Copied to
clipboard

## G = C2×C4×D5order 80 = 24·5

### Direct product of C2×C4 and D5

Aliases: C2×C4×D5, C203C22, C10.2C23, C22.9D10, Dic53C22, D10.8C22, C4(C4×D5), C102(C2×C4), (C2×C20)⋊5C2, C52(C22×C4), C4(C2×Dic5), (C2×C4)Dic5, (C2×Dic5)⋊5C2, C2.1(C22×D5), (C2×C10).9C22, (C22×D5).4C2, (C2×C4)(C2×Dic5), SmallGroup(80,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C4×D5
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5
 Lower central C5 — C2×C4×D5
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D5
G = < a,b,c,d | a2=b4=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 130 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C5, C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C4×D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, D10 [×3], C4×D5 [×2], C22×D5, C2×C4×D5

Smallest permutation representation of C2×C4×D5
On 40 points
Generators in S40
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)

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

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

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

C2×C4×D5 is a maximal subgroup of
D101C8  D10⋊C8  D10.3Q8  C42⋊D5  Dic54D4  D10.12D4  D10⋊D4  C4⋊C47D5  D208C4  D10.13D4  C4⋊D20  D10⋊Q8  D102Q8  C202D4  D103Q8  D5⋊M4(2)  D10.C23
C2×C4×D5 is a maximal quotient of
C42⋊D5  C23.11D10  Dic54D4  Dic53Q8  C4⋊C47D5  D208C4  D20.3C4  D20.2C4

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 D5 D10 D10 C4×D5 kernel C2×C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 D10 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 2 4 2 8

Matrix representation of C2×C4×D5 in GL3(𝔽41) generated by

 40 0 0 0 1 0 0 0 1
,
 1 0 0 0 32 0 0 0 32
,
 1 0 0 0 40 1 0 5 35
,
 40 0 0 0 40 0 0 5 1
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,32,0,0,0,32],[1,0,0,0,40,5,0,1,35],[40,0,0,0,40,5,0,0,1] >;

C2×C4×D5 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_5
% in TeX

G:=Group("C2xC4xD5");
// GroupNames label

G:=SmallGroup(80,36);
// by ID

G=gap.SmallGroup(80,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,42,1604]);
// Polycyclic

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

׿
×
𝔽