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## G = C2×Q8.C18order 288 = 25·32

### Direct product of C2 and Q8.C18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×Q8.C18
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — C2×Q8⋊C9 — C2×Q8.C18
 Lower central Q8 — C2×Q8.C18
 Upper central C1 — C2×C12

Generators and relations for C2×Q8.C18
G = < a,b,c,d | a2=b4=1, c2=d18=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=bc, dcd-1=b >

Subgroups: 225 in 85 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C9, C12, C12, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C18, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×C4○D4, C36, C2×C18, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, Q8⋊C9, C2×C36, C6×C4○D4, C2×Q8⋊C9, Q8.C18, C2×Q8.C18
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C4.A4, C22×A4, C2×C3.A4, C2×C4.A4, Q8.C18, C22×C3.A4, C2×Q8.C18

Smallest permutation representation of C2×Q8.C18
On 144 points
Generators in S144
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 109)(98 110)(99 111)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)
(1 138 19 120)(2 56 20 38)(3 83 21 101)(4 141 22 123)(5 59 23 41)(6 86 24 104)(7 144 25 126)(8 62 26 44)(9 89 27 107)(10 111 28 129)(11 65 29 47)(12 92 30 74)(13 114 31 132)(14 68 32 50)(15 95 33 77)(16 117 34 135)(17 71 35 53)(18 98 36 80)(37 99 55 81)(39 140 57 122)(40 102 58 84)(42 143 60 125)(43 105 61 87)(45 110 63 128)(46 108 64 90)(48 113 66 131)(49 75 67 93)(51 116 69 134)(52 78 70 96)(54 119 72 137)(73 130 91 112)(76 133 94 115)(79 136 97 118)(82 139 100 121)(85 142 103 124)(88 109 106 127)
(1 81 19 99)(2 139 20 121)(3 57 21 39)(4 84 22 102)(5 142 23 124)(6 60 24 42)(7 87 25 105)(8 109 26 127)(9 63 27 45)(10 90 28 108)(11 112 29 130)(12 66 30 48)(13 93 31 75)(14 115 32 133)(15 69 33 51)(16 96 34 78)(17 118 35 136)(18 72 36 54)(37 138 55 120)(38 100 56 82)(40 141 58 123)(41 103 59 85)(43 144 61 126)(44 106 62 88)(46 111 64 129)(47 73 65 91)(49 114 67 132)(50 76 68 94)(52 117 70 135)(53 79 71 97)(74 131 92 113)(77 134 95 116)(80 137 98 119)(83 140 101 122)(86 143 104 125)(89 110 107 128)
(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)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 9A ··· 9F 12A ··· 12H 12I 12J 12K 12L 18A ··· 18R 36A ··· 36X order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 9 ··· 9 12 ··· 12 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 6 6 1 1 1 1 1 1 6 6 1 ··· 1 6 6 6 6 4 ··· 4 1 ··· 1 6 6 6 6 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 C4.A4 Q8.C18 A4 C2×A4 C2×A4 C3.A4 C2×C3.A4 C2×C3.A4 kernel C2×Q8.C18 C2×Q8⋊C9 Q8.C18 C6×C4○D4 C6×Q8 C3×C4○D4 C2×C4○D4 C2×Q8 C4○D4 C6 C2 C2×C12 C12 C2×C6 C2×C4 C4 C22 # reps 1 1 2 2 2 4 6 6 12 12 24 1 2 1 2 4 2

Matrix representation of C2×Q8.C18 in GL3(𝔽37) generated by

 36 0 0 0 36 0 0 0 36
,
 1 0 0 0 1 36 0 2 36
,
 1 0 0 0 6 0 0 12 31
,
 26 0 0 0 25 7 0 23 14
G:=sub<GL(3,GF(37))| [36,0,0,0,36,0,0,0,36],[1,0,0,0,1,2,0,36,36],[1,0,0,0,6,12,0,0,31],[26,0,0,0,25,23,0,7,14] >;

C2×Q8.C18 in GAP, Magma, Sage, TeX

C_2\times Q_8.C_{18}
% in TeX

G:=Group("C2xQ8.C18");
// GroupNames label

G:=SmallGroup(288,347);
// by ID

G=gap.SmallGroup(288,347);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,79,648,172,1153,285,124]);
// Polycyclic

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

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