metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊D4⋊F5, C3⋊F5⋊2D4, C3⋊3(D4×F5), C15⋊4(C4×D4), D6⋊F5⋊5C2, D6⋊4(C2×F5), D10⋊2(C4×S3), D30⋊4(C2×C4), C15⋊D4⋊2C4, C3⋊D20⋊2C4, C15⋊7D4⋊2C4, D5.3(S3×D4), C22⋊F5⋊3S3, (C2×F5).6D6, C22⋊3(S3×F5), Dic3⋊F5⋊4C2, C5⋊(Dic3⋊4D4), (Dic3×F5)⋊4C2, Dic3⋊2(C2×F5), Dic15⋊2(C2×C4), (C6×F5).6C22, C6.31(C22×F5), C30.31(C22×C4), (C22×D5).39D6, (C6×D5).35C23, D5.4(D4⋊2S3), D10.38(C22×S3), (D5×Dic3).8C22, (C2×S3×F5)⋊5C2, (C2×C6)⋊2(C2×F5), (C2×C30)⋊3(C2×C4), (C2×C10)⋊5(C4×S3), C2.31(C2×S3×F5), (C5×C3⋊D4)⋊2C4, C10.31(S3×C2×C4), (S3×C10)⋊4(C2×C4), (C22×C3⋊F5)⋊2C2, (C6×D5)⋊13(C2×C4), (C3×D5).8(C2×D4), (D5×C3⋊D4).2C2, (C3×C22⋊F5)⋊4C2, (C2×S3×D5).4C22, (C5×Dic3)⋊2(C2×C4), (D5×C2×C6).72C22, (C2×C3⋊F5).13C22, (C3×D5).9(C4○D4), SmallGroup(480,1012)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D4⋊F5
G = < a,b,c,d,e | a3=b4=c2=d5=e4=1, bab-1=cac=a-1, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 1092 in 188 conjugacy classes, 52 normal (50 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C3×D5, D15, C30, C30, C4×D4, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C2×F5, C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, Dic15, C3×F5, C3⋊F5, C3⋊F5, S3×D5, C6×D5, C6×D5, S3×C10, D30, C2×C30, C4×F5, C4⋊F5, C22⋊F5, C22⋊F5, D4×D5, C22×F5, Dic3⋊4D4, D5×Dic3, C15⋊D4, C3⋊D20, C5×C3⋊D4, C15⋊7D4, S3×F5, C6×F5, C2×C3⋊F5, C2×C3⋊F5, C2×S3×D5, D5×C2×C6, D4×F5, Dic3×F5, D6⋊F5, Dic3⋊F5, C3×C22⋊F5, D5×C3⋊D4, C2×S3×F5, C22×C3⋊F5, C3⋊D4⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, F5, C4×S3, C22×S3, C4×D4, C2×F5, S3×C2×C4, S3×D4, D4⋊2S3, C22×F5, Dic3⋊4D4, S3×F5, D4×F5, C2×S3×F5, C3⋊D4⋊F5
►On 60 points
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)
(1 31 16 46)(2 32 17 47)(3 33 18 48)(4 34 19 49)(5 35 20 50)(6 41 21 56)(7 42 22 57)(8 43 23 58)(9 44 24 59)(10 45 25 60)(11 36 26 51)(12 37 27 52)(13 38 28 53)(14 39 29 54)(15 40 30 55)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 56)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 53)(44 54)(45 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)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)(31 46)(32 48 35 49)(33 50 34 47)(36 51)(37 53 40 54)(38 55 39 52)(41 56)(42 58 45 59)(43 60 44 57)
G:=sub<Sym(60)| (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60), (1,31,16,46)(2,32,17,47)(3,33,18,48)(4,34,19,49)(5,35,20,50)(6,41,21,56)(7,42,22,57)(8,43,23,58)(9,44,24,59)(10,45,25,60)(11,36,26,51)(12,37,27,52)(13,38,28,53)(14,39,29,54)(15,40,30,55), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,56)(37,57)(38,58)(39,59)(40,60)(41,51)(42,52)(43,53)(44,54)(45,55), (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)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)(31,46)(32,48,35,49)(33,50,34,47)(36,51)(37,53,40,54)(38,55,39,52)(41,56)(42,58,45,59)(43,60,44,57)>;
G:=Group( (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60), (1,31,16,46)(2,32,17,47)(3,33,18,48)(4,34,19,49)(5,35,20,50)(6,41,21,56)(7,42,22,57)(8,43,23,58)(9,44,24,59)(10,45,25,60)(11,36,26,51)(12,37,27,52)(13,38,28,53)(14,39,29,54)(15,40,30,55), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,56)(37,57)(38,58)(39,59)(40,60)(41,51)(42,52)(43,53)(44,54)(45,55), (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)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)(31,46)(32,48,35,49)(33,50,34,47)(36,51)(37,53,40,54)(38,55,39,52)(41,56)(42,58,45,59)(43,60,44,57) );
G=PermutationGroup([[(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45),(46,51,56),(47,52,57),(48,53,58),(49,54,59),(50,55,60)], [(1,31,16,46),(2,32,17,47),(3,33,18,48),(4,34,19,49),(5,35,20,50),(6,41,21,56),(7,42,22,57),(8,43,23,58),(9,44,24,59),(10,45,25,60),(11,36,26,51),(12,37,27,52),(13,38,28,53),(14,39,29,54),(15,40,30,55)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30),(31,46),(32,47),(33,48),(34,49),(35,50),(36,56),(37,57),(38,58),(39,59),(40,60),(41,51),(42,52),(43,53),(44,54),(45,55)], [(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),(7,8,10,9),(12,13,15,14),(17,18,20,19),(22,23,25,24),(27,28,30,29),(31,46),(32,48,35,49),(33,50,34,47),(36,51),(37,53,40,54),(38,55,39,52),(41,56),(42,58,45,59),(43,60,44,57)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20 | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 5 | 5 | 6 | 10 | 30 | 2 | 6 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 4 | 2 | 4 | 10 | 10 | 20 | 4 | 8 | 24 | 20 | 20 | 20 | 20 | 8 | 24 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | C4×S3 | F5 | C2×F5 | C2×F5 | C2×F5 | S3×D4 | D4⋊2S3 | S3×F5 | D4×F5 | C2×S3×F5 | C3⋊D4⋊F5 |
| kernel | C3⋊D4⋊F5 | Dic3×F5 | D6⋊F5 | Dic3⋊F5 | C3×C22⋊F5 | D5×C3⋊D4 | C2×S3×F5 | C22×C3⋊F5 | C15⋊D4 | C3⋊D20 | C5×C3⋊D4 | C15⋊7D4 | C22⋊F5 | C3⋊F5 | C2×F5 | C22×D5 | C3×D5 | D10 | C2×C10 | C3⋊D4 | Dic3 | D6 | C2×C6 | D5 | D5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of C3⋊D4⋊F5 ►in GL8(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 41 | 10 | 0 | 56 | 0 | 0 | 0 | 0 |
| 51 | 20 | 56 | 0 | 0 | 0 | 0 | 0 |
| 0 | 48 | 41 | 10 | 0 | 0 | 0 | 0 |
| 48 | 0 | 51 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 57 | 8 | 60 | 0 | 0 | 0 | 0 | 0 |
| 53 | 4 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(61))| [0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[41,51,0,48,0,0,0,0,10,20,48,0,0,0,0,0,0,56,41,51,0,0,0,0,56,0,10,20,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[1,0,57,53,0,0,0,0,0,1,8,4,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
C3⋊D4⋊F5 in GAP, Magma, Sage, TeX
C_3\rtimes D_4\rtimes F_5
% in TeX
G:=Group("C3:D4:F5"); // GroupNames label
G:=SmallGroup(480,1012);
// by ID
G=gap.SmallGroup(480,1012);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,219,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^5=e^4=1,b*a*b^-1=c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations