metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2F5, C5⋊2C2≀C4, C23⋊F5⋊2C2, (C23×C10)⋊5C4, C23.D5⋊7C4, C23.F5⋊2C2, C23.19(C2×F5), (C2×Dic5).14D4, (C22×D5).14D4, C24⋊2D5.2C2, C2.11(C23⋊F5), C10.21(C23⋊C4), C22.23(C22⋊F5), (C22×C10).50(C2×C4), (C2×C5⋊D4).86C22, (C2×C10).46(C22⋊C4), SmallGroup(320,272)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊2F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, faf-1=abcd, bc=cb, bd=db, be=eb, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >
Subgroups: 506 in 94 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, M4(2), C2×D4, C24, Dic5, F5, D10, C2×C10, C2×C10, C23⋊C4, C4.D4, C22≀C2, C5⋊C8, C2×Dic5, C2×Dic5, C5⋊D4, C2×F5, C22×D5, C22×C10, C22×C10, C2≀C4, C23.D5, C23.D5, C22.F5, C22⋊F5, C2×C5⋊D4, C2×C5⋊D4, C23×C10, C23⋊F5, C23.F5, C24⋊2D5, C24⋊2F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C2≀C4, C22⋊F5, C23⋊F5, C24⋊2F5
Character table of C24⋊2F5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 10O | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 20 | 20 | 40 | 40 | 40 | 4 | 40 | 40 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 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 | 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 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | i | -1 | -i | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -i | -1 | i | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | i | 1 | -i | 1 | -i | i | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | 1 | i | 1 | i | -i | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ12 | 4 | -4 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | -4 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ13 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -2 | -2 | -2 | 0 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -4 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ14 | 4 | 4 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ15 | 4 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ16 | 4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | √5 | -√5 | √5 | -1 | -√5 | -1 | -√5 | √5 | √5 | -√5 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ17 | 4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -√5 | √5 | -√5 | -1 | √5 | -1 | √5 | -√5 | -√5 | √5 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ18 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 1 | 2ζ54+2ζ53+1 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | -√5 | √5 | -1 | √5 | -√5 | complex lifted from C23⋊F5 |
| ρ19 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ52+1 | 2ζ54+1 | 2ζ53+1 | √5 | 2ζ5+1 | -√5 | -2ζ54-1 | -2ζ53-1 | -2ζ52-1 | -2ζ5-1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | 1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | complex faithful |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 1 | 2ζ54+2ζ52+1 | 1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | √5 | -√5 | -1 | -√5 | √5 | complex lifted from C23⋊F5 |
| ρ21 | 4 | -4 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2ζ54-1 | -2ζ53-1 | -2ζ5-1 | -√5 | -2ζ52-1 | √5 | 2ζ53+1 | 2ζ5+1 | 2ζ54+1 | 2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | complex faithful |
| ρ22 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ54+1 | 2ζ53+1 | 2ζ5+1 | -√5 | 2ζ52+1 | √5 | -2ζ53-1 | -2ζ5-1 | -2ζ54-1 | -2ζ52-1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | complex faithful |
| ρ23 | 4 | -4 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2ζ5-1 | -2ζ52-1 | -2ζ54-1 | -√5 | -2ζ53-1 | √5 | 2ζ52+1 | 2ζ54+1 | 2ζ5+1 | 2ζ53+1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | 1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | complex faithful |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 1 | 2ζ53+2ζ5+1 | 1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | √5 | -√5 | -1 | -√5 | √5 | complex lifted from C23⋊F5 |
| ρ25 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 1 | 2ζ52+2ζ5+1 | 1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | -√5 | √5 | -1 | √5 | -√5 | complex lifted from C23⋊F5 |
| ρ26 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ53+1 | 2ζ5+1 | 2ζ52+1 | √5 | 2ζ54+1 | -√5 | -2ζ5-1 | -2ζ52-1 | -2ζ53-1 | -2ζ54-1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | 1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | complex faithful |
| ρ27 | 4 | -4 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2ζ52-1 | -2ζ54-1 | -2ζ53-1 | √5 | -2ζ5-1 | -√5 | 2ζ54+1 | 2ζ53+1 | 2ζ52+1 | 2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | 1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | complex faithful |
| ρ28 | 4 | -4 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -2ζ53-1 | -2ζ5-1 | -2ζ52-1 | √5 | -2ζ54-1 | -√5 | 2ζ5+1 | 2ζ52+1 | 2ζ53+1 | 2ζ54+1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | 1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | complex faithful |
| ρ29 | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 2ζ5+1 | 2ζ52+1 | 2ζ54+1 | -√5 | 2ζ53+1 | √5 | -2ζ52-1 | -2ζ54-1 | -2ζ5-1 | -2ζ53-1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | 1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | complex faithful |
►On 40 points
Generators in S40
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(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)
(1 23)(2 25 5 21)(3 22 4 24)(6 28)(7 30 10 26)(8 27 9 29)(11 38 16 33)(12 40 20 31)(13 37 19 34)(14 39 18 32)(15 36 17 35)
G:=sub<Sym(40)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,23)(2,25,5,21)(3,22,4,24)(6,28)(7,30,10,26)(8,27,9,29)(11,38,16,33)(12,40,20,31)(13,37,19,34)(14,39,18,32)(15,36,17,35)>;
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,23)(2,25,5,21)(3,22,4,24)(6,28)(7,30,10,26)(8,27,9,29)(11,38,16,33)(12,40,20,31)(13,37,19,34)(14,39,18,32)(15,36,17,35) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)], [(1,23),(2,25,5,21),(3,22,4,24),(6,28),(7,30,10,26),(8,27,9,29),(11,38,16,33),(12,40,20,31),(13,37,19,34),(14,39,18,32),(15,36,17,35)]])
Matrix representation of C24⋊2F5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 23 | 5 |
| 0 | 0 | 1 | 18 |
| 18 | 36 | 0 | 0 |
| 40 | 23 | 0 | 0 |
| 0 | 0 | 18 | 36 |
| 0 | 0 | 40 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 5 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 6 | 0 | 0 |
| 1 | 35 | 0 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,23,1,0,0,5,18],[18,40,0,0,36,23,0,0,0,0,18,40,0,0,36,23],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,1,0,0,5,35,0,0,0,0,6,34,0,0,6,0],[0,0,6,1,0,0,6,35,1,0,0,0,0,1,0,0] >;
C24⋊2F5 in GAP, Magma, Sage, TeX
C_2^4\rtimes_2F_5
% in TeX
G:=Group("C2^4:2F5"); // GroupNames label
G:=SmallGroup(320,272);
// by ID
G=gap.SmallGroup(320,272);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,675,297,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*b*c*d,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations
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