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G = C22⋊C4×F5order 320 = 26·5

Direct product of C22⋊C4 and F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22⋊C4×F5, D10⋊C42, (C2×C10)⋊C42, C2.2(D4×F5), D5.2(C4×D4), C10.5(C4×D4), C22⋊F53C4, (C2×F5).8D4, C222(C4×F5), (C22×F5)⋊2C4, C23.D58C4, D10⋊C43C4, D10.59(C2×D4), C10.8(C2×C42), (C23×F5).1C2, C23.27(C2×F5), D10.3Q87C2, D10.42(C4○D4), D10.29(C22×C4), C22.35(C22×F5), D5.2(C42⋊C2), (C23×D5).83C22, (C22×F5).22C22, (C22×D5).268C23, (C2×C4×F5)⋊7C2, C51(C4×C22⋊C4), (C2×C4)⋊7(C2×F5), (C2×C20)⋊9(C2×C4), C2.10(C2×C4×F5), (C2×F5)⋊1(C2×C4), (C5×C22⋊C4)⋊6C4, (D5×C22⋊C4).7C2, D5.1(C2×C22⋊C4), (C2×C22⋊F5).3C2, (C2×Dic5)⋊14(C2×C4), (C2×C4×D5).288C22, (C22×C10).19(C2×C4), (C2×C10).36(C22×C4), (C22×D5).43(C2×C4), SmallGroup(320,1036)

Series: Derived Chief Lower central Upper central

C1C10 — C22⋊C4×F5
C1C5D5D10C22×D5C22×F5C23×F5 — C22⋊C4×F5
C5C10 — C22⋊C4×F5
C1C22C22⋊C4

Generators and relations for C22⋊C4×F5
 G = < a,b,c,d,e | a2=b2=c4=d5=e4=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 1018 in 258 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×14], C22, C22 [×2], C22 [×20], C5, C2×C4 [×2], C2×C4 [×32], C23, C23 [×10], D5 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42 [×4], C22⋊C4, C22⋊C4 [×7], C22×C4 [×14], C24, Dic5 [×2], C20 [×2], F5 [×4], F5 [×6], D10 [×2], D10 [×6], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C23×C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×F5 [×12], C2×F5 [×14], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C4×C22⋊C4, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C4×F5 [×4], C22⋊F5 [×4], C2×C4×D5 [×2], C22×F5 [×2], C22×F5 [×6], C22×F5 [×4], C23×D5, D10.3Q8 [×2], D5×C22⋊C4, C2×C4×F5 [×2], C2×C22⋊F5, C23×F5, C22⋊C4×F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], F5, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C2×F5 [×3], C4×C22⋊C4, C4×F5 [×2], C22×F5, C2×C4×F5, D4×F5 [×2], C22⋊C4×F5

Smallest permutation representation of C22⋊C4×F5
On 40 points
Generators in S40
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(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 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 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)
(1 11)(2 13 5 14)(3 15 4 12)(6 16)(7 18 10 19)(8 20 9 17)(21 31)(22 33 25 34)(23 35 24 32)(26 36)(27 38 30 39)(28 40 29 37)

G:=sub<Sym(40)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30), (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), (1,11)(2,13,5,14)(3,15,4,12)(6,16)(7,18,10,19)(8,20,9,17)(21,31)(22,33,25,34)(23,35,24,32)(26,36)(27,38,30,39)(28,40,29,37)>;

G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30), (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), (1,11)(2,13,5,14)(3,15,4,12)(6,16)(7,18,10,19)(8,20,9,17)(21,31)(22,33,25,34)(23,35,24,32)(26,36)(27,38,30,39)(28,40,29,37) );

G=PermutationGroup([(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(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)], [(1,31,11,21),(2,32,12,22),(3,33,13,23),(4,34,14,24),(5,35,15,25),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30)], [(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)], [(1,11),(2,13,5,14),(3,15,4,12),(6,16),(7,18,10,19),(8,20,9,17),(21,31),(22,33,25,34),(23,35,24,32),(26,36),(27,38,30,39),(28,40,29,37)])

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E···4L4M···4AB 5 10A10B10C10D10E20A20B20C20D
order12222222222244444···44···45101010101020202020
size1111225555101022225···510···104444888888

50 irreducible representations

dim111111111112244448
type+++++++++++
imageC1C2C2C2C2C2C4C4C4C4C4D4C4○D4F5C2×F5C2×F5C4×F5D4×F5
kernelC22⋊C4×F5D10.3Q8D5×C22⋊C4C2×C4×F5C2×C22⋊F5C23×F5D10⋊C4C23.D5C5×C22⋊C4C22⋊F5C22×F5C2×F5D10C22⋊C4C2×C4C23C22C2
# reps121211422884412142

Matrix representation of C22⋊C4×F5 in GL6(𝔽41)

4000000
010000
001000
000100
000010
000001
,
4000000
0400000
001000
000100
000010
000001
,
010000
4000000
0032000
0003200
0000320
0000032
,
100000
010000
0040404040
001000
000100
000010
,
4000000
0400000
0000400
0040000
0000040
0004000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,0,40,0] >;

C22⋊C4×F5 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times F_5
% in TeX

G:=Group("C2^2:C4xF5");
// GroupNames label

G:=SmallGroup(320,1036);
// by ID

G=gap.SmallGroup(320,1036);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=e^4=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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