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## G = C4×C32⋊7D4order 288 = 25·32

### Direct product of C4 and C32⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C4×C32⋊7D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C32⋊7D4 — C4×C32⋊7D4
 Lower central C32 — C3×C6 — C4×C32⋊7D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C4×C327D4
G = < a,b,c,d,e | a4=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 932 in 282 conjugacy classes, 97 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], S3 [×8], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3 [×16], C12 [×8], C12 [×4], D6 [×16], C2×C6 [×12], C2×C6 [×8], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×12], C3⋊D4 [×16], C2×C12 [×8], C2×C12 [×8], C22×S3 [×4], C22×C6 [×4], C4×D4, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×4], Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×4], S3×C2×C4 [×4], C2×C3⋊D4 [×4], C22×C12 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3 [×3], C327D4 [×4], C6×C12 [×2], C6×C12 [×2], C22×C3⋊S3, C2×C62, C4×C3⋊D4 [×4], C4×C3⋊Dic3, C6.Dic6, C6.11D12, C625C4, C2×C4×C3⋊S3, C2×C327D4, C2×C6×C12, C4×C327D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], C23, D6 [×12], C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3 [×8], C3⋊D4 [×8], C22×S3 [×4], C4×D4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], C4○D12 [×4], C2×C3⋊D4 [×4], C4×C3⋊S3 [×2], C327D4 [×2], C22×C3⋊S3, C4×C3⋊D4 [×4], C2×C4×C3⋊S3, C12.59D6, C2×C327D4, C4×C327D4

Smallest permutation representation of C4×C327D4
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 23 15)(2 24 16)(3 21 13)(4 22 14)(5 111 57)(6 112 58)(7 109 59)(8 110 60)(9 29 77)(10 30 78)(11 31 79)(12 32 80)(17 82 115)(18 83 116)(19 84 113)(20 81 114)(25 69 43)(26 70 44)(27 71 41)(28 72 42)(33 49 47)(34 50 48)(35 51 45)(36 52 46)(37 128 122)(38 125 123)(39 126 124)(40 127 121)(53 118 90)(54 119 91)(55 120 92)(56 117 89)(61 75 67)(62 76 68)(63 73 65)(64 74 66)(85 143 93)(86 144 94)(87 141 95)(88 142 96)(97 131 103)(98 132 104)(99 129 101)(100 130 102)(105 139 133)(106 140 134)(107 137 135)(108 138 136)
(1 41 35)(2 42 36)(3 43 33)(4 44 34)(5 67 79)(6 68 80)(7 65 77)(8 66 78)(9 109 63)(10 110 64)(11 111 61)(12 112 62)(13 69 47)(14 70 48)(15 71 45)(16 72 46)(17 124 118)(18 121 119)(19 122 120)(20 123 117)(21 25 49)(22 26 50)(23 27 51)(24 28 52)(29 59 73)(30 60 74)(31 57 75)(32 58 76)(37 92 84)(38 89 81)(39 90 82)(40 91 83)(53 115 126)(54 116 127)(55 113 128)(56 114 125)(85 137 129)(86 138 130)(87 139 131)(88 140 132)(93 107 99)(94 108 100)(95 105 97)(96 106 98)(101 143 135)(102 144 136)(103 141 133)(104 142 134)
(1 99 73 40)(2 100 74 37)(3 97 75 38)(4 98 76 39)(5 117 49 141)(6 118 50 142)(7 119 51 143)(8 120 52 144)(9 116 71 137)(10 113 72 138)(11 114 69 139)(12 115 70 140)(13 131 61 125)(14 132 62 126)(15 129 63 127)(16 130 64 128)(17 26 134 80)(18 27 135 77)(19 28 136 78)(20 25 133 79)(21 103 67 123)(22 104 68 124)(23 101 65 121)(24 102 66 122)(29 83 41 107)(30 84 42 108)(31 81 43 105)(32 82 44 106)(33 95 57 89)(34 96 58 90)(35 93 59 91)(36 94 60 92)(45 85 109 54)(46 86 110 55)(47 87 111 56)(48 88 112 53)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)(13 23)(14 24)(15 21)(16 22)(17 86)(18 87)(19 88)(20 85)(25 45)(26 46)(27 47)(28 48)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 98)(38 99)(39 100)(40 97)(49 71)(50 72)(51 69)(52 70)(53 136)(54 133)(55 134)(56 135)(61 65)(62 66)(63 67)(64 68)(73 75)(74 76)(77 111)(78 112)(79 109)(80 110)(81 93)(82 94)(83 95)(84 96)(89 107)(90 108)(91 105)(92 106)(101 125)(102 126)(103 127)(104 128)(113 142)(114 143)(115 144)(116 141)(117 137)(118 138)(119 139)(120 140)(121 131)(122 132)(123 129)(124 130)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A ··· 6AB 12A ··· 12AF order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 18 18 2 2 2 2 1 1 1 1 2 2 18 ··· 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 kernel C4×C32⋊7D4 C4×C3⋊Dic3 C6.Dic6 C6.11D12 C62⋊5C4 C2×C4×C3⋊S3 C2×C32⋊7D4 C2×C6×C12 C32⋊7D4 C22×C12 C3×C12 C2×C12 C22×C6 C3×C6 C12 C2×C6 C6 # reps 1 1 1 1 1 1 1 1 8 4 2 8 4 2 16 16 16

Matrix representation of C4×C327D4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 12 1 0 0 12 0
,
 0 1 0 0 12 12 0 0 0 0 12 1 0 0 12 0
,
 2 4 0 0 2 11 0 0 0 0 2 9 0 0 11 11
,
 1 0 0 0 12 12 0 0 0 0 12 0 0 0 12 1
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[0,12,0,0,1,12,0,0,0,0,12,12,0,0,1,0],[2,2,0,0,4,11,0,0,0,0,2,11,0,0,9,11],[1,12,0,0,0,12,0,0,0,0,12,12,0,0,0,1] >;

C4×C327D4 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_7D_4
% in TeX

G:=Group("C4xC3^2:7D4");
// GroupNames label

G:=SmallGroup(288,785);
// by ID

G=gap.SmallGroup(288,785);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,2693,9414]);
// Polycyclic

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

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