direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4⋊F5, C60⋊3C4, C12⋊3F5, C20⋊1C12, Dic5⋊3C12, C4⋊(C3×F5), C15⋊3(C4⋊C4), (C2×F5).C6, D5.(C3×Q8), C2.5(C6×F5), (C3×D5).4D4, (C4×D5).4C6, D5.1(C3×D4), (C6×F5).3C2, C6.18(C2×F5), (C3×D5).3Q8, C10.4(C2×C12), C30.18(C2×C4), (C3×Dic5)⋊7C4, D10.5(C2×C6), (D5×C12).11C2, (C6×D5).24C22, C5⋊(C3×C4⋊C4), SmallGroup(240,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊F5
G = < a,b,c,d | a3=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C3×C4⋊F5
►On 60 points
Generators in S60
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)
(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)
(2 3 5 4)(6 8 7 10)(11 13 12 15)(16 18 17 20)(21 23 22 25)(26 28 27 30)(31 38 32 40)(33 37 35 36)(34 39)(41 48 42 50)(43 47 45 46)(44 49)(51 58 52 60)(53 57 55 56)(54 59)
G:=sub<Sym(60)| (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55), (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), (2,3,5,4)(6,8,7,10)(11,13,12,15)(16,18,17,20)(21,23,22,25)(26,28,27,30)(31,38,32,40)(33,37,35,36)(34,39)(41,48,42,50)(43,47,45,46)(44,49)(51,58,52,60)(53,57,55,56)(54,59)>;
G:=Group( (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55), (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), (2,3,5,4)(6,8,7,10)(11,13,12,15)(16,18,17,20)(21,23,22,25)(26,28,27,30)(31,38,32,40)(33,37,35,36)(34,39)(41,48,42,50)(43,47,45,46)(44,49)(51,58,52,60)(53,57,55,56)(54,59) );
G=PermutationGroup([[(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(1,39,9,34),(2,40,10,35),(3,36,6,31),(4,37,7,32),(5,38,8,33),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,56,26,51),(22,57,27,52),(23,58,28,53),(24,59,29,54),(25,60,30,55)], [(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)], [(2,3,5,4),(6,8,7,10),(11,13,12,15),(16,18,17,20),(21,23,22,25),(26,28,27,30),(31,38,32,40),(33,37,35,36),(34,39),(41,48,42,50),(43,47,45,46),(44,49),(51,58,52,60),(53,57,55,56),(54,59)]])
C3×C4⋊F5 is a maximal subgroup of
D60⋊C4 Dic6⋊F5 Dic5.Dic6 Dic5.4Dic6 C4⋊F5⋊3S3 Dic6⋊5F5 D60⋊3C4 C3×D4×F5 C3×Q8×F5
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | ··· | 4F | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 10 | 12A | 12B | 12C | ··· | 12L | 15A | 15B | 20A | 20B | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 2 | 10 | ··· | 10 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | F5 | C2×F5 | C3×F5 | C4⋊F5 | C6×F5 | C3×C4⋊F5 |
| kernel | C3×C4⋊F5 | D5×C12 | C6×F5 | C4⋊F5 | C3×Dic5 | C60 | C4×D5 | C2×F5 | Dic5 | C20 | C3×D5 | C3×D5 | D5 | D5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×C4⋊F5 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 4 | 0 | 0 |
| 4 | 5 | 0 | 0 |
| 1 | 6 | 6 | 2 |
| 5 | 0 | 6 | 1 |
| 1 | 0 | 3 | 2 |
| 5 | 2 | 6 | 1 |
| 2 | 3 | 4 | 6 |
| 0 | 5 | 2 | 6 |
| 6 | 0 | 4 | 4 |
| 2 | 2 | 0 | 2 |
| 5 | 2 | 4 | 0 |
| 0 | 3 | 2 | 2 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,4,1,5,4,5,6,0,0,0,6,6,0,0,2,1],[1,5,2,0,0,2,3,5,3,6,4,2,2,1,6,6],[6,2,5,0,0,2,2,3,4,0,4,2,4,2,0,2] >;
C3×C4⋊F5 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes F_5
% in TeX
G:=Group("C3xC4:F5"); // GroupNames label
G:=SmallGroup(240,114);
// by ID
G=gap.SmallGroup(240,114);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,151,3461,599]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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