direct product, metabelian, soluble, monomial, A-group
Aliases: C12×C3.A4, C62.11C12, (C2×C6)⋊2C36, (C22×C12)⋊C9, C6.8(C6×A4), C3.2(C12×A4), C12.5(C3×A4), (C3×C12).4A4, C22⋊2(C3×C36), C32.3(C4×A4), (C2×C62).11C6, (C22×C6).4C18, C23.2(C3×C18), (C22×C12).3C32, (C2×C6×C12).1C3, (C22×C4)⋊2(C3×C9), C2.1(C6×C3.A4), C6.8(C2×C3.A4), (C3×C6).21(C2×A4), (C2×C6).3(C3×C12), (C6×C3.A4).4C2, (C2×C3.A4).6C6, (C22×C6).4(C3×C6), SmallGroup(432,331)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C12×C3.A4 |
Generators and relations for C12×C3.A4
G = < a,b,c,d,e | a12=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Subgroups: 222 in 102 conjugacy classes, 48 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C23, C9, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, C22×C4, C18, C3×C6, C3×C6, C2×C12, C22×C6, C22×C6, C3×C9, C36, C3.A4, C3×C12, C3×C12, C62, C62, C22×C12, C22×C12, C3×C18, C2×C3.A4, C6×C12, C2×C62, C3×C36, C3×C3.A4, C4×C3.A4, C2×C6×C12, C6×C3.A4, C12×C3.A4
Quotients: C1, C2, C3, C4, C6, C9, C32, C12, A4, C18, C3×C6, C2×A4, C3×C9, C36, C3.A4, C3×C12, C3×A4, C4×A4, C3×C18, C2×C3.A4, C6×A4, C3×C36, C3×C3.A4, C4×C3.A4, C12×A4, C6×C3.A4, C12×C3.A4
►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)
(1 89 55)(2 90 56)(3 91 57)(4 92 58)(5 93 59)(6 94 60)(7 95 49)(8 96 50)(9 85 51)(10 86 52)(11 87 53)(12 88 54)(13 99 33)(14 100 34)(15 101 35)(16 102 36)(17 103 25)(18 104 26)(19 105 27)(20 106 28)(21 107 29)(22 108 30)(23 97 31)(24 98 32)(37 84 71)(38 73 72)(39 74 61)(40 75 62)(41 76 63)(42 77 64)(43 78 65)(44 79 66)(45 80 67)(46 81 68)(47 82 69)(48 83 70)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)
(1 67 22 89 45 108 55 80 30)(2 68 23 90 46 97 56 81 31)(3 69 24 91 47 98 57 82 32)(4 70 13 92 48 99 58 83 33)(5 71 14 93 37 100 59 84 34)(6 72 15 94 38 101 60 73 35)(7 61 16 95 39 102 49 74 36)(8 62 17 96 40 103 50 75 25)(9 63 18 85 41 104 51 76 26)(10 64 19 86 42 105 52 77 27)(11 65 20 87 43 106 53 78 28)(12 66 21 88 44 107 54 79 29)
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), (1,89,55)(2,90,56)(3,91,57)(4,92,58)(5,93,59)(6,94,60)(7,95,49)(8,96,50)(9,85,51)(10,86,52)(11,87,53)(12,88,54)(13,99,33)(14,100,34)(15,101,35)(16,102,36)(17,103,25)(18,104,26)(19,105,27)(20,106,28)(21,107,29)(22,108,30)(23,97,31)(24,98,32)(37,84,71)(38,73,72)(39,74,61)(40,75,62)(41,76,63)(42,77,64)(43,78,65)(44,79,66)(45,80,67)(46,81,68)(47,82,69)(48,83,70), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (1,67,22,89,45,108,55,80,30)(2,68,23,90,46,97,56,81,31)(3,69,24,91,47,98,57,82,32)(4,70,13,92,48,99,58,83,33)(5,71,14,93,37,100,59,84,34)(6,72,15,94,38,101,60,73,35)(7,61,16,95,39,102,49,74,36)(8,62,17,96,40,103,50,75,25)(9,63,18,85,41,104,51,76,26)(10,64,19,86,42,105,52,77,27)(11,65,20,87,43,106,53,78,28)(12,66,21,88,44,107,54,79,29)>;
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), (1,89,55)(2,90,56)(3,91,57)(4,92,58)(5,93,59)(6,94,60)(7,95,49)(8,96,50)(9,85,51)(10,86,52)(11,87,53)(12,88,54)(13,99,33)(14,100,34)(15,101,35)(16,102,36)(17,103,25)(18,104,26)(19,105,27)(20,106,28)(21,107,29)(22,108,30)(23,97,31)(24,98,32)(37,84,71)(38,73,72)(39,74,61)(40,75,62)(41,76,63)(42,77,64)(43,78,65)(44,79,66)(45,80,67)(46,81,68)(47,82,69)(48,83,70), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (1,67,22,89,45,108,55,80,30)(2,68,23,90,46,97,56,81,31)(3,69,24,91,47,98,57,82,32)(4,70,13,92,48,99,58,83,33)(5,71,14,93,37,100,59,84,34)(6,72,15,94,38,101,60,73,35)(7,61,16,95,39,102,49,74,36)(8,62,17,96,40,103,50,75,25)(9,63,18,85,41,104,51,76,26)(10,64,19,86,42,105,52,77,27)(11,65,20,87,43,106,53,78,28)(12,66,21,88,44,107,54,79,29) );
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)], [(1,89,55),(2,90,56),(3,91,57),(4,92,58),(5,93,59),(6,94,60),(7,95,49),(8,96,50),(9,85,51),(10,86,52),(11,87,53),(12,88,54),(13,99,33),(14,100,34),(15,101,35),(16,102,36),(17,103,25),(18,104,26),(19,105,27),(20,106,28),(21,107,29),(22,108,30),(23,97,31),(24,98,32),(37,84,71),(38,73,72),(39,74,61),(40,75,62),(41,76,63),(42,77,64),(43,78,65),(44,79,66),(45,80,67),(46,81,68),(47,82,69),(48,83,70)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108)], [(1,67,22,89,45,108,55,80,30),(2,68,23,90,46,97,56,81,31),(3,69,24,91,47,98,57,82,32),(4,70,13,92,48,99,58,83,33),(5,71,14,93,37,100,59,84,34),(6,72,15,94,38,101,60,73,35),(7,61,16,95,39,102,49,74,36),(8,62,17,96,40,103,50,75,25),(9,63,18,85,41,104,51,76,26),(10,64,19,86,42,105,52,77,27),(11,65,20,87,43,106,53,78,28),(12,66,21,88,44,107,54,79,29)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6H | 6I | ··· | 6X | 9A | ··· | 9R | 12A | ··· | 12P | 12Q | ··· | 12AF | 18A | ··· | 18R | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
144 irreducible representations
| dim | 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 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C36 | A4 | C2×A4 | C3.A4 | C3×A4 | C4×A4 | C2×C3.A4 | C6×A4 | C4×C3.A4 | C12×A4 |
| kernel | C12×C3.A4 | C6×C3.A4 | C4×C3.A4 | C2×C6×C12 | C3×C3.A4 | C2×C3.A4 | C2×C62 | C22×C12 | C3.A4 | C62 | C22×C6 | C2×C6 | C3×C12 | C3×C6 | C12 | C12 | C32 | C6 | C6 | C3 | C3 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 18 | 12 | 4 | 18 | 36 | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 4 |
Matrix representation of C12×C3.A4 ►in GL4(𝔽37) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 26 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 26 | 0 | 0 |
G:=sub<GL(4,GF(37))| [23,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,26,0,0,0,0,26,0,0,0,0,26],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,1],[10,0,0,0,0,0,0,26,0,1,0,0,0,0,1,0] >;
C12×C3.A4 in GAP, Magma, Sage, TeX
C_{12}\times C_3.A_4 % in TeX
G:=Group("C12xC3.A4"); // GroupNames label
G:=SmallGroup(432,331);
// by ID
G=gap.SmallGroup(432,331);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,260,4548,7951]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations