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## G = C22⋊C4×C18order 288 = 25·32

### Direct product of C18 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22⋊C4×C18
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C2×C36 — C9×C22⋊C4 — C22⋊C4×C18
 Lower central C1 — C2 — C22⋊C4×C18
 Upper central C1 — C22×C18 — C22⋊C4×C18

Generators and relations for C22⋊C4×C18
G = < a,b,c,d | a18=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 282 in 198 conjugacy classes, 114 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×4], C23, C23 [×6], C23 [×4], C9, C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C24, C18, C18 [×6], C18 [×4], C2×C12 [×4], C2×C12 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C36 [×4], C2×C18, C2×C18 [×10], C2×C18 [×12], C3×C22⋊C4 [×4], C22×C12 [×2], C23×C6, C2×C36 [×4], C2×C36 [×4], C22×C18, C22×C18 [×6], C22×C18 [×4], C6×C22⋊C4, C9×C22⋊C4 [×4], C22×C36 [×2], C23×C18, C22⋊C4×C18
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×4], C23, C9, C12 [×4], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C18 [×7], C2×C12 [×6], C3×D4 [×4], C22×C6, C2×C22⋊C4, C36 [×4], C2×C18 [×7], C3×C22⋊C4 [×4], C22×C12, C6×D4 [×2], C2×C36 [×6], D4×C9 [×4], C22×C18, C6×C22⋊C4, C9×C22⋊C4 [×4], C22×C36, D4×C18 [×2], C22⋊C4×C18

Smallest permutation representation of C22⋊C4×C18
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)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 91)(16 92)(17 93)(18 94)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(55 142)(56 143)(57 144)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)(73 126)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)
(1 33 121 144)(2 34 122 127)(3 35 123 128)(4 36 124 129)(5 19 125 130)(6 20 126 131)(7 21 109 132)(8 22 110 133)(9 23 111 134)(10 24 112 135)(11 25 113 136)(12 26 114 137)(13 27 115 138)(14 28 116 139)(15 29 117 140)(16 30 118 141)(17 31 119 142)(18 32 120 143)(37 78 67 105)(38 79 68 106)(39 80 69 107)(40 81 70 108)(41 82 71 91)(42 83 72 92)(43 84 55 93)(44 85 56 94)(45 86 57 95)(46 87 58 96)(47 88 59 97)(48 89 60 98)(49 90 61 99)(50 73 62 100)(51 74 63 101)(52 75 64 102)(53 76 65 103)(54 77 66 104)

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 4A ··· 4H 6A ··· 6N 6O ··· 6V 9A ··· 9F 12A ··· 12P 18A ··· 18AP 18AQ ··· 18BN 36A ··· 36AV order 1 2 ··· 2 2 2 2 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 ··· 1 2 2 2 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C9 C12 C18 C18 C18 C36 D4 C3×D4 D4×C9 kernel C22⋊C4×C18 C9×C22⋊C4 C22×C36 C23×C18 C6×C22⋊C4 C22×C18 C3×C22⋊C4 C22×C12 C23×C6 C2×C22⋊C4 C22×C6 C22⋊C4 C22×C4 C24 C23 C2×C18 C2×C6 C22 # reps 1 4 2 1 2 8 8 4 2 6 16 24 12 6 48 4 8 24

Matrix representation of C22⋊C4×C18 in GL4(𝔽37) generated by

 1 0 0 0 0 36 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 36 0 0 0 0 36 30 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 6 0 0 0 0 1 0 0 0 0 7 25 0 0 35 30
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,30,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[6,0,0,0,0,1,0,0,0,0,7,35,0,0,25,30] >;

C22⋊C4×C18 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_{18}
% in TeX

G:=Group("C2^2:C4xC18");
// GroupNames label

G:=SmallGroup(288,165);
// by ID

G=gap.SmallGroup(288,165);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,360]);
// Polycyclic

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

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