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## G = C4⋊C4×C3⋊S3order 288 = 25·32

### Direct product of C4⋊C4 and C3⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊C4×C3⋊S3
G = < a,b,c,d,e | a4=b4=c3=d3=e2=1, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 948 in 276 conjugacy classes, 93 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×6], C22, C22 [×6], S3 [×16], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, C32, Dic3 [×16], C12 [×8], C12 [×8], D6 [×24], C2×C6 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×32], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C2×C4⋊C4, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, Dic3⋊C4 [×8], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], S3×C2×C4 [×12], C4×C3⋊S3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C22×C3⋊S3, S3×C4⋊C4 [×4], C6.Dic6 [×2], C12⋊Dic3, C32×C4⋊C4, C2×C4×C3⋊S3, C2×C4×C3⋊S3 [×2], C4⋊C4×C3⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×12], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C2×C4⋊C4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], S3×D4 [×4], S3×Q8 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, S3×C4⋊C4 [×4], C2×C4×C3⋊S3, D4×C3⋊S3, Q8×C3⋊S3, C4⋊C4×C3⋊S3

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A ··· 4F 4G ··· 4L 6A ··· 6L 12A ··· 12X 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 2 ··· 2 18 ··· 18 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D6 C4×S3 S3×D4 S3×Q8 kernel C4⋊C4×C3⋊S3 C6.Dic6 C12⋊Dic3 C32×C4⋊C4 C2×C4×C3⋊S3 C4×C3⋊S3 C3×C4⋊C4 C2×C3⋊S3 C2×C3⋊S3 C2×C12 C12 C6 C6 # reps 1 2 1 1 3 8 4 2 2 12 16 4 4

Matrix representation of C4⋊C4×C3⋊S3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 3 0 0 0 0 0 8
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 5 8
,
 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 1 0 0 0 0 0 0 1
,
 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 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12

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

C4⋊C4×C3⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C4:C4xC3:S3");
// GroupNames label

G:=SmallGroup(288,748);
// by ID

G=gap.SmallGroup(288,748);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^3=d^3=e^2=1,b*a*b^-1=a^-1,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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