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## G = (D4×C10)⋊C4order 320 = 26·5

### 6th semidirect product of D4×C10 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — (D4×C10)⋊C4
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — D10.C23 — (D4×C10)⋊C4
 Lower central C5 — C10 — C20 — (D4×C10)⋊C4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for (D4×C10)⋊C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a3b2, cbc=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 1018 in 190 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2 [×8], C4 [×2], C4 [×4], C22, C22 [×20], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4 [×8], C23 [×11], D5 [×2], D5 [×3], C10, C10 [×3], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4, C2×D4 [×8], C24, Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×4], D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C5⋊C8 [×2], C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C5×D4, C2×F5 [×2], C22×D5, C22×D5 [×9], C22×C10, C23.37D4, D5⋊C8 [×2], C4.F5, C4×F5, C4⋊F5 [×2], C22.F5, C22⋊F5, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], C2×C5⋊D4, D4×C10, C23×D5, D20⋊C4 [×4], D5⋊M4(2), D10.C23, C2×D4×D5, (D4×C10)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8⋊C22 [×2], C2×F5 [×3], C23.37D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (D4×C10)⋊C4

Smallest permutation representation of (D4×C10)⋊C4
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 19 14 9)(2 20 15 10)(3 16 11 6)(4 17 12 7)(5 18 13 8)(21 39 26 34)(22 40 27 35)(23 31 28 36)(24 32 29 37)(25 33 30 38)
(1 14)(2 15)(3 11)(4 12)(5 13)(21 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)
(1 26 13 29)(2 28 12 27)(3 30 11 25)(4 22 15 23)(5 24 14 21)(6 38 16 33)(7 40 20 31)(8 32 19 39)(9 34 18 37)(10 36 17 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), (1,19,14,9)(2,20,15,10)(3,16,11,6)(4,17,12,7)(5,18,13,8)(21,39,26,34)(22,40,27,35)(23,31,28,36)(24,32,29,37)(25,33,30,38), (1,14)(2,15)(3,11)(4,12)(5,13)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38), (1,26,13,29)(2,28,12,27)(3,30,11,25)(4,22,15,23)(5,24,14,21)(6,38,16,33)(7,40,20,31)(8,32,19,39)(9,34,18,37)(10,36,17,35)>;

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 5 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 8 8 8 8 10 10 10 10 10 10 10 20 20 size 1 1 2 4 4 5 5 10 20 20 2 2 10 10 20 20 20 20 4 20 20 20 20 4 4 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 F5 C8⋊C22 C2×F5 C2×F5 C22⋊F5 C22⋊F5 (D4×C10)⋊C4 kernel (D4×C10)⋊C4 D20⋊C4 D5⋊M4(2) D10.C23 C2×D4×D5 C2×D20 D4×D5 D4×C10 C4×D5 C2×Dic5 C22×D5 C2×D4 D5 C2×C4 D4 C4 C22 C1 # reps 1 4 1 1 1 2 4 2 2 1 1 1 2 1 2 2 2 2

Matrix representation of (D4×C10)⋊C4 in GL8(𝔽41)

 0 35 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 33 34 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 33 5 0 1
,
 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 9 0 0 0 0 0 0 18 40 0 0 0 0 0 0 2 9 1 20 0 0 0 0 4 36 4 40
,
 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 9 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 8 36 4 40
,
 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 7 6 0 0 0 0 0 0 33 34 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 18 40 18 16 0 0 0 0 40 0 0 0 0 0 0 0 33 5 0 1

G:=sub<GL(8,GF(41))| [0,7,0,0,0,0,0,0,35,7,0,0,0,0,0,0,0,0,1,33,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,40,0,0,33,0,0,0,0,0,40,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,18,2,4,0,0,0,0,9,40,9,36,0,0,0,0,0,0,1,4,0,0,0,0,0,0,20,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,9,40,0,36,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,40],[0,0,7,33,0,0,0,0,0,0,6,34,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,18,40,33,0,0,0,0,0,40,0,5,0,0,0,0,1,18,0,0,0,0,0,0,0,16,0,1] >;

(D4×C10)⋊C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})\rtimes C_4
% in TeX

G:=Group("(D4xC10):C4");
// GroupNames label

G:=SmallGroup(320,1105);
// by ID

G=gap.SmallGroup(320,1105);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,1684,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3*b^2,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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