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## G = C9×C4⋊C4order 144 = 24·32

### Direct product of C9 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C9×C4⋊C4
G = < a,b,c | a9=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×C4⋊C4
Regular action 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 83 41 70)(2 84 42 71)(3 85 43 72)(4 86 44 64)(5 87 45 65)(6 88 37 66)(7 89 38 67)(8 90 39 68)(9 82 40 69)(10 124 134 102)(11 125 135 103)(12 126 127 104)(13 118 128 105)(14 119 129 106)(15 120 130 107)(16 121 131 108)(17 122 132 100)(18 123 133 101)(19 112 144 95)(20 113 136 96)(21 114 137 97)(22 115 138 98)(23 116 139 99)(24 117 140 91)(25 109 141 92)(26 110 142 93)(27 111 143 94)(28 60 50 77)(29 61 51 78)(30 62 52 79)(31 63 53 80)(32 55 54 81)(33 56 46 73)(34 57 47 74)(35 58 48 75)(36 59 49 76)
(1 106 34 110)(2 107 35 111)(3 108 36 112)(4 100 28 113)(5 101 29 114)(6 102 30 115)(7 103 31 116)(8 104 32 117)(9 105 33 109)(10 79 138 66)(11 80 139 67)(12 81 140 68)(13 73 141 69)(14 74 142 70)(15 75 143 71)(16 76 144 72)(17 77 136 64)(18 78 137 65)(19 85 131 59)(20 86 132 60)(21 87 133 61)(22 88 134 62)(23 89 135 63)(24 90 127 55)(25 82 128 56)(26 83 129 57)(27 84 130 58)(37 124 52 98)(38 125 53 99)(39 126 54 91)(40 118 46 92)(41 119 47 93)(42 120 48 94)(43 121 49 95)(44 122 50 96)(45 123 51 97)

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

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

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

C9×C4⋊C4 is a maximal subgroup of
C36.Q8  C4.Dic18  C18.Q16  C18.D8  Dic93Q8  C36⋊Q8  Dic9.Q8  C36.3Q8  C4⋊C47D9  D36⋊C4  D18.D4  C4⋊D36  D18⋊Q8  D182Q8  C4⋊C4⋊D9  D4×C36  Q8×C36

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4F 6A ··· 6F 9A ··· 9F 12A ··· 12L 18A ··· 18R 36A ··· 36AJ order 1 2 2 2 3 3 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C9 C12 C18 C36 D4 Q8 C3×D4 C3×Q8 D4×C9 Q8×C9 kernel C9×C4⋊C4 C2×C36 C3×C4⋊C4 C36 C2×C12 C4⋊C4 C12 C2×C4 C4 C18 C18 C6 C6 C2 C2 # reps 1 3 2 4 6 6 8 18 24 1 1 2 2 6 6

Matrix representation of C9×C4⋊C4 in GL3(𝔽37) generated by

 26 0 0 0 7 0 0 0 7
,
 1 0 0 0 1 16 0 23 36
,
 31 0 0 0 32 11 0 1 5
G:=sub<GL(3,GF(37))| [26,0,0,0,7,0,0,0,7],[1,0,0,0,1,23,0,16,36],[31,0,0,0,32,1,0,11,5] >;

C9×C4⋊C4 in GAP, Magma, Sage, TeX

C_9\times C_4\rtimes C_4
% in TeX

G:=Group("C9xC4:C4");
// GroupNames label

G:=SmallGroup(144,22);
// by ID

G=gap.SmallGroup(144,22);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,79,230]);
// Polycyclic

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

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