direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42.12C4, C42.6C12, C12.39M4(2), (C4×C8)⋊2C6, C4⋊C8⋊17C6, (C2×C12)⋊9C8, (C4×C24)⋊7C2, (C2×C4)⋊4C24, (C4×C12).9C4, C4.9(C2×C24), C12.49(C2×C8), C22⋊C8.9C6, C2.3(C22×C24), C6.32(C22×C8), C22.5(C2×C24), (C2×C42).18C6, C42.92(C2×C6), C2.5(C6×M4(2)), (C22×C4).19C12, C23.37(C2×C12), (C22×C12).36C4, C6.51(C2×M4(2)), C4.12(C3×M4(2)), C12.350(C4○D4), (C4×C12).352C22, (C2×C24).360C22, (C2×C12).987C23, C6.59(C42⋊C2), C22.21(C22×C12), (C22×C12).497C22, (C3×C4⋊C8)⋊36C2, (C2×C4×C12).38C2, (C2×C6).23(C2×C8), (C2×C8).64(C2×C6), C4.48(C3×C4○D4), (C2×C4).78(C2×C12), (C2×C12).339(C2×C4), (C3×C22⋊C8).18C2, C2.4(C3×C42⋊C2), (C22×C4).100(C2×C6), (C22×C6).118(C2×C4), (C2×C6).237(C22×C4), (C2×C4).155(C22×C6), SmallGroup(192,864)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.12C4
G = < a,b,c,d | a3=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, cd=dc >
Subgroups: 146 in 118 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C2×C8, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C4×C12, C2×C24, C22×C12, C42.12C4, C4×C24, C3×C22⋊C8, C3×C4⋊C8, C2×C4×C12, C3×C42.12C4
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C12, C2×C6, C2×C8, M4(2), C22×C4, C4○D4, C24, C2×C12, C22×C6, C42⋊C2, C22×C8, C2×M4(2), C2×C24, C3×M4(2), C22×C12, C3×C4○D4, C42.12C4, C3×C42⋊C2, C22×C24, C6×M4(2), C3×C42.12C4
►On 96 points
(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 53 89)(42 54 90)(43 55 91)(44 56 92)(45 49 93)(46 50 94)(47 51 95)(48 52 96)
(1 7 5 3)(2 78 6 74)(4 80 8 76)(9 52 13 56)(10 16 14 12)(11 54 15 50)(17 23 21 19)(18 40 22 36)(20 34 24 38)(25 31 29 27)(26 48 30 44)(28 42 32 46)(33 39 37 35)(41 47 45 43)(49 55 53 51)(57 63 61 59)(58 96 62 92)(60 90 64 94)(65 71 69 67)(66 88 70 84)(68 82 72 86)(73 79 77 75)(81 87 85 83)(89 95 93 91)
(1 41 75 29)(2 42 76 30)(3 43 77 31)(4 44 78 32)(5 45 79 25)(6 46 80 26)(7 47 73 27)(8 48 74 28)(9 68 50 88)(10 69 51 81)(11 70 52 82)(12 71 53 83)(13 72 54 84)(14 65 55 85)(15 66 56 86)(16 67 49 87)(17 91 37 63)(18 92 38 64)(19 93 39 57)(20 94 40 58)(21 95 33 59)(22 96 34 60)(23 89 35 61)(24 90 36 62)
(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)
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)(41,53,89)(42,54,90)(43,55,91)(44,56,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,7,5,3)(2,78,6,74)(4,80,8,76)(9,52,13,56)(10,16,14,12)(11,54,15,50)(17,23,21,19)(18,40,22,36)(20,34,24,38)(25,31,29,27)(26,48,30,44)(28,42,32,46)(33,39,37,35)(41,47,45,43)(49,55,53,51)(57,63,61,59)(58,96,62,92)(60,90,64,94)(65,71,69,67)(66,88,70,84)(68,82,72,86)(73,79,77,75)(81,87,85,83)(89,95,93,91), (1,41,75,29)(2,42,76,30)(3,43,77,31)(4,44,78,32)(5,45,79,25)(6,46,80,26)(7,47,73,27)(8,48,74,28)(9,68,50,88)(10,69,51,81)(11,70,52,82)(12,71,53,83)(13,72,54,84)(14,65,55,85)(15,66,56,86)(16,67,49,87)(17,91,37,63)(18,92,38,64)(19,93,39,57)(20,94,40,58)(21,95,33,59)(22,96,34,60)(23,89,35,61)(24,90,36,62), (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)>;
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)(41,53,89)(42,54,90)(43,55,91)(44,56,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,7,5,3)(2,78,6,74)(4,80,8,76)(9,52,13,56)(10,16,14,12)(11,54,15,50)(17,23,21,19)(18,40,22,36)(20,34,24,38)(25,31,29,27)(26,48,30,44)(28,42,32,46)(33,39,37,35)(41,47,45,43)(49,55,53,51)(57,63,61,59)(58,96,62,92)(60,90,64,94)(65,71,69,67)(66,88,70,84)(68,82,72,86)(73,79,77,75)(81,87,85,83)(89,95,93,91), (1,41,75,29)(2,42,76,30)(3,43,77,31)(4,44,78,32)(5,45,79,25)(6,46,80,26)(7,47,73,27)(8,48,74,28)(9,68,50,88)(10,69,51,81)(11,70,52,82)(12,71,53,83)(13,72,54,84)(14,65,55,85)(15,66,56,86)(16,67,49,87)(17,91,37,63)(18,92,38,64)(19,93,39,57)(20,94,40,58)(21,95,33,59)(22,96,34,60)(23,89,35,61)(24,90,36,62), (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) );
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),(41,53,89),(42,54,90),(43,55,91),(44,56,92),(45,49,93),(46,50,94),(47,51,95),(48,52,96)], [(1,7,5,3),(2,78,6,74),(4,80,8,76),(9,52,13,56),(10,16,14,12),(11,54,15,50),(17,23,21,19),(18,40,22,36),(20,34,24,38),(25,31,29,27),(26,48,30,44),(28,42,32,46),(33,39,37,35),(41,47,45,43),(49,55,53,51),(57,63,61,59),(58,96,62,92),(60,90,64,94),(65,71,69,67),(66,88,70,84),(68,82,72,86),(73,79,77,75),(81,87,85,83),(89,95,93,91)], [(1,41,75,29),(2,42,76,30),(3,43,77,31),(4,44,78,32),(5,45,79,25),(6,46,80,26),(7,47,73,27),(8,48,74,28),(9,68,50,88),(10,69,51,81),(11,70,52,82),(12,71,53,83),(13,72,54,84),(14,65,55,85),(15,66,56,86),(16,67,49,87),(17,91,37,63),(18,92,38,64),(19,93,39,57),(20,94,40,58),(21,95,33,59),(22,96,34,60),(23,89,35,61),(24,90,36,62)], [(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)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4L | 4M | ··· | 4R | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8P | 12A | ··· | 12X | 12Y | ··· | 12AJ | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | M4(2) | C4○D4 | C3×M4(2) | C3×C4○D4 |
| kernel | C3×C42.12C4 | C4×C24 | C3×C22⋊C8 | C3×C4⋊C8 | C2×C4×C12 | C42.12C4 | C4×C12 | C22×C12 | C4×C8 | C22⋊C8 | C4⋊C8 | C2×C42 | C2×C12 | C42 | C22×C4 | C2×C4 | C12 | C12 | C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 8 | 8 | 32 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C42.12C4 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 27 | 0 | 0 |
| 0 | 46 | 63 |
| 0 | 0 | 27 |
| 27 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 10 | 0 | 0 |
| 0 | 51 | 27 |
| 0 | 2 | 22 |
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[27,0,0,0,46,0,0,63,27],[27,0,0,0,72,0,0,0,72],[10,0,0,0,51,2,0,27,22] >;
C3×C42.12C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2._{12}C_4 % in TeX
G:=Group("C3xC4^2.12C4"); // GroupNames label
G:=SmallGroup(192,864);
// by ID
G=gap.SmallGroup(192,864);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,142,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^2,c*d=d*c>;
// generators/relations