direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊F5, D10.4Q8, D10.11D4, D10.10C23, C10⋊(C4⋊C4), D5⋊(C4⋊C4), (C2×C4)⋊3F5, C4⋊2(C2×F5), C20⋊2(C2×C4), (C2×C20)⋊3C4, (C4×D5)⋊4C4, D5.1(C2×D4), D5.1(C2×Q8), Dic5⋊7(C2×C4), (C2×Dic5)⋊8C4, C2.6(C22×F5), D10.16(C2×C4), C10.5(C22×C4), (C22×F5).2C2, (C2×F5).1C22, C22.18(C2×F5), (C4×D5).30C22, (C22×D5).37C22, C5⋊(C2×C4⋊C4), (C2×C4×D5).14C2, (C2×C10).17(C2×C4), SmallGroup(160,204)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊F5
G = < a,b,c,d | a2=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 292 in 92 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, D5, D5, C10, C10, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C4⋊C4, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C4⋊F5, C2×C4×D5, C22×F5, C2×C4⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×F5, C4⋊F5, C22×F5, C2×C4⋊F5
Character table of C2×C4⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -1 | 1 | i | -i | i | -i | -i | i | -i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | 1 | -1 | i | -i | -i | i | -i | i | i | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -1 | -1 | i | -i | i | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | i | 1 | 1 | i | -i | -i | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | 1 | 1 | -i | i | i | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -1 | -1 | -i | i | -i | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | 1 | -1 | -i | i | i | -i | i | -i | -i | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | -1 | 1 | -i | i | -i | i | i | -i | i | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×F5 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | √-5 | √-5 | -√-5 | -√-5 | complex lifted from C4⋊F5 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -√-5 | -√-5 | √-5 | √-5 | complex lifted from C4⋊F5 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | √-5 | -√-5 | -√-5 | √-5 | complex lifted from C4⋊F5 |
| ρ28 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -√-5 | √-5 | √-5 | -√-5 | complex lifted from C4⋊F5 |
►On 40 points
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 21 16 26)(12 22 17 27)(13 23 18 28)(14 24 19 29)(15 25 20 30)
(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)
(2 3 5 4)(6 8 7 10)(11 18 12 20)(13 17 15 16)(14 19)(21 23 22 25)(26 28 27 30)(31 38 32 40)(33 37 35 36)(34 39)
G:=sub<Sym(40)| (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,21,16,26)(12,22,17,27)(13,23,18,28)(14,24,19,29)(15,25,20,30), (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), (2,3,5,4)(6,8,7,10)(11,18,12,20)(13,17,15,16)(14,19)(21,23,22,25)(26,28,27,30)(31,38,32,40)(33,37,35,36)(34,39)>;
G:=Group( (1,24)(2,25)(3,21)(4,22)(5,23)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,21,16,26)(12,22,17,27)(13,23,18,28)(14,24,19,29)(15,25,20,30), (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), (2,3,5,4)(6,8,7,10)(11,18,12,20)(13,17,15,16)(14,19)(21,23,22,25)(26,28,27,30)(31,38,32,40)(33,37,35,36)(34,39) );
G=PermutationGroup([[(1,24),(2,25),(3,21),(4,22),(5,23),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)], [(1,39,9,34),(2,40,10,35),(3,36,6,31),(4,37,7,32),(5,38,8,33),(11,21,16,26),(12,22,17,27),(13,23,18,28),(14,24,19,29),(15,25,20,30)], [(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)], [(2,3,5,4),(6,8,7,10),(11,18,12,20),(13,17,15,16),(14,19),(21,23,22,25),(26,28,27,30),(31,38,32,40),(33,37,35,36),(34,39)]])
C2×C4⋊F5 is a maximal subgroup of
D10.18D8 D10.10D8 M4(2)⋊F5 C42⋊8F5 C42⋊9F5 D10⋊(C4⋊C4) C10.(C4×D4) C4⋊C4×F5 C4⋊C4⋊5F5 C20⋊(C4⋊C4) M4(2)⋊1F5 D10⋊6(C4⋊C4) C2.(D4×F5) (C2×F5)⋊Q8 C4○D4⋊F5 C2×D4×F5 C2×Q8×F5 D5.2+ 1+4
C2×C4⋊F5 is a maximal quotient of
C42.11F5 C20⋊3M4(2) C42.14F5 C42⋊8F5 C42⋊9F5 C20⋊C8⋊C2 D10⋊(C4⋊C4) C4⋊C4.9F5 C20⋊(C4⋊C4) (C2×C8)⋊6F5 (C8×D5).C4 M4(2)⋊1F5 M4(2).1F5 C20⋊8M4(2) D10⋊6(C4⋊C4)
Matrix representation of C2×C4⋊F5 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 5 | 0 | 0 | 0 | 0 |
| 9 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 27 | 27 |
| 0 | 0 | 14 | 7 | 14 | 0 |
| 0 | 0 | 0 | 14 | 7 | 14 |
| 0 | 0 | 27 | 27 | 0 | 34 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 38 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,9,0,0,0,0,5,6,0,0,0,0,0,0,34,14,0,27,0,0,0,7,14,27,0,0,27,14,7,0,0,0,27,0,14,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[9,38,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;
C2×C4⋊F5 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes F_5
% in TeX
G:=Group("C2xC4:F5"); // GroupNames label
G:=SmallGroup(160,204);
// by ID
G=gap.SmallGroup(160,204);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,2309,599]);
// Polycyclic
G:=Group<a,b,c,d|a^2=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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