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## G = C32×C42⋊C2order 288 = 25·32

### Direct product of C32 and C42⋊C2

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×C42⋊C2
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — C32×C22⋊C4 — C32×C42⋊C2
 Lower central C1 — C2 — C32×C42⋊C2
 Upper central C1 — C6×C12 — C32×C42⋊C2

Generators and relations for C32×C42⋊C2
G = < a,b,c,d,e | a3=b3=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, de=ed >

Subgroups: 276 in 228 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×8], C23, C32, C12 [×16], C12 [×16], C2×C6 [×12], C2×C6 [×8], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×C12 [×40], C22×C6 [×4], C42⋊C2, C3×C12 [×4], C3×C12 [×4], C62, C62 [×2], C62 [×2], C4×C12 [×8], C3×C22⋊C4 [×8], C3×C4⋊C4 [×8], C22×C12 [×4], C6×C12 [×2], C6×C12 [×8], C2×C62, C3×C42⋊C2 [×4], C122 [×2], C32×C22⋊C4 [×2], C32×C4⋊C4 [×2], C2×C6×C12, C32×C42⋊C2
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C22×C4, C4○D4 [×2], C3×C6 [×7], C2×C12 [×24], C22×C6 [×4], C42⋊C2, C3×C12 [×4], C62 [×7], C22×C12 [×4], C3×C4○D4 [×8], C6×C12 [×6], C2×C62, C3×C42⋊C2 [×4], C2×C6×C12, C32×C4○D4 [×2], C32×C42⋊C2

Smallest permutation representation of C32×C42⋊C2
On 144 points
Generators in S144
(1 95 87)(2 96 88)(3 93 85)(4 94 86)(5 34 26)(6 35 27)(7 36 28)(8 33 25)(9 89 17)(10 90 18)(11 91 19)(12 92 20)(13 81 21)(14 82 22)(15 83 23)(16 84 24)(29 77 101)(30 78 102)(31 79 103)(32 80 104)(37 105 97)(38 106 98)(39 107 99)(40 108 100)(41 121 49)(42 122 50)(43 123 51)(44 124 52)(45 113 53)(46 114 54)(47 115 55)(48 116 56)(57 75 65)(58 76 66)(59 73 67)(60 74 68)(61 141 133)(62 142 134)(63 143 135)(64 144 136)(69 137 129)(70 138 130)(71 139 131)(72 140 132)(109 125 117)(110 126 118)(111 127 119)(112 128 120)
(1 83 11)(2 84 12)(3 81 9)(4 82 10)(5 78 38)(6 79 39)(7 80 40)(8 77 37)(13 17 85)(14 18 86)(15 19 87)(16 20 88)(21 89 93)(22 90 94)(23 91 95)(24 92 96)(25 29 97)(26 30 98)(27 31 99)(28 32 100)(33 101 105)(34 102 106)(35 103 107)(36 104 108)(41 109 113)(42 110 114)(43 111 115)(44 112 116)(45 49 117)(46 50 118)(47 51 119)(48 52 120)(53 121 125)(54 122 126)(55 123 127)(56 124 128)(57 61 129)(58 62 130)(59 63 131)(60 64 132)(65 133 137)(66 134 138)(67 135 139)(68 136 140)(69 75 141)(70 76 142)(71 73 143)(72 74 144)
(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 59 111 27)(2 60 112 28)(3 57 109 25)(4 58 110 26)(5 94 76 126)(6 95 73 127)(7 96 74 128)(8 93 75 125)(9 129 41 97)(10 130 42 98)(11 131 43 99)(12 132 44 100)(13 133 45 101)(14 134 46 102)(15 135 47 103)(16 136 48 104)(17 137 49 105)(18 138 50 106)(19 139 51 107)(20 140 52 108)(21 141 53 77)(22 142 54 78)(23 143 55 79)(24 144 56 80)(29 81 61 113)(30 82 62 114)(31 83 63 115)(32 84 64 116)(33 85 65 117)(34 86 66 118)(35 87 67 119)(36 88 68 120)(37 89 69 121)(38 90 70 122)(39 91 71 123)(40 92 72 124)
(2 112)(4 110)(5 76)(7 74)(10 42)(12 44)(14 46)(16 48)(18 50)(20 52)(22 54)(24 56)(26 58)(28 60)(30 62)(32 64)(34 66)(36 68)(38 70)(40 72)(78 142)(80 144)(82 114)(84 116)(86 118)(88 120)(90 122)(92 124)(94 126)(96 128)(98 130)(100 132)(102 134)(104 136)(106 138)(108 140)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A 4B 4C 4D 4E ··· 4N 6A ··· 6X 6Y ··· 6AN 12A ··· 12AF 12AG ··· 12DH order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 C4○D4 C3×C4○D4 kernel C32×C42⋊C2 C122 C32×C22⋊C4 C32×C4⋊C4 C2×C6×C12 C3×C42⋊C2 C6×C12 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C2×C12 C3×C6 C6 # reps 1 2 2 2 1 8 8 16 16 16 8 64 4 32

Matrix representation of C32×C42⋊C2 in GL3(𝔽13) generated by

 3 0 0 0 1 0 0 0 1
,
 9 0 0 0 3 0 0 0 3
,
 5 0 0 0 8 11 0 0 5
,
 1 0 0 0 5 0 0 0 5
,
 12 0 0 0 1 0 0 8 12
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[9,0,0,0,3,0,0,0,3],[5,0,0,0,8,0,0,11,5],[1,0,0,0,5,0,0,0,5],[12,0,0,0,1,8,0,0,12] >;

C32×C42⋊C2 in GAP, Magma, Sage, TeX

C_3^2\times C_4^2\rtimes C_2
% in TeX

G:=Group("C3^2xC4^2:C2");
// GroupNames label

G:=SmallGroup(288,814);
// by ID

G=gap.SmallGroup(288,814);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,394]);
// Polycyclic

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

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