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G = C22⋊C4×C3×C6order 288 = 25·32

Direct product of C3×C6 and C22⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22⋊C4×C3×C6
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — C32×C22⋊C4 — C22⋊C4×C3×C6
 Lower central C1 — C2 — C22⋊C4×C3×C6
 Upper central C1 — C2×C62 — C22⋊C4×C3×C6

Generators and relations for C22⋊C4×C3×C6
G = < a,b,c,d,e | a3=b6=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 564 in 396 conjugacy classes, 228 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C3 [×4], C4 [×4], C22, C22 [×10], C22 [×12], C6 [×28], C6 [×16], C2×C4 [×4], C2×C4 [×4], C23, C23 [×6], C23 [×4], C32, C12 [×16], C2×C6 [×44], C2×C6 [×48], C22⋊C4 [×4], C22×C4 [×2], C24, C3×C6, C3×C6 [×6], C3×C6 [×4], C2×C12 [×16], C2×C12 [×16], C22×C6 [×28], C22×C6 [×16], C2×C22⋊C4, C3×C12 [×4], C62, C62 [×10], C62 [×12], C3×C22⋊C4 [×16], C22×C12 [×8], C23×C6 [×4], C6×C12 [×4], C6×C12 [×4], C2×C62, C2×C62 [×6], C2×C62 [×4], C6×C22⋊C4 [×4], C32×C22⋊C4 [×4], C2×C6×C12 [×2], C22×C62, C22⋊C4×C3×C6
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], D4 [×4], C23, C32, C12 [×16], C2×C6 [×28], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C3×C6 [×7], C2×C12 [×24], C3×D4 [×16], C22×C6 [×4], C2×C22⋊C4, C3×C12 [×4], C62 [×7], C3×C22⋊C4 [×16], C22×C12 [×4], C6×D4 [×8], C6×C12 [×6], D4×C32 [×4], C2×C62, C6×C22⋊C4 [×4], C32×C22⋊C4 [×4], C2×C6×C12, D4×C3×C6 [×2], C22⋊C4×C3×C6

Smallest permutation representation of C22⋊C4×C3×C6
On 144 points
Generators in S144
(1 76 111)(2 77 112)(3 78 113)(4 73 114)(5 74 109)(6 75 110)(7 144 107)(8 139 108)(9 140 103)(10 141 104)(11 142 105)(12 143 106)(13 54 26)(14 49 27)(15 50 28)(16 51 29)(17 52 30)(18 53 25)(19 65 56)(20 66 57)(21 61 58)(22 62 59)(23 63 60)(24 64 55)(31 81 68)(32 82 69)(33 83 70)(34 84 71)(35 79 72)(36 80 67)(37 102 45)(38 97 46)(39 98 47)(40 99 48)(41 100 43)(42 101 44)(85 126 134)(86 121 135)(87 122 136)(88 123 137)(89 124 138)(90 125 133)(91 120 128)(92 115 129)(93 116 130)(94 117 131)(95 118 132)(96 119 127)
(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 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 138)(44 133)(45 134)(46 135)(47 136)(48 137)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 121)(98 122)(99 123)(100 124)(101 125)(102 126)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 44)(8 45)(9 46)(10 47)(11 48)(12 43)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(49 66)(50 61)(51 62)(52 63)(53 64)(54 65)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)
(1 107 61 125)(2 108 62 126)(3 103 63 121)(4 104 64 122)(5 105 65 123)(6 106 66 124)(7 58 133 76)(8 59 134 77)(9 60 135 78)(10 55 136 73)(11 56 137 74)(12 57 138 75)(13 94 67 40)(14 95 68 41)(15 96 69 42)(16 91 70 37)(17 92 71 38)(18 93 72 39)(19 88 109 142)(20 89 110 143)(21 90 111 144)(22 85 112 139)(23 86 113 140)(24 87 114 141)(25 130 79 47)(26 131 80 48)(27 132 81 43)(28 127 82 44)(29 128 83 45)(30 129 84 46)(31 100 49 118)(32 101 50 119)(33 102 51 120)(34 97 52 115)(35 98 53 116)(36 99 54 117)

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A ··· 3H 4A ··· 4H 6A ··· 6BD 6BE ··· 6CJ 12A ··· 12BL order 1 2 ··· 2 2 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 C3×D4 kernel C22⋊C4×C3×C6 C32×C22⋊C4 C2×C6×C12 C22×C62 C6×C22⋊C4 C2×C62 C3×C22⋊C4 C22×C12 C23×C6 C22×C6 C62 C2×C6 # reps 1 4 2 1 8 8 32 16 8 64 4 32

Matrix representation of C22⋊C4×C3×C6 in GL4(𝔽13) generated by

 9 0 0 0 0 3 0 0 0 0 9 0 0 0 0 9
,
 4 0 0 0 0 4 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 0 8 1
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 5 0 0 0 0 12 0 0 0 0 8 2 0 0 0 5
G:=sub<GL(4,GF(13))| [9,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[4,0,0,0,0,4,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,8,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,12,0,0,0,0,8,0,0,0,2,5] >;

C22⋊C4×C3×C6 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_3\times C_6
% in TeX

G:=Group("C2^2:C4xC3xC6");
// GroupNames label

G:=SmallGroup(288,812);
// by ID

G=gap.SmallGroup(288,812);
# by ID

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

G:=Group<a,b,c,d,e|a^3=b^6=c^2=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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

׿
×
𝔽