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G = Dic9⋊C4order 144 = 24·32

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic9⋊C4, C18.5D4, C18.1Q8, C6.3Dic6, C2.1Dic18, C22.4D18, C91(C4⋊C4), C6.8(C4×S3), (C2×C4).1D9, C2.4(C4×D9), C18.4(C2×C4), (C2×C36).1C2, (C2×C12).1S3, (C2×C6).20D6, C2.1(C9⋊D4), C3.(Dic3⋊C4), C6.12(C3⋊D4), (C2×C18).4C22, (C2×Dic9).1C2, SmallGroup(144,12)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic9⋊C4
 G = < a,b,c | a18=c4=1, b2=a9, bab-1=a-1, ac=ca, cbc-1=a9b >

2C4
9C4
9C4
18C4
9C2×C4
9C2×C4
2C12
3Dic3
3Dic3
6Dic3
9C4⋊C4
3C2×Dic3
3C2×Dic3
2C36
2Dic9
3Dic3⋊C4

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

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

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

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

Dic9⋊C4 is a maximal subgroup of
C4×Dic18  C36.6Q8  C422D9  C423D9  C23.16D18  C222Dic18  C23.8D18  Dic94D4  C23.9D18  D18⋊D4  Dic93Q8  C36⋊Q8  Dic9.Q8  C36.3Q8  C4⋊C4×D9  D18.D4  D18⋊Q8  C4⋊C4⋊D9  C36.49D4  C4×C9⋊D4  C23.28D18  C23.23D18  Dic9⋊D4  Dic9⋊Q8  D183Q8  Dic27⋊C4  Dic9⋊Dic3  C18.Dic6  Dic9⋊C12  C6.Dic18
Dic9⋊C4 is a maximal quotient of
C36.Q8  C4.Dic18  Dic9⋊C8  C36.53D4  C18.C42  Dic27⋊C4  Dic9⋊Dic3  C18.Dic6  C6.Dic18

42 conjugacy classes

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

42 irreducible representations

dim1111222222222222
type+++++-++-+-
imageC1C2C2C4S3D4Q8D6D9Dic6C4×S3C3⋊D4D18Dic18C4×D9C9⋊D4
kernelDic9⋊C4C2×Dic9C2×C36Dic9C2×C12C18C18C2×C6C2×C4C6C6C6C22C2C2C2
# reps1214111132223666

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

26000
241000
001131
00617
,
14300
92300
00730
002330
,
6000
0600
003227
00105
G:=sub<GL(4,GF(37))| [26,24,0,0,0,10,0,0,0,0,11,6,0,0,31,17],[14,9,0,0,3,23,0,0,0,0,7,23,0,0,30,30],[6,0,0,0,0,6,0,0,0,0,32,10,0,0,27,5] >;

Dic9⋊C4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(144,12);
// by ID

G=gap.SmallGroup(144,12);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,2404,208,3461]);
// Polycyclic

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

Export

Subgroup lattice of Dic9⋊C4 in TeX

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