direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C42.C22, C8⋊C4⋊6C6, C6.24C4≀C2, (C6×D4).4C4, (C6×Q8).4C4, (C2×D4).1C12, C42.1(C2×C6), (C2×Q8).2C12, (C2×C12).445D4, C4.4D4.1C6, (C4×C12).241C22, C6.12(C4.D4), C2.6(C3×C4≀C2), (C2×C4).9(C2×C12), (C3×C8⋊C4)⋊16C2, (C2×C4).97(C3×D4), C2.3(C3×C4.D4), (C2×C12).176(C2×C4), (C3×C4.4D4).10C2, C22.37(C3×C22⋊C4), (C2×C6).124(C22⋊C4), SmallGroup(192,135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.C22
G = < a,b,c,d,e | a3=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe=b-1, cd=dc, ece=b2c-1, ede=b-1c2d >
Subgroups: 146 in 70 conjugacy classes, 30 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4, C4.4D4, C4×C12, C3×C22⋊C4, C2×C24, C6×D4, C6×Q8, C42.C22, C3×C8⋊C4, C3×C4.4D4, C3×C42.C22
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C4.D4, C4≀C2, C3×C22⋊C4, C42.C22, C3×C4.D4, C3×C4≀C2, C3×C42.C22
►On 96 points
Generators in S96
(1 31 23)(2 32 24)(3 25 17)(4 26 18)(5 27 19)(6 28 20)(7 29 21)(8 30 22)(9 65 57)(10 66 58)(11 67 59)(12 68 60)(13 69 61)(14 70 62)(15 71 63)(16 72 64)(33 56 41)(34 49 42)(35 50 43)(36 51 44)(37 52 45)(38 53 46)(39 54 47)(40 55 48)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 39 79 63)(2 36 80 60)(3 33 73 57)(4 38 74 62)(5 35 75 59)(6 40 76 64)(7 37 77 61)(8 34 78 58)(9 25 56 89)(10 30 49 94)(11 27 50 91)(12 32 51 96)(13 29 52 93)(14 26 53 90)(15 31 54 95)(16 28 55 92)(17 41 81 65)(18 46 82 70)(19 43 83 67)(20 48 84 72)(21 45 85 69)(22 42 86 66)(23 47 87 71)(24 44 88 68)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)
(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)
(2 60)(3 77)(4 34)(6 64)(7 73)(8 38)(9 13)(10 90)(11 50)(12 32)(14 94)(15 54)(16 28)(17 85)(18 42)(20 72)(21 81)(22 46)(24 68)(25 93)(26 49)(29 89)(30 53)(33 37)(35 59)(36 80)(39 63)(40 76)(41 45)(43 67)(44 88)(47 71)(48 84)(51 96)(52 56)(55 92)(57 61)(58 74)(62 78)(65 69)(66 82)(70 86)
G:=sub<Sym(96)| (1,31,23)(2,32,24)(3,25,17)(4,26,18)(5,27,19)(6,28,20)(7,29,21)(8,30,22)(9,65,57)(10,66,58)(11,67,59)(12,68,60)(13,69,61)(14,70,62)(15,71,63)(16,72,64)(33,56,41)(34,49,42)(35,50,43)(36,51,44)(37,52,45)(38,53,46)(39,54,47)(40,55,48)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,39,79,63)(2,36,80,60)(3,33,73,57)(4,38,74,62)(5,35,75,59)(6,40,76,64)(7,37,77,61)(8,34,78,58)(9,25,56,89)(10,30,49,94)(11,27,50,91)(12,32,51,96)(13,29,52,93)(14,26,53,90)(15,31,54,95)(16,28,55,92)(17,41,81,65)(18,46,82,70)(19,43,83,67)(20,48,84,72)(21,45,85,69)(22,42,86,66)(23,47,87,71)(24,44,88,68), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (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), (2,60)(3,77)(4,34)(6,64)(7,73)(8,38)(9,13)(10,90)(11,50)(12,32)(14,94)(15,54)(16,28)(17,85)(18,42)(20,72)(21,81)(22,46)(24,68)(25,93)(26,49)(29,89)(30,53)(33,37)(35,59)(36,80)(39,63)(40,76)(41,45)(43,67)(44,88)(47,71)(48,84)(51,96)(52,56)(55,92)(57,61)(58,74)(62,78)(65,69)(66,82)(70,86)>;
G:=Group( (1,31,23)(2,32,24)(3,25,17)(4,26,18)(5,27,19)(6,28,20)(7,29,21)(8,30,22)(9,65,57)(10,66,58)(11,67,59)(12,68,60)(13,69,61)(14,70,62)(15,71,63)(16,72,64)(33,56,41)(34,49,42)(35,50,43)(36,51,44)(37,52,45)(38,53,46)(39,54,47)(40,55,48)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,39,79,63)(2,36,80,60)(3,33,73,57)(4,38,74,62)(5,35,75,59)(6,40,76,64)(7,37,77,61)(8,34,78,58)(9,25,56,89)(10,30,49,94)(11,27,50,91)(12,32,51,96)(13,29,52,93)(14,26,53,90)(15,31,54,95)(16,28,55,92)(17,41,81,65)(18,46,82,70)(19,43,83,67)(20,48,84,72)(21,45,85,69)(22,42,86,66)(23,47,87,71)(24,44,88,68), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (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), (2,60)(3,77)(4,34)(6,64)(7,73)(8,38)(9,13)(10,90)(11,50)(12,32)(14,94)(15,54)(16,28)(17,85)(18,42)(20,72)(21,81)(22,46)(24,68)(25,93)(26,49)(29,89)(30,53)(33,37)(35,59)(36,80)(39,63)(40,76)(41,45)(43,67)(44,88)(47,71)(48,84)(51,96)(52,56)(55,92)(57,61)(58,74)(62,78)(65,69)(66,82)(70,86) );
G=PermutationGroup([[(1,31,23),(2,32,24),(3,25,17),(4,26,18),(5,27,19),(6,28,20),(7,29,21),(8,30,22),(9,65,57),(10,66,58),(11,67,59),(12,68,60),(13,69,61),(14,70,62),(15,71,63),(16,72,64),(33,56,41),(34,49,42),(35,50,43),(36,51,44),(37,52,45),(38,53,46),(39,54,47),(40,55,48),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(1,39,79,63),(2,36,80,60),(3,33,73,57),(4,38,74,62),(5,35,75,59),(6,40,76,64),(7,37,77,61),(8,34,78,58),(9,25,56,89),(10,30,49,94),(11,27,50,91),(12,32,51,96),(13,29,52,93),(14,26,53,90),(15,31,54,95),(16,28,55,92),(17,41,81,65),(18,46,82,70),(19,43,83,67),(20,48,84,72),(21,45,85,69),(22,42,86,66),(23,47,87,71),(24,44,88,68)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96)], [(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)], [(2,60),(3,77),(4,34),(6,64),(7,73),(8,38),(9,13),(10,90),(11,50),(12,32),(14,94),(15,54),(16,28),(17,85),(18,42),(20,72),(21,81),(22,46),(24,68),(25,93),(26,49),(29,89),(30,53),(33,37),(35,59),(36,80),(39,63),(40,76),(41,45),(43,67),(44,88),(47,71),(48,84),(51,96),(52,56),(55,92),(57,61),(58,74),(62,78),(65,69),(66,82),(70,86)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 8A | ··· | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | ··· | 1 | 8 | 8 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | C3×D4 | C4≀C2 | C3×C4≀C2 | C4.D4 | C3×C4.D4 |
| kernel | C3×C42.C22 | C3×C8⋊C4 | C3×C4.4D4 | C42.C22 | C6×D4 | C6×Q8 | C8⋊C4 | C4.4D4 | C2×D4 | C2×Q8 | C2×C12 | C2×C4 | C6 | C2 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 8 | 16 | 1 | 2 |
Matrix representation of C3×C42.C22 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 2 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 13 | 59 | 0 | 0 |
| 59 | 13 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 47 | 0 |
| 0 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[0,27,0,0,27,0,0,0,0,0,72,2,0,0,72,1],[0,1,0,0,1,0,0,0,0,0,46,0,0,0,0,46],[13,59,0,0,59,13,0,0,0,0,0,47,0,0,60,0],[0,27,0,0,46,0,0,0,0,0,1,0,0,0,1,72] >;
C3×C42.C22 in GAP, Magma, Sage, TeX
C_3\times C_4^2.C_2^2
% in TeX
G:=Group("C3xC4^2.C2^2"); // GroupNames label
G:=SmallGroup(192,135);
// by ID
G=gap.SmallGroup(192,135);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1522,248,2951,102]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=e^2=1,d^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^2,e*b*e=b^-1,c*d=d*c,e*c*e=b^2*c^-1,e*d*e=b^-1*c^2*d>;
// generators/relations