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G = C4⋊Dic9order 144 = 24·32

The semidirect product of C4 and Dic9 acting via Dic9/C18=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic9, C361C4, C2.1D36, C6.4D12, C18.4D4, C18.2Q8, C6.4Dic6, C2.2Dic18, C12.2Dic3, C22.5D18, C92(C4⋊C4), (C2×C4).3D9, (C2×C12).5S3, C18.8(C2×C4), (C2×C36).2C2, (C2×C6).21D6, C3.(C4⋊Dic3), C2.4(C2×Dic9), C6.9(C2×Dic3), (C2×C18).5C22, (C2×Dic9).2C2, SmallGroup(144,13)

Series: Derived Chief Lower central Upper central

C1C18 — C4⋊Dic9
C1C3C9C18C2×C18C2×Dic9 — C4⋊Dic9
C9C18 — C4⋊Dic9
C1C22C2×C4

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

18C4
18C4
9C2×C4
9C2×C4
6Dic3
6Dic3
9C4⋊C4
3C2×Dic3
3C2×Dic3
2Dic9
2Dic9
3C4⋊Dic3

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

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

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

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

C4⋊Dic9 is a maximal subgroup of
C36.Q8  C4.Dic18  C36.45D4  C8⋊Dic9  C721C4  C2.D72  D4⋊Dic9  Q82Dic9  C4×Dic18  C362Q8  C36.6Q8  C4×D36  C222Dic18  C23.8D18  C23.9D18  C22.4D36  C36⋊Q8  Dic9.Q8  C36.3Q8  C4⋊C4×D9  C4⋊C47D9  D182Q8  C4⋊C4⋊D9  C36.49D4  C23.26D18  C367D4  D4×Dic9  C362D4  Q8×Dic9  D183Q8  C4⋊Dic27  Dic3⋊Dic9  C36⋊C12  C36⋊Dic3
C4⋊Dic9 is a maximal quotient of
C36⋊C8  C72.C4  C8⋊Dic9  C721C4  C18.C42  C4⋊Dic27  Dic3⋊Dic9  C36⋊Dic3

42 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F6A6B6C9A9B9C12A12B12C12D18A···18I36A···36L
order122234444446669991212121218···1836···36
size11112221818181822222222222···22···2

42 irreducible representations

dim1111222222222222
type+++++--++-+-+-+
imageC1C2C2C4S3D4Q8Dic3D6D9Dic6D12Dic9D18Dic18D36
kernelC4⋊Dic9C2×Dic9C2×C36C36C2×C12C18C18C12C2×C6C2×C4C6C6C4C22C2C2
# reps1214111213226366

Matrix representation of C4⋊Dic9 in GL3(𝔽37) generated by

100
0310
006
,
3600
0330
009
,
3100
001
010
G:=sub<GL(3,GF(37))| [1,0,0,0,31,0,0,0,6],[36,0,0,0,33,0,0,0,9],[31,0,0,0,0,1,0,1,0] >;

C4⋊Dic9 in GAP, Magma, Sage, TeX

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

G:=Group("C4:Dic9");
// GroupNames label

G:=SmallGroup(144,13);
// by ID

G=gap.SmallGroup(144,13);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,55,2404,208,3461]);
// Polycyclic

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

Export

Subgroup lattice of C4⋊Dic9 in TeX

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