direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C4⋊C4, C4⋊C36, C36⋊3C4, C18.3Q8, C12.4C12, C18.13D4, C2.(Q8×C9), C2.2(D4×C9), C6.3(C3×Q8), (C2×C12).2C6, (C2×C36).7C2, (C2×C4).1C18, C2.2(C2×C36), C6.13(C3×D4), C18.11(C2×C4), C6.11(C2×C12), C22.3(C2×C18), (C2×C18).14C22, C3.(C3×C4⋊C4), (C3×C4⋊C4).C3, (C2×C6).17(C2×C6), (C2×C18)○(C3×C4⋊C4), SmallGroup(144,22)
Series: Derived ►Chief ►Lower central ►Upper central
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 90 42 70)(2 82 43 71)(3 83 44 72)(4 84 45 64)(5 85 37 65)(6 86 38 66)(7 87 39 67)(8 88 40 68)(9 89 41 69)(10 124 127 104)(11 125 128 105)(12 126 129 106)(13 118 130 107)(14 119 131 108)(15 120 132 100)(16 121 133 101)(17 122 134 102)(18 123 135 103)(19 110 144 95)(20 111 136 96)(21 112 137 97)(22 113 138 98)(23 114 139 99)(24 115 140 91)(25 116 141 92)(26 117 142 93)(27 109 143 94)(28 60 48 75)(29 61 49 76)(30 62 50 77)(31 63 51 78)(32 55 52 79)(33 56 53 80)(34 57 54 81)(35 58 46 73)(36 59 47 74)
(1 106 34 117)(2 107 35 109)(3 108 36 110)(4 100 28 111)(5 101 29 112)(6 102 30 113)(7 103 31 114)(8 104 32 115)(9 105 33 116)(10 79 140 68)(11 80 141 69)(12 81 142 70)(13 73 143 71)(14 74 144 72)(15 75 136 64)(16 76 137 65)(17 77 138 66)(18 78 139 67)(19 83 131 59)(20 84 132 60)(21 85 133 61)(22 86 134 62)(23 87 135 63)(24 88 127 55)(25 89 128 56)(26 90 129 57)(27 82 130 58)(37 121 49 97)(38 122 50 98)(39 123 51 99)(40 124 52 91)(41 125 53 92)(42 126 54 93)(43 118 46 94)(44 119 47 95)(45 120 48 96)
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), (1,90,42,70)(2,82,43,71)(3,83,44,72)(4,84,45,64)(5,85,37,65)(6,86,38,66)(7,87,39,67)(8,88,40,68)(9,89,41,69)(10,124,127,104)(11,125,128,105)(12,126,129,106)(13,118,130,107)(14,119,131,108)(15,120,132,100)(16,121,133,101)(17,122,134,102)(18,123,135,103)(19,110,144,95)(20,111,136,96)(21,112,137,97)(22,113,138,98)(23,114,139,99)(24,115,140,91)(25,116,141,92)(26,117,142,93)(27,109,143,94)(28,60,48,75)(29,61,49,76)(30,62,50,77)(31,63,51,78)(32,55,52,79)(33,56,53,80)(34,57,54,81)(35,58,46,73)(36,59,47,74), (1,106,34,117)(2,107,35,109)(3,108,36,110)(4,100,28,111)(5,101,29,112)(6,102,30,113)(7,103,31,114)(8,104,32,115)(9,105,33,116)(10,79,140,68)(11,80,141,69)(12,81,142,70)(13,73,143,71)(14,74,144,72)(15,75,136,64)(16,76,137,65)(17,77,138,66)(18,78,139,67)(19,83,131,59)(20,84,132,60)(21,85,133,61)(22,86,134,62)(23,87,135,63)(24,88,127,55)(25,89,128,56)(26,90,129,57)(27,82,130,58)(37,121,49,97)(38,122,50,98)(39,123,51,99)(40,124,52,91)(41,125,53,92)(42,126,54,93)(43,118,46,94)(44,119,47,95)(45,120,48,96)>;
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), (1,90,42,70)(2,82,43,71)(3,83,44,72)(4,84,45,64)(5,85,37,65)(6,86,38,66)(7,87,39,67)(8,88,40,68)(9,89,41,69)(10,124,127,104)(11,125,128,105)(12,126,129,106)(13,118,130,107)(14,119,131,108)(15,120,132,100)(16,121,133,101)(17,122,134,102)(18,123,135,103)(19,110,144,95)(20,111,136,96)(21,112,137,97)(22,113,138,98)(23,114,139,99)(24,115,140,91)(25,116,141,92)(26,117,142,93)(27,109,143,94)(28,60,48,75)(29,61,49,76)(30,62,50,77)(31,63,51,78)(32,55,52,79)(33,56,53,80)(34,57,54,81)(35,58,46,73)(36,59,47,74), (1,106,34,117)(2,107,35,109)(3,108,36,110)(4,100,28,111)(5,101,29,112)(6,102,30,113)(7,103,31,114)(8,104,32,115)(9,105,33,116)(10,79,140,68)(11,80,141,69)(12,81,142,70)(13,73,143,71)(14,74,144,72)(15,75,136,64)(16,76,137,65)(17,77,138,66)(18,78,139,67)(19,83,131,59)(20,84,132,60)(21,85,133,61)(22,86,134,62)(23,87,135,63)(24,88,127,55)(25,89,128,56)(26,90,129,57)(27,82,130,58)(37,121,49,97)(38,122,50,98)(39,123,51,99)(40,124,52,91)(41,125,53,92)(42,126,54,93)(43,118,46,94)(44,119,47,95)(45,120,48,96) );
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)], [(1,90,42,70),(2,82,43,71),(3,83,44,72),(4,84,45,64),(5,85,37,65),(6,86,38,66),(7,87,39,67),(8,88,40,68),(9,89,41,69),(10,124,127,104),(11,125,128,105),(12,126,129,106),(13,118,130,107),(14,119,131,108),(15,120,132,100),(16,121,133,101),(17,122,134,102),(18,123,135,103),(19,110,144,95),(20,111,136,96),(21,112,137,97),(22,113,138,98),(23,114,139,99),(24,115,140,91),(25,116,141,92),(26,117,142,93),(27,109,143,94),(28,60,48,75),(29,61,49,76),(30,62,50,77),(31,63,51,78),(32,55,52,79),(33,56,53,80),(34,57,54,81),(35,58,46,73),(36,59,47,74)], [(1,106,34,117),(2,107,35,109),(3,108,36,110),(4,100,28,111),(5,101,29,112),(6,102,30,113),(7,103,31,114),(8,104,32,115),(9,105,33,116),(10,79,140,68),(11,80,141,69),(12,81,142,70),(13,73,143,71),(14,74,144,72),(15,75,136,64),(16,76,137,65),(17,77,138,66),(18,78,139,67),(19,83,131,59),(20,84,132,60),(21,85,133,61),(22,86,134,62),(23,87,135,63),(24,88,127,55),(25,89,128,56),(26,90,129,57),(27,82,130,58),(37,121,49,97),(38,122,50,98),(39,123,51,99),(40,124,52,91),(41,125,53,92),(42,126,54,93),(43,118,46,94),(44,119,47,95),(45,120,48,96)]])
C9×C4⋊C4 is a maximal subgroup of
C36.Q8 C4.Dic18 C18.Q16 C18.D8 Dic9⋊3Q8 C36⋊Q8 Dic9.Q8 C36.3Q8 C4⋊C4⋊7D9 D36⋊C4 D18.D4 C4⋊D36 D18⋊Q8 D18⋊2Q8 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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