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G = C4×Dic9order 144 = 24·32

Direct product of C4 and Dic9

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×Dic9, C9⋊C42, C362C4, C12.6Dic3, C22.3D18, C6.7(C4×S3), (C2×C4).6D9, C2.2(C4×D9), C3.(C4×Dic3), (C2×C36).6C2, C18.3(C2×C4), (C2×C6).19D6, (C2×C12).14S3, C2.2(C2×Dic9), C6.8(C2×Dic3), (C2×C18).3C22, (C2×Dic9).4C2, SmallGroup(144,11)

Series: Derived Chief Lower central Upper central

C1C9 — C4×Dic9
C1C3C9C18C2×C18C2×Dic9 — C4×Dic9
C9 — C4×Dic9
C1C2×C4

Generators and relations for C4×Dic9
 G = < a,b,c | a4=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

9C4
9C4
9C4
9C4
9C2×C4
9C2×C4
3Dic3
3Dic3
3Dic3
3Dic3
9C42
3C2×Dic3
3C2×Dic3
3C4×Dic3

Smallest permutation representation of C4×Dic9
Regular action on 144 points
Generators in S144
(1 38 20 119)(2 39 21 120)(3 40 22 121)(4 41 23 122)(5 42 24 123)(6 43 25 124)(7 44 26 125)(8 45 27 126)(9 46 28 109)(10 47 29 110)(11 48 30 111)(12 49 31 112)(13 50 32 113)(14 51 33 114)(15 52 34 115)(16 53 35 116)(17 54 36 117)(18 37 19 118)(55 99 128 84)(56 100 129 85)(57 101 130 86)(58 102 131 87)(59 103 132 88)(60 104 133 89)(61 105 134 90)(62 106 135 73)(63 107 136 74)(64 108 137 75)(65 91 138 76)(66 92 139 77)(67 93 140 78)(68 94 141 79)(69 95 142 80)(70 96 143 81)(71 97 144 82)(72 98 127 83)
(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 76 10 85)(2 75 11 84)(3 74 12 83)(4 73 13 82)(5 90 14 81)(6 89 15 80)(7 88 16 79)(8 87 17 78)(9 86 18 77)(19 92 28 101)(20 91 29 100)(21 108 30 99)(22 107 31 98)(23 106 32 97)(24 105 33 96)(25 104 34 95)(26 103 35 94)(27 102 36 93)(37 66 46 57)(38 65 47 56)(39 64 48 55)(40 63 49 72)(41 62 50 71)(42 61 51 70)(43 60 52 69)(44 59 53 68)(45 58 54 67)(109 130 118 139)(110 129 119 138)(111 128 120 137)(112 127 121 136)(113 144 122 135)(114 143 123 134)(115 142 124 133)(116 141 125 132)(117 140 126 131)

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

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

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

C4×Dic9 is a maximal subgroup of
Dic9⋊C8  C72⋊C4  Dic18⋊C4  Q83Dic9  C42×D9  C422D9  C23.16D18  C23.8D18  Dic94D4  Dic9.D4  Dic93Q8  C36⋊Q8  Dic9.Q8  C36.3Q8  C4⋊C47D9  D36⋊C4  C4⋊C4⋊D9  C23.26D18  C36.17D4  C36⋊D4  Dic9⋊Q8  C36.23D4
C4×Dic9 is a maximal quotient of
C42.D9  C72⋊C4  C18.C42

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E···4L6A6B6C9A9B9C12A12B12C12D18A···18I36A···36L
order1222344444···46669991212121218···1836···36
size1111211119···922222222222···22···2

48 irreducible representations

dim1111122222222
type++++-++-+
imageC1C2C2C4C4S3Dic3D6D9C4×S3Dic9D18C4×D9
kernelC4×Dic9C2×Dic9C2×C36Dic9C36C2×C12C12C2×C6C2×C4C6C4C22C2
# reps12184121346312

Matrix representation of C4×Dic9 in GL4(𝔽37) generated by

1000
0600
0010
0001
,
36000
0100
003120
001711
,
6000
0100
00532
002732
G:=sub<GL(4,GF(37))| [1,0,0,0,0,6,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,1,0,0,0,0,31,17,0,0,20,11],[6,0,0,0,0,1,0,0,0,0,5,27,0,0,32,32] >;

C4×Dic9 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_9
% in TeX

G:=Group("C4xDic9");
// GroupNames label

G:=SmallGroup(144,11);
// by ID

G=gap.SmallGroup(144,11);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,55,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^4=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×Dic9 in TeX

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