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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 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×6], Q8 [×7], C23, C23, C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C2×C8.C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], 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 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 31)(18 28)(19 25)(20 30)(21 27)(22 32)(23 29)(24 26)

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

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

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

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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