direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42⋊C2, C42⋊4C6, C4⋊C4⋊6C6, C12○(C4⋊C4), (C2×C4)⋊4C12, (C4×C12)⋊2C2, (C2×C12)⋊9C4, C4.9(C2×C12), C12○(C22⋊C4), C12.46(C2×C4), C22⋊C4.3C6, (C22×C4).6C6, C23.9(C2×C6), C6.38(C4○D4), C22.5(C2×C12), (C2×C6).72C23, C2.3(C22×C12), C6.31(C22×C4), (C22×C12).14C2, (C2×C12).79C22, C22.6(C22×C6), (C22×C6).25C22, C4○(C3×C4⋊C4), C12○(C3×C4⋊C4), C4○(C3×C22⋊C4), (C3×C4⋊C4)⋊15C2, C12○(C3×C22⋊C4), C2.1(C3×C4○D4), (C2×C6).22(C2×C4), (C2×C4).14(C2×C6), (C3×C22⋊C4).6C2, SmallGroup(96,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊C2
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×C12, C2×C12, C22×C6, C42⋊C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C3×C42⋊C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C4○D4, C2×C12, C22×C6, C42⋊C2, C22×C12, C3×C4○D4, C3×C42⋊C2
►On 48 points
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 27 45)(22 28 46)(23 25 47)(24 26 48)
(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)
(1 23 39 15)(2 24 40 16)(3 21 37 13)(4 22 38 14)(5 34 28 42)(6 35 25 43)(7 36 26 44)(8 33 27 41)(9 45 17 29)(10 46 18 30)(11 47 19 31)(12 48 20 32)
(1 3)(2 38)(4 40)(5 26)(6 8)(7 28)(9 11)(10 20)(12 18)(13 15)(14 24)(16 22)(17 19)(21 23)(25 27)(29 31)(30 48)(32 46)(33 35)(34 44)(36 42)(37 39)(41 43)(45 47)
G:=sub<Sym(48)| (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,27,45)(22,28,46)(23,25,47)(24,26,48), (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), (1,23,39,15)(2,24,40,16)(3,21,37,13)(4,22,38,14)(5,34,28,42)(6,35,25,43)(7,36,26,44)(8,33,27,41)(9,45,17,29)(10,46,18,30)(11,47,19,31)(12,48,20,32), (1,3)(2,38)(4,40)(5,26)(6,8)(7,28)(9,11)(10,20)(12,18)(13,15)(14,24)(16,22)(17,19)(21,23)(25,27)(29,31)(30,48)(32,46)(33,35)(34,44)(36,42)(37,39)(41,43)(45,47)>;
G:=Group( (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,27,45)(22,28,46)(23,25,47)(24,26,48), (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), (1,23,39,15)(2,24,40,16)(3,21,37,13)(4,22,38,14)(5,34,28,42)(6,35,25,43)(7,36,26,44)(8,33,27,41)(9,45,17,29)(10,46,18,30)(11,47,19,31)(12,48,20,32), (1,3)(2,38)(4,40)(5,26)(6,8)(7,28)(9,11)(10,20)(12,18)(13,15)(14,24)(16,22)(17,19)(21,23)(25,27)(29,31)(30,48)(32,46)(33,35)(34,44)(36,42)(37,39)(41,43)(45,47) );
G=PermutationGroup([[(1,35,11),(2,36,12),(3,33,9),(4,34,10),(5,30,14),(6,31,15),(7,32,16),(8,29,13),(17,37,41),(18,38,42),(19,39,43),(20,40,44),(21,27,45),(22,28,46),(23,25,47),(24,26,48)], [(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)], [(1,23,39,15),(2,24,40,16),(3,21,37,13),(4,22,38,14),(5,34,28,42),(6,35,25,43),(7,36,26,44),(8,33,27,41),(9,45,17,29),(10,46,18,30),(11,47,19,31),(12,48,20,32)], [(1,3),(2,38),(4,40),(5,26),(6,8),(7,28),(9,11),(10,20),(12,18),(13,15),(14,24),(16,22),(17,19),(21,23),(25,27),(29,31),(30,48),(32,46),(33,35),(34,44),(36,42),(37,39),(41,43),(45,47)]])
C3×C42⋊C2 is a maximal subgroup of
C42⋊3Dic3 C12.2C42 (C2×C12).Q8 C4⋊C4.232D6 C4⋊C4.233D6 C12.5C42 C4⋊C4.234D6 C42.43D6 C42.187D6 C4⋊C4⋊36D6 C4.(C2×D12) C4⋊C4.236D6 C4⋊C4.237D6 C42⋊6D6 (C2×D12)⋊13C4 C42.87D6 C42.88D6 C42.89D6 C42.90D6 C42⋊9D6 C42.188D6 C42.91D6 C42⋊10D6 C42⋊11D6 C42.92D6 C42⋊12D6 C42.93D6 C42.94D6 C42.95D6 C42.96D6 C42.97D6 C42.98D6 C42.99D6 C42.100D6 C12×C4○D4
C3×C42⋊C2 is a maximal quotient of
C12×C22⋊C4 C12×C4⋊C4
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12H | 12I | ··· | 12AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | C4○D4 | C3×C4○D4 |
| kernel | C3×C42⋊C2 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 4 | 8 |
Matrix representation of C3×C42⋊C2 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 5 | 0 | 0 |
| 0 | 8 | 11 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 5 | 1 |
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[5,0,0,0,8,0,0,11,5],[1,0,0,0,5,0,0,0,5],[1,0,0,0,12,5,0,0,1] >;
C3×C42⋊C2 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes C_2
% in TeX
G:=Group("C3xC4^2:C2"); // GroupNames label
G:=SmallGroup(96,164);
// by ID
G=gap.SmallGroup(96,164);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,122]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations