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G = (C22×F5)⋊C4  order 320 = 26·5

The semidirect product of C22×F5 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C22×F5)⋊C4, C22⋊C4⋊3F5, C22⋊F5⋊1C4, D5.(C23⋊C4), C23.D5⋊5C4, D10.2(C4⋊C4), C22.6(C4×F5), C23.5(C2×F5), (C2×C10).1C42, C22.8(C4⋊F5), (C22×D5).6Q8, C5⋊1(C23.9D4), D10.2(C22⋊C4), (C22×D5).141D4, C2.5(D10.3Q8), (C23×D5).81C22, C22.16(C22⋊F5), C10.3(C2.C42), (C5×C22⋊C4)⋊3C4, (C2×C10).1(C4⋊C4), (D5×C22⋊C4).4C2, (C2×C22⋊F5).1C2, (C22×C10).10(C2×C4), (C22×D5).34(C2×C4), (C2×C10).16(C22⋊C4), SmallGroup(320,204)

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 2A2B2C2D2E2F2G2H2I4A4B4C···4L 5 10A10B10C10D10E20A20B20C20D
order1222222222444···45101010101020202020
size11222551010104420···204444888888

32 irreducible representations

dim1111111224444448
type++++-+++++
imageC1C2C2C4C4C4C4D4Q8F5C23⋊C4C2×F5C4×F5C4⋊F5C22⋊F5(C22×F5)⋊C4
kernel(C22×F5)⋊C4D5×C22⋊C4C2×C22⋊F5C23.D5C5×C22⋊C4C22⋊F5C22×F5C22×D5C22×D5C22⋊C4D5C23C22C22C22C1
# reps1122244311212222

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

400000000
040000000
004000000
000400000
000013200
000004000
000000132
000000040
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
401000000
535000000
354040340000
26770000
00001000
00000100
00000010
00000001
,
001400000
71270000
004000000
35404000000
000000132
000000040
000040900
00000100
,
10000000
01000000
3184000000
3180400000
00001000
0000234000
000000132
0000002340

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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