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## G = C2×C8.C22order 64 = 26

### Direct product of C2 and C8.C22

direct product, p-group, metabelian, nilpotent (class 3), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C8.C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C8.C22
 Lower central C1 — C2 — C4 — C2×C8.C22
 Upper central C1 — C22 — C22×C4 — C2×C8.C22
 Jennings C1 — C2 — C2 — C4 — C2×C8.C22

Generators and relations for C2×C8.C22
G = < a,b,c,d | a2=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 185 in 129 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C2×C8.C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, C2×C8.C22

Character table of C2×C8.C22

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 1 1 -1 linear of order 2 ρ3 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ6 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ7 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ9 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 linear of order 2 ρ10 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ11 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 linear of order 2 ρ12 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ13 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 1 -1 1 -1 linear of order 2 ρ14 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 linear of order 2 ρ15 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 -1 linear of order 2 ρ16 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ17 2 2 2 2 -2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C2×C8.C22
On 32 points
Generators in S32
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)(25 29)(26 32)(28 30)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)

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

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

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

Matrix representation of C2×C8.C22 in GL6(𝔽17)

 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
,
 10 2 0 0 0 0 9 7 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 0 0 5 12 12 5 0 0 5 5 12 12
,
 1 0 0 0 0 0 7 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 1 0 16 0 0 0 0 16 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 2 0 0 0 0 16 0 2 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(17))| [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],[10,9,0,0,0,0,2,7,0,0,0,0,0,0,5,5,5,5,0,0,12,5,12,5,0,0,0,0,12,12,0,0,0,0,5,12],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,16,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,2,0,1,0,0,0,0,2,0,1] >;

C2×C8.C22 in GAP, Magma, Sage, TeX

C_2\times C_8.C_2^2
% in TeX

G:=Group("C2xC8.C2^2");
// GroupNames label

G:=SmallGroup(64,255);
// by ID

G=gap.SmallGroup(64,255);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,217,199,650,1444,730,88]);
// Polycyclic

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

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