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## G = C5×C8⋊C22order 160 = 25·5

### Direct product of C5 and C8⋊C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C8⋊C22
 Chief series C1 — C2 — C4 — C20 — C5×D4 — C5×D8 — C5×C8⋊C22
 Lower central C1 — C2 — C4 — C5×C8⋊C22
 Upper central C1 — C10 — C2×C20 — C5×C8⋊C22

Generators and relations for C5×C8⋊C22
G = < a,b,c,d | a5=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, D4, Q8, C23, C10, C10, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C8⋊C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×C10, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, C5×C8⋊C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C8⋊C22, C5×D4, C22×C10, D4×C10, C5×C8⋊C22

Smallest permutation representation of C5×C8⋊C22
On 40 points
Generators in S40
(1 18 31 35 14)(2 19 32 36 15)(3 20 25 37 16)(4 21 26 38 9)(5 22 27 39 10)(6 23 28 40 11)(7 24 29 33 12)(8 17 30 34 13)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)

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

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

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

C5×C8⋊C22 is a maximal subgroup of   D2018D4  M4(2).D10  M4(2).13D10  D20.38D4  SD16⋊D10  D85D10  D86D10

55 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10T 20A ··· 20H 20I 20J 20K 20L 40A ··· 40H order 1 2 2 2 2 2 4 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 4 2 2 4 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 C5×D4 C5×D4 C8⋊C22 C5×C8⋊C22 kernel C5×C8⋊C22 C5×M4(2) C5×D8 C5×SD16 D4×C10 C5×C4○D4 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 C20 C2×C10 C4 C22 C5 C1 # reps 1 1 2 2 1 1 4 4 8 8 4 4 1 1 4 4 1 4

Matrix representation of C5×C8⋊C22 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 0 1 0 0 0 0 40 0 1 0 0 1 0 0 0
,
 1 0 0 0 0 40 0 0 0 0 0 40 0 0 40 0
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,0,1,0,0,1,0,1,0,0,0,0,40,0,0],[1,0,0,0,0,40,0,0,0,0,0,40,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40] >;

C5×C8⋊C22 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes C_2^2
% in TeX

G:=Group("C5xC8:C2^2");
// GroupNames label

G:=SmallGroup(160,197);
// by ID

G=gap.SmallGroup(160,197);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,1514,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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