direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C4⋊C4×3- 1+2, C36⋊3C12, (C2×C36).9C6, (C6×C12).4C6, C18.3(C3×Q8), C6.25(C6×C12), C12.6(C3×C12), (C3×C12).4C12, C18.13(C3×D4), C18.12(C2×C12), C4⋊(C4×3- 1+2), C6.5(Q8×C32), C62.36(C2×C6), (C2×C6).31C62, C6.17(D4×C32), C2.(Q8×3- 1+2), (C4×3- 1+2)⋊3C4, C2.2(D4×3- 1+2), (C2×3- 1+2).3Q8, (C2×3- 1+2).13D4, C22.4(C22×3- 1+2), (C22×3- 1+2).15C22, (C9×C4⋊C4)⋊C3, C9⋊3(C3×C4⋊C4), C32.(C3×C4⋊C4), (C32×C4⋊C4).C3, (C3×C6).7(C3×Q8), (C2×C12).6(C3×C6), (C3×C6).31(C3×D4), C3.3(C32×C4⋊C4), (C2×C18).17(C2×C6), (C3×C6).31(C2×C12), (C3×C4⋊C4).3C32, C2.4(C2×C4×3- 1+2), (C2×C4×3- 1+2).9C2, (C2×C4).1(C2×3- 1+2), (C2×3- 1+2).12(C2×C4), SmallGroup(432,208)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C6 — C2×C6 — C62 — C22×3- 1+2 — C2×C4×3- 1+2 — C4⋊C4×3- 1+2 |
Generators and relations for C4⋊C4×3- 1+2
G = < a,b,c,d | a4=b4=c9=d3=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 150 in 104 conjugacy classes, 77 normal (28 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C4⋊C4, C18, C3×C6, C2×C12, C2×C12, C2×C12, 3- 1+2, C36, C36, C2×C18, C3×C12, C3×C12, C62, C3×C4⋊C4, C3×C4⋊C4, C2×3- 1+2, C2×C36, C6×C12, C6×C12, C4×3- 1+2, C4×3- 1+2, C22×3- 1+2, C9×C4⋊C4, C32×C4⋊C4, C2×C4×3- 1+2, C2×C4×3- 1+2, C4⋊C4×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4⋊C4, C3×C6, C2×C12, C3×D4, C3×Q8, 3- 1+2, C3×C12, C62, C3×C4⋊C4, C2×3- 1+2, C6×C12, D4×C32, Q8×C32, C4×3- 1+2, C22×3- 1+2, C32×C4⋊C4, C2×C4×3- 1+2, D4×3- 1+2, Q8×3- 1+2, C4⋊C4×3- 1+2
►On 144 points
Generators in S144
(1 88 40 70)(2 89 41 71)(3 90 42 72)(4 82 43 64)(5 83 44 65)(6 84 45 66)(7 85 37 67)(8 86 38 68)(9 87 39 69)(10 123 127 105)(11 124 128 106)(12 125 129 107)(13 126 130 108)(14 118 131 100)(15 119 132 101)(16 120 133 102)(17 121 134 103)(18 122 135 104)(19 113 140 99)(20 114 141 91)(21 115 142 92)(22 116 143 93)(23 117 144 94)(24 109 136 95)(25 110 137 96)(26 111 138 97)(27 112 139 98)(28 59 46 73)(29 60 47 74)(30 61 48 75)(31 62 49 76)(32 63 50 77)(33 55 51 78)(34 56 52 79)(35 57 53 80)(36 58 54 81)
(1 106 34 115)(2 107 35 116)(3 108 36 117)(4 100 28 109)(5 101 29 110)(6 102 30 111)(7 103 31 112)(8 104 32 113)(9 105 33 114)(10 78 141 69)(11 79 142 70)(12 80 143 71)(13 81 144 72)(14 73 136 64)(15 74 137 65)(16 75 138 66)(17 76 139 67)(18 77 140 68)(19 86 135 63)(20 87 127 55)(21 88 128 56)(22 89 129 57)(23 90 130 58)(24 82 131 59)(25 83 132 60)(26 84 133 61)(27 85 134 62)(37 121 49 98)(38 122 50 99)(39 123 51 91)(40 124 52 92)(41 125 53 93)(42 126 54 94)(43 118 46 95)(44 119 47 96)(45 120 48 97)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(55 58 61)(57 63 60)(65 71 68)(66 69 72)(74 80 77)(75 78 81)(83 89 86)(84 87 90)(91 94 97)(93 99 96)(101 107 104)(102 105 108)(110 116 113)(111 114 117)(119 125 122)(120 123 126)(127 130 133)(129 135 132)(137 143 140)(138 141 144)
G:=sub<Sym(144)| (1,88,40,70)(2,89,41,71)(3,90,42,72)(4,82,43,64)(5,83,44,65)(6,84,45,66)(7,85,37,67)(8,86,38,68)(9,87,39,69)(10,123,127,105)(11,124,128,106)(12,125,129,107)(13,126,130,108)(14,118,131,100)(15,119,132,101)(16,120,133,102)(17,121,134,103)(18,122,135,104)(19,113,140,99)(20,114,141,91)(21,115,142,92)(22,116,143,93)(23,117,144,94)(24,109,136,95)(25,110,137,96)(26,111,138,97)(27,112,139,98)(28,59,46,73)(29,60,47,74)(30,61,48,75)(31,62,49,76)(32,63,50,77)(33,55,51,78)(34,56,52,79)(35,57,53,80)(36,58,54,81), (1,106,34,115)(2,107,35,116)(3,108,36,117)(4,100,28,109)(5,101,29,110)(6,102,30,111)(7,103,31,112)(8,104,32,113)(9,105,33,114)(10,78,141,69)(11,79,142,70)(12,80,143,71)(13,81,144,72)(14,73,136,64)(15,74,137,65)(16,75,138,66)(17,76,139,67)(18,77,140,68)(19,86,135,63)(20,87,127,55)(21,88,128,56)(22,89,129,57)(23,90,130,58)(24,82,131,59)(25,83,132,60)(26,84,133,61)(27,85,134,62)(37,121,49,98)(38,122,50,99)(39,123,51,91)(40,124,52,92)(41,125,53,93)(42,126,54,94)(43,118,46,95)(44,119,47,96)(45,120,48,97), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(65,71,68)(66,69,72)(74,80,77)(75,78,81)(83,89,86)(84,87,90)(91,94,97)(93,99,96)(101,107,104)(102,105,108)(110,116,113)(111,114,117)(119,125,122)(120,123,126)(127,130,133)(129,135,132)(137,143,140)(138,141,144)>;
G:=Group( (1,88,40,70)(2,89,41,71)(3,90,42,72)(4,82,43,64)(5,83,44,65)(6,84,45,66)(7,85,37,67)(8,86,38,68)(9,87,39,69)(10,123,127,105)(11,124,128,106)(12,125,129,107)(13,126,130,108)(14,118,131,100)(15,119,132,101)(16,120,133,102)(17,121,134,103)(18,122,135,104)(19,113,140,99)(20,114,141,91)(21,115,142,92)(22,116,143,93)(23,117,144,94)(24,109,136,95)(25,110,137,96)(26,111,138,97)(27,112,139,98)(28,59,46,73)(29,60,47,74)(30,61,48,75)(31,62,49,76)(32,63,50,77)(33,55,51,78)(34,56,52,79)(35,57,53,80)(36,58,54,81), (1,106,34,115)(2,107,35,116)(3,108,36,117)(4,100,28,109)(5,101,29,110)(6,102,30,111)(7,103,31,112)(8,104,32,113)(9,105,33,114)(10,78,141,69)(11,79,142,70)(12,80,143,71)(13,81,144,72)(14,73,136,64)(15,74,137,65)(16,75,138,66)(17,76,139,67)(18,77,140,68)(19,86,135,63)(20,87,127,55)(21,88,128,56)(22,89,129,57)(23,90,130,58)(24,82,131,59)(25,83,132,60)(26,84,133,61)(27,85,134,62)(37,121,49,98)(38,122,50,99)(39,123,51,91)(40,124,52,92)(41,125,53,93)(42,126,54,94)(43,118,46,95)(44,119,47,96)(45,120,48,97), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(65,71,68)(66,69,72)(74,80,77)(75,78,81)(83,89,86)(84,87,90)(91,94,97)(93,99,96)(101,107,104)(102,105,108)(110,116,113)(111,114,117)(119,125,122)(120,123,126)(127,130,133)(129,135,132)(137,143,140)(138,141,144) );
G=PermutationGroup([[(1,88,40,70),(2,89,41,71),(3,90,42,72),(4,82,43,64),(5,83,44,65),(6,84,45,66),(7,85,37,67),(8,86,38,68),(9,87,39,69),(10,123,127,105),(11,124,128,106),(12,125,129,107),(13,126,130,108),(14,118,131,100),(15,119,132,101),(16,120,133,102),(17,121,134,103),(18,122,135,104),(19,113,140,99),(20,114,141,91),(21,115,142,92),(22,116,143,93),(23,117,144,94),(24,109,136,95),(25,110,137,96),(26,111,138,97),(27,112,139,98),(28,59,46,73),(29,60,47,74),(30,61,48,75),(31,62,49,76),(32,63,50,77),(33,55,51,78),(34,56,52,79),(35,57,53,80),(36,58,54,81)], [(1,106,34,115),(2,107,35,116),(3,108,36,117),(4,100,28,109),(5,101,29,110),(6,102,30,111),(7,103,31,112),(8,104,32,113),(9,105,33,114),(10,78,141,69),(11,79,142,70),(12,80,143,71),(13,81,144,72),(14,73,136,64),(15,74,137,65),(16,75,138,66),(17,76,139,67),(18,77,140,68),(19,86,135,63),(20,87,127,55),(21,88,128,56),(22,89,129,57),(23,90,130,58),(24,82,131,59),(25,83,132,60),(26,84,133,61),(27,85,134,62),(37,121,49,98),(38,122,50,99),(39,123,51,91),(40,124,52,92),(41,125,53,93),(42,126,54,94),(43,118,46,95),(44,119,47,96),(45,120,48,97)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(38,44,41),(39,42,45),(47,53,50),(48,51,54),(55,58,61),(57,63,60),(65,71,68),(66,69,72),(74,80,77),(75,78,81),(83,89,86),(84,87,90),(91,94,97),(93,99,96),(101,107,104),(102,105,108),(110,116,113),(111,114,117),(119,125,122),(120,123,126),(127,130,133),(129,135,132),(137,143,140),(138,141,144)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 6A | ··· | 6F | 6G | ··· | 6L | 9A | ··· | 9F | 12A | ··· | 12L | 12M | ··· | 12X | 18A | ··· | 18R | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 2 | ··· | 2 | 6 | ··· | 6 | 3 | ··· | 3 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | C3×D4 | C3×Q8 | 3- 1+2 | C2×3- 1+2 | C4×3- 1+2 | D4×3- 1+2 | Q8×3- 1+2 |
| kernel | C4⋊C4×3- 1+2 | C2×C4×3- 1+2 | C9×C4⋊C4 | C32×C4⋊C4 | C4×3- 1+2 | C2×C36 | C6×C12 | C36 | C3×C12 | C2×3- 1+2 | C2×3- 1+2 | C18 | C18 | C3×C6 | C3×C6 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 3 | 6 | 2 | 4 | 18 | 6 | 24 | 8 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of C4⋊C4×3- 1+2 ►in GL5(𝔽37)
| 1 | 9 | 0 | 0 | 0 |
| 8 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 8 | 36 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 31 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 |
| 0 | 0 | 10 | 36 | 10 |
| 0 | 0 | 25 | 2 | 0 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 |
| 0 | 0 | 15 | 0 | 26 |
G:=sub<GL(5,GF(37))| [1,8,0,0,0,9,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,8,0,0,0,0,36,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[10,0,0,0,0,0,10,0,0,0,0,0,1,10,25,0,0,9,36,2,0,0,0,10,0],[26,0,0,0,0,0,26,0,0,0,0,0,1,1,15,0,0,0,10,0,0,0,0,0,26] >;
C4⋊C4×3- 1+2 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times 3_-^{1+2} % in TeX
G:=Group("C4:C4xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,208);
// by ID
G=gap.SmallGroup(432,208);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,260,772,1109]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^3=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations