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## G = C42×C3⋊S3order 288 = 25·32

### Direct product of C42 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C42×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C42×C3⋊S3
 Lower central C32 — C42×C3⋊S3
 Upper central C1 — C42

Generators and relations for C42×C3⋊S3
G = < a,b,c,d,e | a4=b4=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 948 in 324 conjugacy classes, 129 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×6], C4 [×6], C22, C22 [×6], S3 [×16], C6 [×12], C2×C4 [×3], C2×C4 [×15], C23, C32, Dic3 [×24], C12 [×24], D6 [×24], C2×C6 [×4], C42, C42 [×3], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×48], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C2×C42, C3⋊Dic3 [×6], C3×C12 [×6], C2×C3⋊S3 [×6], C62, C4×Dic3 [×12], C4×C12 [×4], S3×C2×C4 [×12], C4×C3⋊S3 [×12], C2×C3⋊Dic3 [×3], C6×C12 [×3], C22×C3⋊S3, S3×C42 [×4], C4×C3⋊Dic3 [×3], C122, C2×C4×C3⋊S3 [×3], C42×C3⋊S3
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3 [×4], C2×C4 [×18], C23, D6 [×12], C42 [×4], C22×C4 [×3], C3⋊S3, C4×S3 [×24], C22×S3 [×4], C2×C42, C2×C3⋊S3 [×3], S3×C2×C4 [×12], C4×C3⋊S3 [×6], C22×C3⋊S3, S3×C42 [×4], C2×C4×C3⋊S3 [×3], C42×C3⋊S3

Smallest permutation representation of C42×C3⋊S3
On 144 points
Generators in S144
(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 9 54 77)(2 10 55 78)(3 11 56 79)(4 12 53 80)(5 75 129 104)(6 76 130 101)(7 73 131 102)(8 74 132 103)(13 71 123 28)(14 72 124 25)(15 69 121 26)(16 70 122 27)(17 133 41 108)(18 134 42 105)(19 135 43 106)(20 136 44 107)(21 68 120 46)(22 65 117 47)(23 66 118 48)(24 67 119 45)(29 139 114 109)(30 140 115 110)(31 137 116 111)(32 138 113 112)(33 85 143 126)(34 86 144 127)(35 87 141 128)(36 88 142 125)(37 93 61 50)(38 94 62 51)(39 95 63 52)(40 96 64 49)(57 99 89 84)(58 100 90 81)(59 97 91 82)(60 98 92 83)
(1 42 15)(2 43 16)(3 44 13)(4 41 14)(5 120 86)(6 117 87)(7 118 88)(8 119 85)(9 105 69)(10 106 70)(11 107 71)(12 108 72)(17 124 53)(18 121 54)(19 122 55)(20 123 56)(21 127 129)(22 128 130)(23 125 131)(24 126 132)(25 80 133)(26 77 134)(27 78 135)(28 79 136)(29 96 60)(30 93 57)(31 94 58)(32 95 59)(33 103 67)(34 104 68)(35 101 65)(36 102 66)(37 84 110)(38 81 111)(39 82 112)(40 83 109)(45 143 74)(46 144 75)(47 141 76)(48 142 73)(49 92 114)(50 89 115)(51 90 116)(52 91 113)(61 99 140)(62 100 137)(63 97 138)(64 98 139)
(1 117 51)(2 118 52)(3 119 49)(4 120 50)(5 115 14)(6 116 15)(7 113 16)(8 114 13)(9 47 38)(10 48 39)(11 45 40)(12 46 37)(17 127 57)(18 128 58)(19 125 59)(20 126 60)(21 93 53)(22 94 54)(23 95 55)(24 96 56)(25 104 140)(26 101 137)(27 102 138)(28 103 139)(29 123 132)(30 124 129)(31 121 130)(32 122 131)(33 98 136)(34 99 133)(35 100 134)(36 97 135)(41 86 89)(42 87 90)(43 88 91)(44 85 92)(61 80 68)(62 77 65)(63 78 66)(64 79 67)(69 76 111)(70 73 112)(71 74 109)(72 75 110)(81 105 141)(82 106 142)(83 107 143)(84 108 144)
(1 54)(2 55)(3 56)(4 53)(5 57)(6 58)(7 59)(8 60)(9 77)(10 78)(11 79)(12 80)(13 20)(14 17)(15 18)(16 19)(21 50)(22 51)(23 52)(24 49)(25 108)(26 105)(27 106)(28 107)(29 85)(30 86)(31 87)(32 88)(33 109)(34 110)(35 111)(36 112)(37 68)(38 65)(39 66)(40 67)(41 124)(42 121)(43 122)(44 123)(45 64)(46 61)(47 62)(48 63)(69 134)(70 135)(71 136)(72 133)(73 97)(74 98)(75 99)(76 100)(81 101)(82 102)(83 103)(84 104)(89 129)(90 130)(91 131)(92 132)(93 120)(94 117)(95 118)(96 119)(113 125)(114 126)(115 127)(116 128)(137 141)(138 142)(139 143)(140 144)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A ··· 4L 4M ··· 4X 6A ··· 6L 12A ··· 12AV order 1 2 2 2 2 2 2 2 3 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 9 9 9 9 2 2 2 2 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 S3 D6 C4×S3 kernel C42×C3⋊S3 C4×C3⋊Dic3 C122 C2×C4×C3⋊S3 C4×C3⋊S3 C4×C12 C2×C12 C12 # reps 1 3 1 3 24 4 12 48

Matrix representation of C42×C3⋊S3 in GL6(𝔽13)

 12 0 0 0 0 0 0 8 0 0 0 0 0 0 1 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 0 0 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 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 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 1 0 0 0 0 0 12

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

C42×C3⋊S3 in GAP, Magma, Sage, TeX

C_4^2\times C_3\rtimes S_3
% in TeX

G:=Group("C4^2xC3:S3");
// GroupNames label

G:=SmallGroup(288,728);
// by ID

G=gap.SmallGroup(288,728);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,58,2693,9414]);
// Polycyclic

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

׿
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