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## G = C22⋊2Dic18order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C22⋊2Dic18
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C22×Dic9 — C22⋊2Dic18
 Lower central C9 — C2×C18 — C22⋊2Dic18
 Upper central C1 — C22 — C22⋊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 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 18J ··· 18O 36A ··· 36L order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 4 4 18 18 18 18 36 36 2 2 2 4 4 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + - + + - + - + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 C4○D4 D9 Dic6 D18 D18 Dic18 S3×D4 D4⋊2S3 D4×D9 D4⋊2D9 kernel C22⋊2Dic18 Dic9⋊C4 C4⋊Dic9 C18.D4 C9×C22⋊C4 C2×Dic18 C22×Dic9 C3×C22⋊C4 Dic9 C2×C18 C2×C12 C22×C6 C18 C22⋊C4 C2×C6 C2×C4 C23 C22 C6 C6 C2 C2 # reps 1 2 1 1 1 1 1 1 2 2 2 1 2 3 4 6 3 12 1 1 3 3

Matrix representation of C222Dic18 in GL6(𝔽37)

 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 1 0 0 0 0 36 0 0 0 0 0 0 0 0 1 0 0 0 0 36 0 0 0 0 0 0 0 11 20 0 0 0 0 17 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 6 0 0 0 0 17 26

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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