direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C36, C6.4(C6×A4), (C2×C18)⋊6C12, C3.A4⋊5C12, C12.1(C3×A4), C3.1(C12×A4), C2.1(A4×C18), (C6×A4).10C6, (C3×A4).3C12, (A4×C18).4C2, (C2×A4).2C18, (C12×A4).2C3, C18.14(C2×A4), (C22×C36)⋊1C3, C22⋊1(C3×C36), C23.1(C3×C18), (C22×C18).8C6, (C22×C12).1C32, (C4×C3.A4)⋊3C3, (C22×C4)⋊1(C3×C9), (C2×C6).1(C3×C12), (C2×C3.A4).5C6, (C22×C6).2(C3×C6), SmallGroup(432,325)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C36 |
Generators and relations for A4×C36
G = < a,b,c,d | a36=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Subgroups: 186 in 81 conjugacy classes, 39 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, C23, C9, C9, C32, C12, C12, A4, C2×C6, C2×C6, C22×C4, C18, C18, C3×C6, C2×C12, C2×A4, C22×C6, C3×C9, C36, C36, C3.A4, C2×C18, C2×C18, C3×C12, C3×A4, C4×A4, C22×C12, C3×C18, C2×C36, C2×C3.A4, C22×C18, C6×A4, C3×C36, C9×A4, C4×C3.A4, C22×C36, C12×A4, A4×C18, A4×C36
Quotients: C1, C2, C3, C4, C6, C9, C32, C12, A4, C18, C3×C6, C2×A4, C3×C9, C36, C3×C12, C3×A4, C4×A4, C3×C18, C6×A4, C3×C36, C9×A4, C12×A4, A4×C18, A4×C36
►On 108 points
Generators in S108
(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)
(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)
(1 59 77)(2 60 78)(3 61 79)(4 62 80)(5 63 81)(6 64 82)(7 65 83)(8 66 84)(9 67 85)(10 68 86)(11 69 87)(12 70 88)(13 71 89)(14 72 90)(15 37 91)(16 38 92)(17 39 93)(18 40 94)(19 41 95)(20 42 96)(21 43 97)(22 44 98)(23 45 99)(24 46 100)(25 47 101)(26 48 102)(27 49 103)(28 50 104)(29 51 105)(30 52 106)(31 53 107)(32 54 108)(33 55 73)(34 56 74)(35 57 75)(36 58 76)
G:=sub<Sym(108)| (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), (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,59,77)(2,60,78)(3,61,79)(4,62,80)(5,63,81)(6,64,82)(7,65,83)(8,66,84)(9,67,85)(10,68,86)(11,69,87)(12,70,88)(13,71,89)(14,72,90)(15,37,91)(16,38,92)(17,39,93)(18,40,94)(19,41,95)(20,42,96)(21,43,97)(22,44,98)(23,45,99)(24,46,100)(25,47,101)(26,48,102)(27,49,103)(28,50,104)(29,51,105)(30,52,106)(31,53,107)(32,54,108)(33,55,73)(34,56,74)(35,57,75)(36,58,76)>;
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), (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,59,77)(2,60,78)(3,61,79)(4,62,80)(5,63,81)(6,64,82)(7,65,83)(8,66,84)(9,67,85)(10,68,86)(11,69,87)(12,70,88)(13,71,89)(14,72,90)(15,37,91)(16,38,92)(17,39,93)(18,40,94)(19,41,95)(20,42,96)(21,43,97)(22,44,98)(23,45,99)(24,46,100)(25,47,101)(26,48,102)(27,49,103)(28,50,104)(29,51,105)(30,52,106)(31,53,107)(32,54,108)(33,55,73)(34,56,74)(35,57,75)(36,58,76) );
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)], [(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108)], [(1,59,77),(2,60,78),(3,61,79),(4,62,80),(5,63,81),(6,64,82),(7,65,83),(8,66,84),(9,67,85),(10,68,86),(11,69,87),(12,70,88),(13,71,89),(14,72,90),(15,37,91),(16,38,92),(17,39,93),(18,40,94),(19,41,95),(20,42,96),(21,43,97),(22,44,98),(23,45,99),(24,46,100),(25,47,101),(26,48,102),(27,49,103),(28,50,104),(29,51,105),(30,52,106),(31,53,107),(32,54,108),(33,55,73),(34,56,74),(35,57,75),(36,58,76)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6L | 9A | ··· | 9F | 9G | ··· | 9R | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12T | 18A | ··· | 18F | 18G | ··· | 18R | 18S | ··· | 18AD | 36A | ··· | 36L | 36M | ··· | 36X | 36Y | ··· | 36AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C4 | C6 | C6 | C6 | C9 | C12 | C12 | C12 | C18 | C36 | A4 | C2×A4 | C3×A4 | C4×A4 | C6×A4 | C9×A4 | C12×A4 | A4×C18 | A4×C36 |
| kernel | A4×C36 | A4×C18 | C4×C3.A4 | C22×C36 | C12×A4 | C9×A4 | C2×C3.A4 | C22×C18 | C6×A4 | C4×A4 | C3.A4 | C2×C18 | C3×A4 | C2×A4 | A4 | C36 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 18 | 8 | 4 | 4 | 18 | 36 | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 |
Matrix representation of A4×C36 ►in GL3(𝔽37) generated by
| 24 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 24 |
| 1 | 0 | 0 |
| 26 | 36 | 0 |
| 10 | 0 | 36 |
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 27 | 0 | 1 |
| 26 | 35 | 0 |
| 0 | 11 | 1 |
| 0 | 27 | 0 |
G:=sub<GL(3,GF(37))| [24,0,0,0,24,0,0,0,24],[1,26,10,0,36,0,0,0,36],[36,0,27,0,36,0,0,0,1],[26,0,0,35,11,27,0,1,0] >;
A4×C36 in GAP, Magma, Sage, TeX
A_4\times C_{36} % in TeX
G:=Group("A4xC36"); // GroupNames label
G:=SmallGroup(432,325);
// by ID
G=gap.SmallGroup(432,325);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,142,4548,7951]);
// Polycyclic
G:=Group<a,b,c,d|a^36=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations