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## G = C22⋊C4×F5order 320 = 26·5

### Direct product of C22⋊C4 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22⋊C4×F5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C23×F5 — C22⋊C4×F5
 Lower central C5 — C10 — C22⋊C4×F5
 Upper central C1 — C22 — C22⋊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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4L 4M ··· 4AB 5 10A 10B 10C 10D 10E 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 5 10 10 10 10 10 20 20 20 20 size 1 1 1 1 2 2 5 5 5 5 10 10 2 2 2 2 5 ··· 5 10 ··· 10 4 4 4 4 8 8 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 8 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 D4 C4○D4 F5 C2×F5 C2×F5 C4×F5 D4×F5 kernel C22⋊C4×F5 D10.3Q8 D5×C22⋊C4 C2×C4×F5 C2×C22⋊F5 C23×F5 D10⋊C4 C23.D5 C5×C22⋊C4 C22⋊F5 C22×F5 C2×F5 D10 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 1 2 1 1 4 2 2 8 8 4 4 1 2 1 4 2

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

 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 1 0 0 0 0 40 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 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 40 0 0

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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