3 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 6 2 ··· 2 4 4 4 4 1 1 6 2 ··· 2 4 4 4 4 6 ··· 6 18 18 54 54 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 S3 S3 Dic3 D6 Dic3 M4(2) C4×S3 C8⋊S3 C4.Dic3 S32 S3×Dic3 D6.Dic3 kernel C33⋊7M4(2) C3×C32⋊4C8 C33⋊7C8 S3×C3×C12 C32×Dic3 S3×C3×C6 C32⋊4C8 S3×C12 C3×Dic3 C3×C12 S3×C6 C33 C3×C6 C32 C32 C12 C6 C3 # reps 1 1 1 1 2 2 1 4 4 5 4 2 2 4 16 4 4 8

Matrix representation of C337M4(2) in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72
,
 15 56 0 0 0 0 0 0 32 58 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 51 20 0 0 0 0 0 0 16 22 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[15,32,0,0,0,0,0,0,56,58,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[51,16,0,0,0,0,0,0,20,22,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C337M4(2) in GAP, Magma, Sage, TeX

`C_3^3\rtimes_7M_4(2)`
`% in TeX`

`G:=Group("C3^3:7M4(2)");`
`// GroupNames label`

`G:=SmallGroup(432,433);`
`// by ID`

`G=gap.SmallGroup(432,433);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;`
`// generators/relations`

׿
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