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## G = C5×D4⋊4D4order 320 = 26·5

### Direct product of C5 and D4⋊4D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4⋊4D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — D4×C10 — C5×C8⋊C22 — C5×D4⋊4D4
 Lower central C1 — C2 — C2×C4 — C5×D4⋊4D4
 Upper central C1 — C10 — C2×C20 — C5×D4⋊4D4

Generators and relations for C5×D44D4
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 370 in 168 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C10, C10, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C22×C10, D44D4, C4×C20, C5×M4(2), C5×D8, C5×SD16, D4×C10, D4×C10, C5×C4○D4, C5×C4○D4, C5×C4.D4, C5×C4≀C2, C5×C41D4, C5×C8⋊C22, C5×2+ 1+4, C5×D44D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C5×D4, C22×C10, D44D4, D4×C10, C5×C22≀C2, C5×D44D4

Smallest permutation representation of C5×D44D4
On 40 points
Generators in S40
(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 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 16 36 11)(7 17 37 12)(8 18 38 13)(9 19 39 14)(10 20 40 15)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(26 36)(27 37)(28 38)(29 39)(30 40)
(6 11 36 16)(7 12 37 17)(8 13 38 18)(9 14 39 19)(10 15 40 20)
(6 16)(7 17)(8 18)(9 19)(10 20)(11 36)(12 37)(13 38)(14 39)(15 40)(26 31)(27 32)(28 33)(29 34)(30 35)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10X 10Y 10Z 10AA 10AB 20A ··· 20H 20I ··· 20X 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 4 4 4 4 8 2 2 4 4 4 4 1 1 1 1 8 8 1 1 1 1 2 2 2 2 4 ··· 4 8 8 8 8 2 ··· 2 4 ··· 4 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D4 C5×D4 C5×D4 C5×D4 D4⋊4D4 C5×D4⋊4D4 kernel C5×D4⋊4D4 C5×C4.D4 C5×C4≀C2 C5×C4⋊1D4 C5×C8⋊C22 C5×2+ 1+4 D4⋊4D4 C4.D4 C4≀C2 C4⋊1D4 C8⋊C22 2+ 1+4 C5×D4 C5×Q8 C22×C10 D4 Q8 C23 C5 C1 # reps 1 1 2 1 2 1 4 4 8 4 8 4 2 2 2 8 8 8 2 8

Matrix representation of C5×D44D4 in GL6(𝔽41)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 40 40 1 2 0 0 0 1 40 40
,
 40 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 1 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 40 1 1
,
 40 1 0 0 0 0 39 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 40 39 0 0 0 0 1 1
,
 40 0 0 0 0 0 39 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 1 1 40 39 0 0 0 40 0 1

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,40,0,0,0,1,0,40,1,0,0,0,0,1,40,0,0,0,0,2,40],[40,0,0,0,0,0,1,1,0,0,0,0,0,0,40,0,0,0,0,0,40,0,1,40,0,0,1,1,0,1,0,0,2,0,0,1],[40,39,0,0,0,0,1,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,40,1,40,0,0,0,0,40,0,0,0,0,0,39,1] >;

C5×D44D4 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_4D_4
% in TeX

G:=Group("C5xD4:4D4");
// GroupNames label

G:=SmallGroup(320,954);
// by ID

G=gap.SmallGroup(320,954);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,7004,3511,1768,172,5052]);
// Polycyclic

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

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