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G = (C2×D4)⋊7F5  order 320 = 26·5

5th semidirect product of C2×D4 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D4)⋊7F5, (D4×C10)⋊3C4, D5⋊(C23⋊C4), C23⋊F5⋊3C2, C23⋊1(C2×F5), D10.5(C2×D4), (C23×D5)⋊7C4, C22⋊F5⋊2C22, D10.D4⋊3C2, (C22×D5).67D4, C22.1(C22⋊F5), D10.43(C22⋊C4), C22.11(C22×F5), (C2×D20).139C22, (C23×D5).87C22, (C22×D5).147C23, (C2×C4×D5)⋊4C4, C5⋊2(C2×C23⋊C4), (C2×C4)⋊1(C2×F5), (C2×D4×D5).7C2, (C2×C20)⋊2(C2×C4), (C2×C5⋊D4)⋊3C4, (C2×C22⋊F5)⋊2C2, (C22×C10)⋊2(C2×C4), (C2×Dic5)⋊3(C2×C4), (C22×D5)⋊4(C2×C4), C2.18(C2×C22⋊F5), C10.17(C2×C22⋊C4), (C2×C10).1(C22⋊C4), (C2×C10).73(C22×C4), (C2×C5⋊D4).87C22, SmallGroup(320,1108)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — (C2×D4)⋊7F5
Chief series
C1 — C5 — C10 — D10 — C22×D5 — C22⋊F5 — C2×C22⋊F5 — (C2×D4)⋊7F5
Lower central
C5 — C10 — C2×C10 — (C2×D4)⋊7F5

Subgroups: 1162 in 210 conjugacy classes, 50 normal (28 characteristic)
C1, C2, C2 [×10], C4 [×6], C22, C22 [×2], C22 [×22], C5, C2×C4, C2×C4 [×11], D4 [×8], C23 [×2], C23 [×13], D5 [×2], D5 [×4], C10, C10 [×4], C22⋊C4 [×6], C22×C4 [×3], C2×D4, C2×D4 [×7], C24 [×2], Dic5, C20, F5 [×4], D10 [×2], D10 [×2], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×3], C23⋊C4 [×4], C2×C22⋊C4 [×2], C22×D4, C4×D5 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C2×F5 [×8], C22×D5 [×3], C22×D5 [×4], C22×D5 [×6], C22×C10 [×2], C2×C23⋊C4, C22⋊F5 [×4], C22⋊F5 [×2], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4 [×2], D4×C10, C22×F5 [×2], C23×D5 [×2], D10.D4 [×2], C23⋊F5 [×2], C2×C22⋊F5 [×2], C2×D4×D5, (C2×D4)⋊7F5

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C23⋊C4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C23⋊C4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×D4)⋊7F5

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=b-1, bd=db, cd=dc, ece-1=b2c, ede-1=d3 >

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

10000000
01000000
00100000
00010000
0000283600
000091300
0000002836
000000913
,
10000000
01000000
00100000
00010000
000000400
000000040
00001000
00000100
,
400000000
040000000
004000000
000400000
000013500
0000322800
0000002836
000000913
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
400000000
000400000
040000000
11110000
000040000
000038100
000000135
000000728

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,9,0,0,0,0,0,0,36,13,0,0,0,0,0,0,0,0,28,9,0,0,0,0,0,0,36,13],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,13,32,0,0,0,0,0,0,5,28,0,0,0,0,0,0,0,0,28,9,0,0,0,0,0,0,36,13],[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],[40,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1,0,0,0,0,0,0,0,0,40,38,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,7,0,0,0,0,0,0,5,28] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B···4J 5 10A10B10C10D10E10F10G20A20B
order12222222222244···45101010101010102020
size1122245510101020420···204444888888

32 irreducible representations

dim1111111112444448
type++++++++++++
imageC1C2C2C2C2C4C4C4C4D4F5C23⋊C4C2×F5C2×F5C22⋊F5(C2×D4)⋊7F5
kernel(C2×D4)⋊7F5D10.D4C23⋊F5C2×C22⋊F5C2×D4×D5C2×C4×D5C2×C5⋊D4D4×C10C23×D5C22×D5C2×D4D5C2×C4C23C22C1
# reps1222122224121242

In GAP, Magma, Sage, TeX 

(C_2\times D_4)\rtimes_7F_5
 
% in TeX
 
G:=Group("(C2xD4):7F5");
 
// GroupNames label
 
G:=SmallGroup(320,1108);
 
// by ID
 
G=gap.SmallGroup(320,1108);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,1684,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=e^4=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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