direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42.6C4, C42.12C12, C12.35M4(2), C4⋊C8⋊13C6, C8⋊C4⋊7C6, (C4×C12).10C4, C22⋊C8.7C6, C42.60(C2×C6), (C2×C42).19C6, C2.8(C6×M4(2)), C4.8(C3×M4(2)), C23.38(C2×C12), (C22×C4).16C12, (C22×C12).37C4, (C2×C6).18M4(2), C6.52(C2×M4(2)), C12.351(C4○D4), (C2×C24).326C22, (C2×C12).988C23, (C4×C12).301C22, C22.6(C3×M4(2)), C6.60(C42⋊C2), C22.45(C22×C12), (C22×C12).498C22, (C3×C4⋊C8)⋊32C2, (C2×C4×C12).39C2, (C2×C8).50(C2×C6), (C3×C8⋊C4)⋊21C2, C4.49(C3×C4○D4), (C2×C4).60(C2×C12), (C2×C12).290(C2×C4), (C3×C22⋊C8).16C2, (C22×C4).101(C2×C6), (C2×C6).238(C22×C4), (C2×C4).156(C22×C6), (C22×C6).119(C2×C4), C2.11(C3×C42⋊C2), SmallGroup(192,865)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.6C4
G = < a,b,c,d | a3=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, dcd-1=b2c >
Subgroups: 146 in 110 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C2×C8, C22×C4, C24, C2×C12, C2×C12, C22×C6, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C4×C12, C2×C24, C22×C12, C42.6C4, C3×C8⋊C4, C3×C22⋊C8, C3×C4⋊C8, C2×C4×C12, C3×C42.6C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, M4(2), C22×C4, C4○D4, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C3×M4(2), C22×C12, C3×C4○D4, C42.6C4, C3×C42⋊C2, C6×M4(2), C3×C42.6C4
►On 96 points
(1 65 17)(2 66 18)(3 67 19)(4 68 20)(5 69 21)(6 70 22)(7 71 23)(8 72 24)(9 32 61)(10 25 62)(11 26 63)(12 27 64)(13 28 57)(14 29 58)(15 30 59)(16 31 60)(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 92 13 96)(10 16 14 12)(11 94 15 90)(17 23 21 19)(18 38 22 34)(20 40 24 36)(25 31 29 27)(26 46 30 42)(28 48 32 44)(33 39 37 35)(41 47 45 43)(49 55 53 51)(50 59 54 63)(52 61 56 57)(58 64 62 60)(65 71 69 67)(66 86 70 82)(68 88 72 84)(73 79 77 75)(81 87 85 83)(89 95 93 91)
(1 41 75 31)(2 46 76 28)(3 43 77 25)(4 48 78 30)(5 45 79 27)(6 42 80 32)(7 47 73 29)(8 44 74 26)(9 22 90 40)(10 19 91 37)(11 24 92 34)(12 21 93 39)(13 18 94 36)(14 23 95 33)(15 20 96 38)(16 17 89 35)(49 87 64 69)(50 84 57 66)(51 81 58 71)(52 86 59 68)(53 83 60 65)(54 88 61 70)(55 85 62 67)(56 82 63 72)
(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)| (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,32,61)(10,25,62)(11,26,63)(12,27,64)(13,28,57)(14,29,58)(15,30,59)(16,31,60)(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)(9,92,13,96)(10,16,14,12)(11,94,15,90)(17,23,21,19)(18,38,22,34)(20,40,24,36)(25,31,29,27)(26,46,30,42)(28,48,32,44)(33,39,37,35)(41,47,45,43)(49,55,53,51)(50,59,54,63)(52,61,56,57)(58,64,62,60)(65,71,69,67)(66,86,70,82)(68,88,72,84)(73,79,77,75)(81,87,85,83)(89,95,93,91), (1,41,75,31)(2,46,76,28)(3,43,77,25)(4,48,78,30)(5,45,79,27)(6,42,80,32)(7,47,73,29)(8,44,74,26)(9,22,90,40)(10,19,91,37)(11,24,92,34)(12,21,93,39)(13,18,94,36)(14,23,95,33)(15,20,96,38)(16,17,89,35)(49,87,64,69)(50,84,57,66)(51,81,58,71)(52,86,59,68)(53,83,60,65)(54,88,61,70)(55,85,62,67)(56,82,63,72), (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( (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,32,61)(10,25,62)(11,26,63)(12,27,64)(13,28,57)(14,29,58)(15,30,59)(16,31,60)(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)(9,92,13,96)(10,16,14,12)(11,94,15,90)(17,23,21,19)(18,38,22,34)(20,40,24,36)(25,31,29,27)(26,46,30,42)(28,48,32,44)(33,39,37,35)(41,47,45,43)(49,55,53,51)(50,59,54,63)(52,61,56,57)(58,64,62,60)(65,71,69,67)(66,86,70,82)(68,88,72,84)(73,79,77,75)(81,87,85,83)(89,95,93,91), (1,41,75,31)(2,46,76,28)(3,43,77,25)(4,48,78,30)(5,45,79,27)(6,42,80,32)(7,47,73,29)(8,44,74,26)(9,22,90,40)(10,19,91,37)(11,24,92,34)(12,21,93,39)(13,18,94,36)(14,23,95,33)(15,20,96,38)(16,17,89,35)(49,87,64,69)(50,84,57,66)(51,81,58,71)(52,86,59,68)(53,83,60,65)(54,88,61,70)(55,85,62,67)(56,82,63,72), (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([[(1,65,17),(2,66,18),(3,67,19),(4,68,20),(5,69,21),(6,70,22),(7,71,23),(8,72,24),(9,32,61),(10,25,62),(11,26,63),(12,27,64),(13,28,57),(14,29,58),(15,30,59),(16,31,60),(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),(9,92,13,96),(10,16,14,12),(11,94,15,90),(17,23,21,19),(18,38,22,34),(20,40,24,36),(25,31,29,27),(26,46,30,42),(28,48,32,44),(33,39,37,35),(41,47,45,43),(49,55,53,51),(50,59,54,63),(52,61,56,57),(58,64,62,60),(65,71,69,67),(66,86,70,82),(68,88,72,84),(73,79,77,75),(81,87,85,83),(89,95,93,91)], [(1,41,75,31),(2,46,76,28),(3,43,77,25),(4,48,78,30),(5,45,79,27),(6,42,80,32),(7,47,73,29),(8,44,74,26),(9,22,90,40),(10,19,91,37),(11,24,92,34),(12,21,93,39),(13,18,94,36),(14,23,95,33),(15,20,96,38),(16,17,89,35),(49,87,64,69),(50,84,57,66),(51,81,58,71),(52,86,59,68),(53,83,60,65),(54,88,61,70),(55,85,62,67),(56,82,63,72)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12AB | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 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 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | M4(2) | C4○D4 | M4(2) | C3×M4(2) | C3×C4○D4 | C3×M4(2) |
| kernel | C3×C42.6C4 | C3×C8⋊C4 | C3×C22⋊C8 | C3×C4⋊C8 | C2×C4×C12 | C42.6C4 | C4×C12 | C22×C12 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C2×C42 | C42 | C22×C4 | C12 | C12 | C2×C6 | C4 | C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C3×C42.6C4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 1 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,0,27,0,0,0,0,1,0,0,0,0,72],[72,0,0,0,0,1,0,0,0,0,27,0,0,0,0,27],[0,27,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C42.6C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2._6C_4
% in TeX
G:=Group("C3xC4^2.6C4"); // GroupNames label
G:=SmallGroup(192,865);
// by ID
G=gap.SmallGroup(192,865);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,142,124]);
// 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,d*c*d^-1=b^2*c>;
// generators/relations