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G = C242F5order 320 = 26·5

1st semidirect product of C24 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C242F5, C52C2≀C4, C23⋊F52C2, (C23×C10)⋊5C4, C23.D57C4, C23.F52C2, C23.19(C2×F5), (C2×Dic5).14D4, (C22×D5).14D4, C242D5.2C2, 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

C1C22×C10 — C242F5
C1C5C10C2×C10C22×D5C2×C5⋊D4C23⋊F5 — C242F5
C5C10C2×C10C22×C10 — C242F5
C1C2C22C23C24

Generators and relations for C242F5
 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 [×5], C4 [×3], C22, C22 [×12], C5, C8, C2×C4 [×3], D4 [×3], C23, C23 [×5], D5, C10, C10 [×4], C22⋊C4 [×3], M4(2), C2×D4 [×2], C24, Dic5 [×2], F5, D10 [×2], C2×C10, C2×C10 [×10], C23⋊C4, C4.D4, C22≀C2, C5⋊C8, C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×F5, C22×D5, C22×C10, C22×C10 [×4], C2≀C4, C23.D5, C23.D5, C22.F5, C22⋊F5, C2×C5⋊D4, C2×C5⋊D4, C23×C10, C23⋊F5, C23.F5, C242D5, C242F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C2≀C4, C22⋊F5, C23⋊F5, C242F5

Character table of C242F5

 class 12A2B2C2D2E2F4A4B4C4D58A8B10A10B10C10D10E10F10G10H10I10J10K10L10M10N10O
 size 112444202040404044040444444444444444
ρ111111111111111111111111111111    trivial
ρ211111111-11-11-1-1111111111111111    linear of order 2
ρ3111-11-111-1-1-1111-1-1-11-11-1-1-1-111111    linear of order 2
ρ4111-11-1111-111-1-1-1-1-11-11-1-1-1-111111    linear of order 2
ρ5111111-1-1i-1-i1i-i111111111111111    linear of order 4
ρ6111111-1-1-i-1i1-ii111111111111111    linear of order 4
ρ7111-11-1-1-1i1-i1-ii-1-1-11-11-1-1-1-111111    linear of order 4
ρ8111-11-1-1-1-i1i1i-i-1-1-11-11-1-1-1-111111    linear of order 4
ρ92220-202-20002000002020000-2-22-2-2    orthogonal lifted from D4
ρ102220-20-220002000002020000-2-22-2-2    orthogonal lifted from D4
ρ1144-400000000400000-40-4000000400    orthogonal lifted from C23⋊C4
ρ124-4020-200000400222020-2-2-2-200-400    orthogonal lifted from C2≀C4
ρ134-40-20200000400-2-2-20-20222200-400    orthogonal lifted from C2≀C4
ρ14444-44-400000-100111-11-11111-1-1-1-1-1    orthogonal lifted from C2×F5
ρ1544444400000-100-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ164440-4000000-1005-55-1-5-1-555-511-111    orthogonal lifted from C22⋊F5
ρ174440-4000000-100-55-5-15-15-5-5511-111    orthogonal lifted from C22⋊F5
ρ1844-400000000-10053+2ζ5+152+2ζ5+154+2ζ52+1154+2ζ53+1152+2ζ5+154+2ζ52+153+2ζ5+154+2ζ53+1-55-15-5    complex lifted from C23⋊F5
ρ194-40-20200000-10052+154+153+155+1-5-2ζ54-1-2ζ53-1-2ζ52-1-2ζ5-154+2ζ53+154+2ζ52+1153+2ζ5+152+2ζ5+1    complex faithful
ρ2044-400000000-10054+2ζ53+153+2ζ5+152+2ζ5+1154+2ζ52+1153+2ζ5+152+2ζ5+154+2ζ53+154+2ζ52+15-5-1-55    complex lifted from C23⋊F5
ρ214-4020-200000-100-2ζ54-1-2ζ53-1-2ζ5-1-5-2ζ52-1553+15+154+152+153+2ζ5+154+2ζ53+1152+2ζ5+154+2ζ52+1    complex faithful
ρ224-40-20200000-10054+153+15+1-552+15-2ζ53-1-2ζ5-1-2ζ54-1-2ζ52-153+2ζ5+154+2ζ53+1152+2ζ5+154+2ζ52+1    complex faithful
ρ234-4020-200000-100-2ζ5-1-2ζ52-1-2ζ54-1-5-2ζ53-1552+154+15+153+154+2ζ52+152+2ζ5+1154+2ζ53+153+2ζ5+1    complex faithful
ρ2444-400000000-10052+2ζ5+154+2ζ52+154+2ζ53+1153+2ζ5+1154+2ζ52+154+2ζ53+152+2ζ5+153+2ζ5+15-5-1-55    complex lifted from C23⋊F5
ρ2544-400000000-10054+2ζ52+154+2ζ53+153+2ζ5+1152+2ζ5+1154+2ζ53+153+2ζ5+154+2ζ52+152+2ζ5+1-55-15-5    complex lifted from C23⋊F5
ρ264-40-20200000-10053+15+152+1554+1-5-2ζ5-1-2ζ52-1-2ζ53-1-2ζ54-152+2ζ5+153+2ζ5+1154+2ζ52+154+2ζ53+1    complex faithful
ρ274-4020-200000-100-2ζ52-1-2ζ54-1-2ζ53-15-2ζ5-1-554+153+152+15+154+2ζ53+154+2ζ52+1153+2ζ5+152+2ζ5+1    complex faithful
ρ284-4020-200000-100-2ζ53-1-2ζ5-1-2ζ52-15-2ζ54-1-55+152+153+154+152+2ζ5+153+2ζ5+1154+2ζ52+154+2ζ53+1    complex faithful
ρ294-40-20200000-1005+152+154+1-553+15-2ζ52-1-2ζ54-1-2ζ5-1-2ζ53-154+2ζ52+152+2ζ5+1154+2ζ53+153+2ζ5+1    complex faithful

Smallest permutation representation of C242F5
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 C242F5 in GL4(𝔽41) generated by

40000
04000
00235
00118
,
183600
402300
001836
004023
,
1000
0100
00400
00040
,
40000
04000
00400
00040
,
40500
13500
0066
00340
,
0010
0001
6600
13500
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] >;

C242F5 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

Export

Character table of C242F5 in TeX

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