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G = C222Dic18order 288 = 25·32

The semidirect product of C22 and Dic18 acting via Dic18/Dic9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic9.6D4, C222Dic18, C23.17D18, (C2×C18)⋊Q8, C2.6(D4×D9), C4⋊Dic92C2, (C2×C4).5D18, (C2×C12).1D6, C6.76(S3×D4), C91(C22⋊Q8), C18.4(C2×Q8), Dic9⋊C44C2, C18.16(C2×D4), C22⋊C4.1D9, (C2×C6).3Dic6, (C2×Dic18)⋊2C2, (C2×C36).1C22, C2.6(C2×Dic18), C6.31(C2×Dic6), (C22×C6).38D6, C18.21(C4○D4), C2.6(D42D9), (C2×C18).19C23, C6.73(D42S3), C18.D4.2C2, C3.(Dic3.D4), (C22×C18).8C22, (C22×Dic9).3C2, C22.39(C22×D9), (C2×Dic9).24C22, (C3×C22⋊C4).2S3, (C9×C22⋊C4).1C2, (C2×C6).176(C22×S3), SmallGroup(288,88)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C222Dic18
C1C3C9C18C2×C18C2×Dic9C22×Dic9 — C222Dic18
C9C2×C18 — C222Dic18
C1C22C22⋊C4

Generators and relations for C222Dic18
 G = < a,b,c,d | a2=b2=c36=1, d2=c18, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 420 in 111 conjugacy classes, 46 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C9, Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, C18 [×3], C18 [×2], Dic6 [×2], C2×Dic3 [×6], C2×C12 [×2], C22×C6, C22⋊Q8, Dic9 [×2], Dic9 [×3], C36 [×2], C2×C18, C2×C18 [×2], C2×C18 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, Dic18 [×2], C2×Dic9 [×4], C2×Dic9 [×2], C2×C36 [×2], C22×C18, Dic3.D4, Dic9⋊C4 [×2], C4⋊Dic9, C18.D4, C9×C22⋊C4, C2×Dic18, C22×Dic9, C222Dic18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, D9, Dic6 [×2], C22×S3, C22⋊Q8, D18 [×3], C2×Dic6, S3×D4, D42S3, Dic18 [×2], C22×D9, Dic3.D4, C2×Dic18, D4×D9, D42D9, C222Dic18

Smallest permutation representation of C222Dic18
On 144 points
Generators in S144
(1 19)(2 81)(3 21)(4 83)(5 23)(6 85)(7 25)(8 87)(9 27)(10 89)(11 29)(12 91)(13 31)(14 93)(15 33)(16 95)(17 35)(18 97)(20 99)(22 101)(24 103)(26 105)(28 107)(30 73)(32 75)(34 77)(36 79)(37 115)(38 56)(39 117)(40 58)(41 119)(42 60)(43 121)(44 62)(45 123)(46 64)(47 125)(48 66)(49 127)(50 68)(51 129)(52 70)(53 131)(54 72)(55 133)(57 135)(59 137)(61 139)(63 141)(65 143)(67 109)(69 111)(71 113)(74 92)(76 94)(78 96)(80 98)(82 100)(84 102)(86 104)(88 106)(90 108)(110 128)(112 130)(114 132)(116 134)(118 136)(120 138)(122 140)(124 142)(126 144)
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)
(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 66 19 48)(2 65 20 47)(3 64 21 46)(4 63 22 45)(5 62 23 44)(6 61 24 43)(7 60 25 42)(8 59 26 41)(9 58 27 40)(10 57 28 39)(11 56 29 38)(12 55 30 37)(13 54 31 72)(14 53 32 71)(15 52 33 70)(16 51 34 69)(17 50 35 68)(18 49 36 67)(73 115 91 133)(74 114 92 132)(75 113 93 131)(76 112 94 130)(77 111 95 129)(78 110 96 128)(79 109 97 127)(80 144 98 126)(81 143 99 125)(82 142 100 124)(83 141 101 123)(84 140 102 122)(85 139 103 121)(86 138 104 120)(87 137 105 119)(88 136 106 118)(89 135 107 117)(90 134 108 116)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order122222344444444666669991212121218···1818···1836···36
size1111222441818181836362224422244442···24···44···4

54 irreducible representations

dim1111111222222222224444
type+++++++++-+++-++-+-+-
imageC1C2C2C2C2C2C2S3D4Q8D6D6C4○D4D9Dic6D18D18Dic18S3×D4D42S3D4×D9D42D9
kernelC222Dic18Dic9⋊C4C4⋊Dic9C18.D4C9×C22⋊C4C2×Dic18C22×Dic9C3×C22⋊C4Dic9C2×C18C2×C12C22×C6C18C22⋊C4C2×C6C2×C4C23C22C6C6C2C2
# reps12111111222123463121133

Matrix representation of C222Dic18 in GL6(𝔽37)

3600000
010000
0036000
0003600
000010
000001
,
3600000
0360000
001000
000100
000010
000001
,
010000
3600000
000100
0036000
00001120
00001731
,
3100000
060000
006000
0003100
0000116
00001726

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,11,17,0,0,0,0,20,31],[31,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,31,0,0,0,0,0,0,11,17,0,0,0,0,6,26] >;

C222Dic18 in GAP, Magma, Sage, TeX

C_2^2\rtimes_2{\rm Dic}_{18}
% in TeX

G:=Group("C2^2:2Dic18");
// GroupNames label

G:=SmallGroup(288,88);
// by ID

G=gap.SmallGroup(288,88);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,254,219,58,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=c^18,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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