direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4.C42, C12.29C42, M4(2).2C12, C4.3(C4×C12), (C2×C8).5C12, (C2×C24).15C4, (C22×C8).5C6, (C2×C12).508D4, C23.7(C3×Q8), (C22×C24).6C2, C6.9(C8.C4), (C22×C6).18Q8, (C2×M4(2)).7C6, (C3×M4(2)).6C4, (C6×M4(2)).19C2, C12.104(C22⋊C4), C6.25(C2.C42), (C22×C12).572C22, (C2×C6).60(C4⋊C4), (C2×C4).41(C2×C12), (C2×C4).113(C3×D4), C2.3(C3×C8.C4), C22.17(C3×C4⋊C4), C4.26(C3×C22⋊C4), (C2×C12).262(C2×C4), (C22×C4).112(C2×C6), C2.6(C3×C2.C42), SmallGroup(192,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.C42
G = < a,b,c,d | a3=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c >
Subgroups: 122 in 90 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C8, C2×M4(2), C2×C24, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C4.C42, C22×C24, C6×M4(2), C3×C4.C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C8.C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C4.C42, C3×C2.C42, C3×C8.C4, C3×C4.C42
►On 96 points
Generators in S96
(1 71 23)(2 72 24)(3 65 17)(4 66 18)(5 67 19)(6 68 20)(7 69 21)(8 70 22)(9 58 26)(10 59 27)(11 60 28)(12 61 29)(13 62 30)(14 63 31)(15 64 32)(16 57 25)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 55 89)(42 56 90)(43 49 91)(44 50 92)(45 51 93)(46 52 94)(47 53 95)(48 54 96)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 47 45 43)(42 44 46 48)(49 55 53 51)(50 52 54 56)(57 63 61 59)(58 60 62 64)(65 71 69 67)(66 68 70 72)(73 79 77 75)(74 76 78 80)(81 87 85 83)(82 84 86 88)(89 95 93 91)(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)
(1 43 75 27 5 47 79 31)(2 42 80 30 6 46 76 26)(3 45 77 29 7 41 73 25)(4 44 74 32 8 48 78 28)(9 72 56 88 13 68 52 84)(10 67 53 87 14 71 49 83)(11 66 50 82 15 70 54 86)(12 69 55 81 16 65 51 85)(17 93 37 61 21 89 33 57)(18 92 34 64 22 96 38 60)(19 95 39 63 23 91 35 59)(20 94 36 58 24 90 40 62)
G:=sub<Sym(96)| (1,71,23)(2,72,24)(3,65,17)(4,66,18)(5,67,19)(6,68,20)(7,69,21)(8,70,22)(9,58,26)(10,59,27)(11,60,28)(12,61,29)(13,62,30)(14,63,31)(15,64,32)(16,57,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,47,45,43)(42,44,46,48)(49,55,53,51)(50,52,54,56)(57,63,61,59)(58,60,62,64)(65,71,69,67)(66,68,70,72)(73,79,77,75)(74,76,78,80)(81,87,85,83)(82,84,86,88)(89,95,93,91)(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), (1,43,75,27,5,47,79,31)(2,42,80,30,6,46,76,26)(3,45,77,29,7,41,73,25)(4,44,74,32,8,48,78,28)(9,72,56,88,13,68,52,84)(10,67,53,87,14,71,49,83)(11,66,50,82,15,70,54,86)(12,69,55,81,16,65,51,85)(17,93,37,61,21,89,33,57)(18,92,34,64,22,96,38,60)(19,95,39,63,23,91,35,59)(20,94,36,58,24,90,40,62)>;
G:=Group( (1,71,23)(2,72,24)(3,65,17)(4,66,18)(5,67,19)(6,68,20)(7,69,21)(8,70,22)(9,58,26)(10,59,27)(11,60,28)(12,61,29)(13,62,30)(14,63,31)(15,64,32)(16,57,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (1,7,5,3)(2,4,6,8)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,39,37,35)(34,36,38,40)(41,47,45,43)(42,44,46,48)(49,55,53,51)(50,52,54,56)(57,63,61,59)(58,60,62,64)(65,71,69,67)(66,68,70,72)(73,79,77,75)(74,76,78,80)(81,87,85,83)(82,84,86,88)(89,95,93,91)(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), (1,43,75,27,5,47,79,31)(2,42,80,30,6,46,76,26)(3,45,77,29,7,41,73,25)(4,44,74,32,8,48,78,28)(9,72,56,88,13,68,52,84)(10,67,53,87,14,71,49,83)(11,66,50,82,15,70,54,86)(12,69,55,81,16,65,51,85)(17,93,37,61,21,89,33,57)(18,92,34,64,22,96,38,60)(19,95,39,63,23,91,35,59)(20,94,36,58,24,90,40,62) );
G=PermutationGroup([[(1,71,23),(2,72,24),(3,65,17),(4,66,18),(5,67,19),(6,68,20),(7,69,21),(8,70,22),(9,58,26),(10,59,27),(11,60,28),(12,61,29),(13,62,30),(14,63,31),(15,64,32),(16,57,25),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,55,89),(42,56,90),(43,49,91),(44,50,92),(45,51,93),(46,52,94),(47,53,95),(48,54,96)], [(1,7,5,3),(2,4,6,8),(9,11,13,15),(10,16,14,12),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32),(33,39,37,35),(34,36,38,40),(41,47,45,43),(42,44,46,48),(49,55,53,51),(50,52,54,56),(57,63,61,59),(58,60,62,64),(65,71,69,67),(66,68,70,72),(73,79,77,75),(74,76,78,80),(81,87,85,83),(82,84,86,88),(89,95,93,91),(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)], [(1,43,75,27,5,47,79,31),(2,42,80,30,6,46,76,26),(3,45,77,29,7,41,73,25),(4,44,74,32,8,48,78,28),(9,72,56,88,13,68,52,84),(10,67,53,87,14,71,49,83),(11,66,50,82,15,70,54,86),(12,69,55,81,16,65,51,85),(17,93,37,61,21,89,33,57),(18,92,34,64,22,96,38,60),(19,95,39,63,23,91,35,59),(20,94,36,58,24,90,40,62)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P | 24Q | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | C8.C4 | C3×C8.C4 |
| kernel | C3×C4.C42 | C22×C24 | C6×M4(2) | C4.C42 | C2×C24 | C3×M4(2) | C22×C8 | C2×M4(2) | C2×C8 | M4(2) | C2×C12 | C22×C6 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 8 | 2 | 4 | 8 | 16 | 3 | 1 | 6 | 2 | 8 | 16 |
Matrix representation of C3×C4.C42 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 27 |
| 27 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 27 | 0 |
| 46 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 63 |
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[1,0,0,0,46,0,0,0,27],[27,0,0,0,0,27,0,1,0],[46,0,0,0,22,0,0,0,63] >;
C3×C4.C42 in GAP, Magma, Sage, TeX
C_3\times C_4.C_4^2
% in TeX
G:=Group("C3xC4.C4^2"); // GroupNames label
G:=SmallGroup(192,147);
// by ID
G=gap.SmallGroup(192,147);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,248,172,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations