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## G = C2×Q8⋊C9order 144 = 24·32

### Direct product of C2 and Q8⋊C9

Aliases: C2×Q8⋊C9, Q8⋊C18, C6.SL2(𝔽3), (C2×Q8)⋊C9, (C6×Q8).C3, C6.5(C2×A4), (C2×C6).4A4, (C3×Q8).2C6, C3.(C2×SL2(𝔽3)), C22.2(C3.A4), C2.2(C2×C3.A4), SmallGroup(144,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×Q8⋊C9
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — C2×Q8⋊C9
 Lower central Q8 — C2×Q8⋊C9
 Upper central C1 — C2×C6

Generators and relations for C2×Q8⋊C9
G = < a,b,c,d | a2=b4=d9=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Smallest permutation representation of C2×Q8⋊C9
Regular action on 144 points
Generators in S144
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 28)(9 29)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 91)(18 92)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 109)(26 110)(27 111)(37 119)(38 120)(39 121)(40 122)(41 123)(42 124)(43 125)(44 126)(45 118)(46 106)(47 107)(48 108)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 81)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 133)(65 134)(66 135)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)(82 144)(83 136)(84 137)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)
(1 133 85 119)(2 114 86 76)(3 104 87 91)(4 127 88 122)(5 117 89 79)(6 107 90 94)(7 130 82 125)(8 111 83 73)(9 101 84 97)(10 68 46 41)(11 35 47 143)(12 55 48 26)(13 71 49 44)(14 29 50 137)(15 58 51 20)(16 65 52 38)(17 32 53 140)(18 61 54 23)(19 72 57 45)(21 139 59 31)(22 66 60 39)(24 142 62 34)(25 69 63 42)(27 136 56 28)(30 64 138 37)(33 67 141 40)(36 70 144 43)(74 118 112 132)(75 102 113 98)(77 121 115 135)(78 105 116 92)(80 124 109 129)(81 108 110 95)(93 128 106 123)(96 131 100 126)(99 134 103 120)
(1 113 85 75)(2 103 86 99)(3 135 87 121)(4 116 88 78)(5 106 89 93)(6 129 90 124)(7 110 82 81)(8 100 83 96)(9 132 84 118)(10 34 46 142)(11 63 47 25)(12 70 48 43)(13 28 49 136)(14 57 50 19)(15 64 51 37)(16 31 52 139)(17 60 53 22)(18 67 54 40)(20 138 58 30)(21 65 59 38)(23 141 61 33)(24 68 62 41)(26 144 55 36)(27 71 56 44)(29 72 137 45)(32 66 140 39)(35 69 143 42)(73 126 111 131)(74 101 112 97)(76 120 114 134)(77 104 115 91)(79 123 117 128)(80 107 109 94)(92 127 105 122)(95 130 108 125)(98 133 102 119)
(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)| (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,28)(9,29)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,91)(18,92)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,109)(26,110)(27,111)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,118)(46,106)(47,107)(48,108)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,81)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,133)(65,134)(66,135)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(82,144)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143), (1,133,85,119)(2,114,86,76)(3,104,87,91)(4,127,88,122)(5,117,89,79)(6,107,90,94)(7,130,82,125)(8,111,83,73)(9,101,84,97)(10,68,46,41)(11,35,47,143)(12,55,48,26)(13,71,49,44)(14,29,50,137)(15,58,51,20)(16,65,52,38)(17,32,53,140)(18,61,54,23)(19,72,57,45)(21,139,59,31)(22,66,60,39)(24,142,62,34)(25,69,63,42)(27,136,56,28)(30,64,138,37)(33,67,141,40)(36,70,144,43)(74,118,112,132)(75,102,113,98)(77,121,115,135)(78,105,116,92)(80,124,109,129)(81,108,110,95)(93,128,106,123)(96,131,100,126)(99,134,103,120), (1,113,85,75)(2,103,86,99)(3,135,87,121)(4,116,88,78)(5,106,89,93)(6,129,90,124)(7,110,82,81)(8,100,83,96)(9,132,84,118)(10,34,46,142)(11,63,47,25)(12,70,48,43)(13,28,49,136)(14,57,50,19)(15,64,51,37)(16,31,52,139)(17,60,53,22)(18,67,54,40)(20,138,58,30)(21,65,59,38)(23,141,61,33)(24,68,62,41)(26,144,55,36)(27,71,56,44)(29,72,137,45)(32,66,140,39)(35,69,143,42)(73,126,111,131)(74,101,112,97)(76,120,114,134)(77,104,115,91)(79,123,117,128)(80,107,109,94)(92,127,105,122)(95,130,108,125)(98,133,102,119), (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( (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,28)(9,29)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,91)(18,92)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,109)(26,110)(27,111)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,118)(46,106)(47,107)(48,108)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,81)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,133)(65,134)(66,135)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(82,144)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143), (1,133,85,119)(2,114,86,76)(3,104,87,91)(4,127,88,122)(5,117,89,79)(6,107,90,94)(7,130,82,125)(8,111,83,73)(9,101,84,97)(10,68,46,41)(11,35,47,143)(12,55,48,26)(13,71,49,44)(14,29,50,137)(15,58,51,20)(16,65,52,38)(17,32,53,140)(18,61,54,23)(19,72,57,45)(21,139,59,31)(22,66,60,39)(24,142,62,34)(25,69,63,42)(27,136,56,28)(30,64,138,37)(33,67,141,40)(36,70,144,43)(74,118,112,132)(75,102,113,98)(77,121,115,135)(78,105,116,92)(80,124,109,129)(81,108,110,95)(93,128,106,123)(96,131,100,126)(99,134,103,120), (1,113,85,75)(2,103,86,99)(3,135,87,121)(4,116,88,78)(5,106,89,93)(6,129,90,124)(7,110,82,81)(8,100,83,96)(9,132,84,118)(10,34,46,142)(11,63,47,25)(12,70,48,43)(13,28,49,136)(14,57,50,19)(15,64,51,37)(16,31,52,139)(17,60,53,22)(18,67,54,40)(20,138,58,30)(21,65,59,38)(23,141,61,33)(24,68,62,41)(26,144,55,36)(27,71,56,44)(29,72,137,45)(32,66,140,39)(35,69,143,42)(73,126,111,131)(74,101,112,97)(76,120,114,134)(77,104,115,91)(79,123,117,128)(80,107,109,94)(92,127,105,122)(95,130,108,125)(98,133,102,119), (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([[(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,28),(9,29),(10,93),(11,94),(12,95),(13,96),(14,97),(15,98),(16,99),(17,91),(18,92),(19,112),(20,113),(21,114),(22,115),(23,116),(24,117),(25,109),(26,110),(27,111),(37,119),(38,120),(39,121),(40,122),(41,123),(42,124),(43,125),(44,126),(45,118),(46,106),(47,107),(48,108),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,81),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,133),(65,134),(66,135),(67,127),(68,128),(69,129),(70,130),(71,131),(72,132),(82,144),(83,136),(84,137),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143)], [(1,133,85,119),(2,114,86,76),(3,104,87,91),(4,127,88,122),(5,117,89,79),(6,107,90,94),(7,130,82,125),(8,111,83,73),(9,101,84,97),(10,68,46,41),(11,35,47,143),(12,55,48,26),(13,71,49,44),(14,29,50,137),(15,58,51,20),(16,65,52,38),(17,32,53,140),(18,61,54,23),(19,72,57,45),(21,139,59,31),(22,66,60,39),(24,142,62,34),(25,69,63,42),(27,136,56,28),(30,64,138,37),(33,67,141,40),(36,70,144,43),(74,118,112,132),(75,102,113,98),(77,121,115,135),(78,105,116,92),(80,124,109,129),(81,108,110,95),(93,128,106,123),(96,131,100,126),(99,134,103,120)], [(1,113,85,75),(2,103,86,99),(3,135,87,121),(4,116,88,78),(5,106,89,93),(6,129,90,124),(7,110,82,81),(8,100,83,96),(9,132,84,118),(10,34,46,142),(11,63,47,25),(12,70,48,43),(13,28,49,136),(14,57,50,19),(15,64,51,37),(16,31,52,139),(17,60,53,22),(18,67,54,40),(20,138,58,30),(21,65,59,38),(23,141,61,33),(24,68,62,41),(26,144,55,36),(27,71,56,44),(29,72,137,45),(32,66,140,39),(35,69,143,42),(73,126,111,131),(74,101,112,97),(76,120,114,134),(77,104,115,91),(79,123,117,128),(80,107,109,94),(92,127,105,122),(95,130,108,125),(98,133,102,119)], [(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)]])

C2×Q8⋊C9 is a maximal subgroup of   Q8⋊Dic9  Q8.D18  2- 1+4⋊C9  C18×SL2(𝔽3)

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 6A ··· 6F 9A ··· 9F 12A 12B 12C 12D 18A ··· 18R order 1 2 2 2 3 3 4 4 6 ··· 6 9 ··· 9 12 12 12 12 18 ··· 18 size 1 1 1 1 1 1 6 6 1 ··· 1 4 ··· 4 6 6 6 6 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 type + + - + + image C1 C2 C3 C6 C9 C18 SL2(𝔽3) SL2(𝔽3) Q8⋊C9 A4 C2×A4 C3.A4 C2×C3.A4 kernel C2×Q8⋊C9 Q8⋊C9 C6×Q8 C3×Q8 C2×Q8 Q8 C6 C6 C2 C2×C6 C6 C22 C2 # reps 1 1 2 2 6 6 2 4 12 1 1 2 2

Matrix representation of C2×Q8⋊C9 in GL4(𝔽37) generated by

 36 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 31 12 0 0 0 6
,
 1 0 0 0 0 1 0 0 0 0 31 0 0 0 31 6
,
 10 0 0 0 0 33 0 0 0 0 23 4 0 0 35 4
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,31,0,0,0,12,6],[1,0,0,0,0,1,0,0,0,0,31,31,0,0,0,6],[10,0,0,0,0,33,0,0,0,0,23,35,0,0,4,4] >;

C2×Q8⋊C9 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes C_9
% in TeX

G:=Group("C2xQ8:C9");
// GroupNames label

G:=SmallGroup(144,35);
// by ID

G=gap.SmallGroup(144,35);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,43,441,117,820,202,88]);
// Polycyclic

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

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