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G = C32×D4⋊C4order 288 = 25·32

Direct product of C32 and D4⋊C4

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×D4⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×C12 — C32×C4⋊C4 — C32×D4⋊C4
 Lower central C1 — C2 — C4 — C32×D4⋊C4
 Upper central C1 — C62 — C6×C12 — C32×D4⋊C4

Generators and relations for C32×D4⋊C4
G = < a,b,c,d,e | a3=b3=c4=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=ece-1=c-1, ede-1=cd >

Subgroups: 276 in 150 conjugacy classes, 84 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4, C22, C22 [×4], C6 [×12], C6 [×8], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C32, C12 [×8], C12 [×4], C2×C6 [×4], C2×C6 [×16], C4⋊C4, C2×C8, C2×D4, C3×C6 [×3], C3×C6 [×2], C24 [×4], C2×C12 [×4], C2×C12 [×4], C3×D4 [×8], C3×D4 [×4], C22×C6 [×4], D4⋊C4, C3×C12 [×2], C3×C12, C62, C62 [×4], C3×C4⋊C4 [×4], C2×C24 [×4], C6×D4 [×4], C3×C24, C6×C12, C6×C12, D4×C32 [×2], D4×C32, C2×C62, C3×D4⋊C4 [×4], C32×C4⋊C4, C6×C24, D4×C3×C6, C32×D4⋊C4
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4 [×2], C32, C12 [×8], C2×C6 [×4], C22⋊C4, D8, SD16, C3×C6 [×3], C2×C12 [×4], C3×D4 [×8], D4⋊C4, C3×C12 [×2], C62, C3×C22⋊C4 [×4], C3×D8 [×4], C3×SD16 [×4], C6×C12, D4×C32 [×2], C3×D4⋊C4 [×4], C32×C22⋊C4, C32×D8, C32×SD16, C32×D4⋊C4

Smallest permutation representation of C32×D4⋊C4
On 144 points
Generators in S144
(1 72 40)(2 69 37)(3 70 38)(4 71 39)(5 114 82)(6 115 83)(7 116 84)(8 113 81)(9 76 41)(10 73 42)(11 74 43)(12 75 44)(13 53 45)(14 54 46)(15 55 47)(16 56 48)(17 57 49)(18 58 50)(19 59 51)(20 60 52)(21 61 29)(22 62 30)(23 63 31)(24 64 32)(25 65 33)(26 66 34)(27 67 35)(28 68 36)(77 141 109)(78 142 110)(79 143 111)(80 144 112)(85 125 117)(86 126 118)(87 127 119)(88 128 120)(89 129 121)(90 130 122)(91 131 123)(92 132 124)(93 133 101)(94 134 102)(95 135 103)(96 136 104)(97 137 105)(98 138 106)(99 139 107)(100 140 108)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 138 130)(6 139 131)(7 140 132)(8 137 129)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 76 65)(58 73 66)(59 74 67)(60 75 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(81 97 89)(82 98 90)(83 99 91)(84 100 92)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 121 113)(106 122 114)(107 123 115)(108 124 116)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(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 4)(2 3)(5 7)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 28)(26 27)(29 30)(31 32)(33 36)(34 35)(37 38)(39 40)(41 44)(42 43)(45 46)(47 48)(49 52)(50 51)(53 54)(55 56)(57 60)(58 59)(61 62)(63 64)(65 68)(66 67)(69 70)(71 72)(73 74)(75 76)(78 80)(82 84)(86 88)(90 92)(94 96)(98 100)(102 104)(106 108)(110 112)(114 116)(118 120)(122 124)(126 128)(130 132)(134 136)(138 140)(142 144)
(1 84 11 80)(2 83 12 79)(3 82 9 78)(4 81 10 77)(5 76 142 70)(6 75 143 69)(7 74 144 72)(8 73 141 71)(13 91 20 87)(14 90 17 86)(15 89 18 85)(16 92 19 88)(21 99 28 95)(22 98 25 94)(23 97 26 93)(24 100 27 96)(29 107 36 103)(30 106 33 102)(31 105 34 101)(32 108 35 104)(37 115 44 111)(38 114 41 110)(39 113 42 109)(40 116 43 112)(45 123 52 119)(46 122 49 118)(47 121 50 117)(48 124 51 120)(53 131 60 127)(54 130 57 126)(55 129 58 125)(56 132 59 128)(61 139 68 135)(62 138 65 134)(63 137 66 133)(64 140 67 136)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A 4B 4C 4D 6A ··· 6X 6Y ··· 6AN 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12AF 24A ··· 24AF order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 1 ··· 1 2 2 4 4 1 ··· 1 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D8 SD16 C3×D4 C3×D4 C3×D8 C3×SD16 kernel C32×D4⋊C4 C32×C4⋊C4 C6×C24 D4×C3×C6 C3×D4⋊C4 D4×C32 C3×C4⋊C4 C2×C24 C6×D4 C3×D4 C3×C12 C62 C3×C6 C3×C6 C12 C2×C6 C6 C6 # reps 1 1 1 1 8 4 8 8 8 32 1 1 2 2 8 8 16 16

Matrix representation of C32×D4⋊C4 in GL3(𝔽73) generated by

 8 0 0 0 8 0 0 0 8
,
 8 0 0 0 64 0 0 0 64
,
 1 0 0 0 0 1 0 72 0
,
 72 0 0 0 0 1 0 1 0
,
 27 0 0 0 57 16 0 16 16
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[8,0,0,0,64,0,0,0,64],[1,0,0,0,0,72,0,1,0],[72,0,0,0,0,1,0,1,0],[27,0,0,0,57,16,0,16,16] >;

C32×D4⋊C4 in GAP, Magma, Sage, TeX

C_3^2\times D_4\rtimes C_4
% in TeX

G:=Group("C3^2xD4:C4");
// GroupNames label

G:=SmallGroup(288,320);
// by ID

G=gap.SmallGroup(288,320);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,6304,3161,172]);
// Polycyclic

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

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