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G = C337M4(2)  order 432 = 24·33

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

metabelian, supersoluble, monomial

Aliases: C337M4(2), C12.70S32, (S3×C12).3S3, C337C89C2, D6.(C3⋊Dic3), C324C813S3, (C3×C12).165D6, (S3×C6).7Dic3, C6.29(S3×Dic3), Dic3.(C3⋊Dic3), C33(D6.Dic3), C3214(C8⋊S3), C31(C12.58D6), (C3×Dic3).3Dic3, C325(C4.Dic3), (C32×Dic3).3C4, (C32×C12).67C22, (S3×C3×C6).6C4, C4.25(S3×C3⋊S3), (S3×C3×C12).1C2, C12.40(C2×C3⋊S3), (C3×C6).90(C4×S3), C2.3(S3×C3⋊Dic3), C6.2(C2×C3⋊Dic3), (C4×S3).2(C3⋊S3), (C3×C324C8)⋊11C2, (C32×C6).34(C2×C4), (C3×C6).36(C2×Dic3), SmallGroup(432,433)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C337M4(2)
C1C3C32C33C32×C6C32×C12S3×C3×C12 — C337M4(2)
C33C32×C6 — C337M4(2)
C1C4

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 2A2B3A···3E3F3G3H3I4A4B4C6A···6E6F6G6H6I6J···6Q8A8B8C8D12A···12J12K···12R12S···12Z24A24B24C24D
order1223···333334446···666666···6888812···1212···1212···1224242424
size1162···244441162···244446···6181854542···24···46···618181818

66 irreducible representations

dim111111222222222444
type++++++-+-+-
imageC1C2C2C2C4C4S3S3Dic3D6Dic3M4(2)C4×S3C8⋊S3C4.Dic3S32S3×Dic3D6.Dic3
kernelC337M4(2)C3×C324C8C337C8S3×C3×C12C32×Dic3S3×C3×C6C324C8S3×C12C3×Dic3C3×C12S3×C6C33C3×C6C32C32C12C6C3
# reps1111221445422416448

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

10000000
01000000
006400000
00080000
000007200
000017200
00000010
00000001
,
10000000
01000000
006400000
00080000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000072
000000172
,
1556000000
3258000000
00010000
00100000
000002700
000027000
00000001
00000010
,
5120000000
1622000000
00100000
00010000
000072000
000007200
00000001
00000010

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

׿
×
𝔽