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G = C4○D4⋊F5  order 320 = 26·5

2nd semidirect product of C4○D4 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D4⋊2F5, D4⋊6(C2×F5), Q8⋊6(C2×F5), C4○D20⋊2C4, D20⋊6(C2×C4), D4⋊2D5⋊8C4, D20⋊C4⋊4C2, Q8⋊F5⋊4C2, Q8⋊2D5⋊8C4, C4⋊F5.9C22, Dic10⋊6(C2×C4), (C4×D5).114D4, D5⋊M4(2)⋊6C2, D5⋊C8.8C22, C5⋊(C23.36D4), C4.22(C22×F5), D10.100(C2×D4), C20.22(C22×C4), D5.5(C8⋊C22), Dic5.10(C2×D4), (C4×D5).44C23, (D4×D5).14C22, (C22×D5).72D4, C4.33(C22⋊F5), (Q8×D5).12C22, C20.31(C22⋊C4), (C2×Dic5).124D4, D5.5(C8.C22), C22.5(C22⋊F5), D10.17(C22⋊C4), Dic5.49(C22⋊C4), (C2×C4⋊F5)⋊3C2, (C5×C4○D4)⋊2C4, (C5×D4)⋊6(C2×C4), (C5×Q8)⋊6(C2×C4), (D5×C4○D4).6C2, (C2×C4).43(C2×F5), (C2×C20).67(C2×C4), (C4×D5).28(C2×C4), C2.35(C2×C22⋊F5), C10.34(C2×C22⋊C4), (C2×C4×D5).211C22, (C2×C10).5(C22⋊C4), SmallGroup(320,1131)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — C4○D4⋊F5
Chief series
C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — C2×C4⋊F5 — C4○D4⋊F5
Lower central
C5 — C10 — C20 — C4○D4⋊F5
Upper central
C1 — C2 — C2×C4 — C4○D4

Generators and relations for C4○D4⋊F5
 G = < a,b,c,d,e | a4=c2=d5=e4=1, b2=a2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, ebe-1=abc, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 730 in 162 conjugacy classes, 50 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5, C22×D5, C22×D5, C23.36D4, D5⋊C8, C4.F5, C4⋊F5, C4⋊F5, C22.F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, C22×F5, D20⋊C4, Q8⋊F5, D5⋊M4(2), C2×C4⋊F5, D5×C4○D4, C4○D4⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C2×C22⋊C4, C8⋊C22, C8.C22, C2×F5, C23.36D4, C22⋊F5, C22×F5, C2×C22⋊F5, C4○D4⋊F5

Smallest permutation representation of C4○D4⋊F5
►On 40 points
Generators in S40
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

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

(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)

(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)

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F···4J 5 8A8B8C8D10A10B10C10D20A20B20C20D20E
order12222222444444···458888101010102020202020
size1124551020224101020···20420202020488844888

32 irreducible representations

dim1111111111222444444448
type+++++++++++-+++++
imageC1C2C2C2C2C2C4C4C4C4D4D4D4F5C8⋊C22C8.C22C2×F5C2×F5C2×F5C22⋊F5C22⋊F5C4○D4⋊F5
kernelC4○D4⋊F5D20⋊C4Q8⋊F5D5⋊M4(2)C2×C4⋊F5D5×C4○D4C4○D20D4⋊2D5Q8⋊2D5C5×C4○D4C4×D5C2×Dic5C22×D5C4○D4D5D5C2×C4D4Q8C4C22C1
# reps1221112222211111111222

Matrix representation of C4○D4⋊F5 ►in GL8(𝔽41)

10000000
01000000
00100000
00010000
0000403600
000025100
000001040
000025110
,
10000000
01000000
00100000
00010000
00001050
000000401
0000160400
00001640400
,
10000000
01000000
00100000
00010000
000040000
000004000
000025010
000025001
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
10000000
00010000
01000000
404040400000
00001000
0000164000
00000001
00000010

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,25,0,25,0,0,0,0,36,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,0,0,40,0,0,0,0,5,40,40,40,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,25,25,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C4○D4⋊F5 in GAP, Magma, Sage, TeX 

C_4\circ D_4\rtimes F_5
 
% in TeX
 
G:=Group("C4oD4:F5");
 
// GroupNames label
 
G:=SmallGroup(320,1131);
 
// by ID
 
G=gap.SmallGroup(320,1131);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,1684,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=c^2=d^5=e^4=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=a^2*b,b*d=d*b,e*b*e^-1=a*b*c,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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