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## G = (C2×F5)⋊D4order 320 = 26·5

### The semidirect product of C2×F5 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×F5)⋊D4
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C23×F5 — (C2×F5)⋊D4
 Lower central C5 — C2×C10 — (C2×F5)⋊D4
 Upper central C1 — C22 — C2×D4

Generators and relations for (C2×F5)⋊D4
G = < a,b,c,d,e | a2=b5=c4=d4=e2=1, ab=ba, dcd-1=ac=ca, ad=da, ae=ea, cbc-1=b3, bd=db, be=eb, ce=ec, ede=d-1 >

Subgroups: 1354 in 286 conjugacy classes, 64 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×2], C22 [×30], C5, C2×C4, C2×C4 [×25], D4 [×8], C23 [×2], C23 [×17], D5 [×2], D5 [×2], D5 [×3], C10, C10 [×2], C10 [×3], C22⋊C4 [×6], C22×C4 [×11], C2×D4, C2×D4 [×7], C24 [×2], Dic5, C20, F5 [×6], D10 [×2], D10 [×6], D10 [×17], C2×C10, C2×C10 [×2], C2×C10 [×5], C2.C42 [×2], C2×C22⋊C4 [×3], C23×C4, C22×D4, C4×D5 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C2×F5 [×4], C2×F5 [×18], C22×D5 [×3], C22×D5 [×4], C22×D5 [×10], C22×C10 [×2], C23.23D4, C22⋊F5 [×6], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4 [×2], D4×C10, C22×F5 [×4], C22×F5 [×6], C23×D5 [×2], D10.3Q8 [×2], C2×C22⋊F5, C2×C22⋊F5 [×2], C2×D4×D5, C23×F5, (C2×F5)⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], F5, C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×F5 [×3], C23.23D4, C22⋊F5 [×2], C22×F5, D4×F5 [×2], C2×C22⋊F5, (C2×F5)⋊D4

Smallest permutation representation of (C2×F5)⋊D4
On 40 points
Generators in S40
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(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)
(1 19)(2 16 5 17)(3 18 4 20)(6 13 7 15)(8 12 10 11)(9 14)(21 28 22 30)(23 27 25 26)(24 29)(31 38 32 40)(33 37 35 36)(34 39)
(1 24 9 29)(2 25 10 30)(3 21 6 26)(4 22 7 27)(5 23 8 28)(11 31 16 36)(12 32 17 37)(13 33 18 38)(14 34 19 39)(15 35 20 40)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 C4○D4 F5 C2×F5 C2×F5 C22⋊F5 D4×F5 kernel (C2×F5)⋊D4 D10.3Q8 C2×C22⋊F5 C2×D4×D5 C23×F5 C2×D20 C2×C5⋊D4 D4×C10 C2×F5 C22×D5 D10 C2×D4 C2×C4 C23 C22 C2 # reps 1 2 3 1 1 2 4 2 4 4 4 1 1 2 4 2

Matrix representation of (C2×F5)⋊D4 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 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 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40
,
 9 0 0 0 0 0 0 0 14 32 0 0 0 0 0 0 0 0 9 32 0 0 0 0 0 0 0 32 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 1 0 0 0 0 0 0 40 40 40 40
,
 28 5 0 0 0 0 0 0 32 13 0 0 0 0 0 0 0 0 40 1 0 0 0 0 0 0 39 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 40 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

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40],[9,14,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,32,32,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40],[28,32,0,0,0,0,0,0,5,13,0,0,0,0,0,0,0,0,40,39,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

(C2×F5)⋊D4 in GAP, Magma, Sage, TeX

(C_2\times F_5)\rtimes D_4
% in TeX

G:=Group("(C2xF5):D4");
// GroupNames label

G:=SmallGroup(320,1117);
// by ID

G=gap.SmallGroup(320,1117);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,6278,1595]);
// Polycyclic

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

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