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## G = (C22×F5)⋊C4order 320 = 26·5

### The semidirect product of C22×F5 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C22×F5)⋊C4
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — C2×C22⋊F5 — (C22×F5)⋊C4
 Lower central C5 — C10 — C2×C10 — (C22×F5)⋊C4
 Upper central C1 — C2 — C23 — C22⋊C4

Generators and relations for (C22×F5)⋊C4
G = < a,b,c,d,e | a2=b2=c5=d4=e4=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, dcd-1=c3, ce=ec, ede-1=abd >

Subgroups: 802 in 142 conjugacy classes, 36 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, C23, D5, D5, C10, C10, C22⋊C4, C22⋊C4, C22×C4, C24, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C22×D5, C22×C10, C23.9D4, D10⋊C4, C23.D5, C5×C22⋊C4, C22⋊F5, C22⋊F5, C2×C4×D5, C22×F5, C22×F5, C23×D5, D5×C22⋊C4, C2×C22⋊F5, (C22×F5)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C23⋊C4, C2×F5, C23.9D4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, (C22×F5)⋊C4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C ··· 4L 5 10A 10B 10C 10D 10E 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 2 2 4 4 4 ··· 4 5 10 10 10 10 10 20 20 20 20 size 1 1 2 2 2 5 5 10 10 10 4 4 20 ··· 20 4 4 4 4 8 8 8 8 8 8

32 irreducible representations

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

Matrix representation of (C22×F5)⋊C4 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 32 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 32 0 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 40 1 0 0 0 0 0 0 5 35 0 0 0 0 0 0 35 40 40 34 0 0 0 0 2 6 7 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 1 40 0 0 0 0 7 1 2 7 0 0 0 0 0 0 40 0 0 0 0 0 35 40 40 0 0 0 0 0 0 0 0 0 0 0 1 32 0 0 0 0 0 0 0 40 0 0 0 0 40 9 0 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 18 40 0 0 0 0 0 3 18 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 23 40 0 0 0 0 0 0 0 0 1 32 0 0 0 0 0 0 23 40

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,5,35,2,0,0,0,0,1,35,40,6,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,7,0,35,0,0,0,0,0,1,0,40,0,0,0,0,1,2,40,40,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,1,0,0,0,0,1,0,0,0,0,0,0,0,32,40,0,0],[1,0,3,3,0,0,0,0,0,1,18,18,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,23,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,23,0,0,0,0,0,0,32,40] >;

(C22×F5)⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("(C2^2xF5):C4");
// GroupNames label

G:=SmallGroup(320,204);
// by ID

G=gap.SmallGroup(320,204);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,851,6278,3156]);
// Polycyclic

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

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