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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

Aliases: C22⋊C4×C3×C6, C62.142D4, C23.9C62, C62.284C23, (C2×C4)⋊3C62, (C2×C62)⋊7C4, C6.81(C6×D4), C223(C6×C12), (C22×C12)⋊7C6, (C22×C6)⋊5C12, C6222(C2×C4), C233(C3×C12), C24.2(C3×C6), (C6×C12)⋊30C22, (C23×C6).11C6, C6.36(C22×C12), C22.4(C2×C62), (C22×C62).1C2, (C2×C62).85C22, C22.12(D4×C32), (C2×C6×C12)⋊5C2, C2.1(D4×C3×C6), C2.1(C2×C6×C12), (C2×C6)⋊10(C2×C12), (C2×C12)⋊11(C2×C6), (C22×C4)⋊3(C3×C6), (C2×C6).70(C3×D4), (C3×C6).298(C2×D4), (C2×C6).90(C22×C6), (C22×C6).77(C2×C6), (C3×C6).128(C22×C4), SmallGroup(288,812)

Series: Derived Chief Lower central Upper central

C1C2 — C22⋊C4×C3×C6
C1C2C22C2×C6C62C6×C12C32×C22⋊C4 — C22⋊C4×C3×C6
C1C2 — C22⋊C4×C3×C6
C1C2×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···2G2H2I2J2K3A···3H4A···4H6A···6BD6BE···6CJ12A···12BL
order12···222223···34···46···66···612···12
size11···122221···12···21···12···22···2

180 irreducible representations

dim111111111122
type+++++
imageC1C2C2C2C3C4C6C6C6C12D4C3×D4
kernelC22⋊C4×C3×C6C32×C22⋊C4C2×C6×C12C22×C62C6×C22⋊C4C2×C62C3×C22⋊C4C22×C12C23×C6C22×C6C62C2×C6
# reps1421883216864432

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

9000
0300
0090
0009
,
4000
0400
00120
00012
,
1000
01200
00120
0081
,
1000
0100
00120
00012
,
5000
01200
0082
0005
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

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