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## G = C2×C4.10D4order 64 = 26

### Direct product of C2 and C4.10D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4.10D4
 Lower central C1 — C2 — C22 — C2×C4.10D4
 Upper central C1 — C22 — C22×C4 — C2×C4.10D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4.10D4

Generators and relations for C2×C4.10D4
G = < a,b,c,d | a2=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 105 in 73 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4.10D4, C2×M4(2), C22×Q8, C2×C4.10D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C2×C4.10D4

Character table of C2×C4.10D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 2 2 2 4 4 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 i -i i -i i -i i -i linear of order 4 ρ10 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 -i -i -i i i -i i i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -i -i i i i i -i -i linear of order 4 ρ12 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 i -i -i -i i i -i i linear of order 4 ρ13 1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 -i i -i i -i i -i i linear of order 4 ρ14 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 i i i -i -i i -i -i linear of order 4 ρ15 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 i i -i -i -i -i i i linear of order 4 ρ16 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -i i i i -i -i i -i linear of order 4 ρ17 2 -2 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 2 2 -2 -2 0 0 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 C4.10D4, 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 C4.10D4, Schur index 2

Smallest permutation representation of C2×C4.10D4
On 32 points
Generators in S32
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 20 13 24)(10 17 14 21)(11 22 15 18)(12 19 16 23)
(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)
(1 20 31 9 5 24 27 13)(2 16 28 19 6 12 32 23)(3 18 25 15 7 22 29 11)(4 14 30 17 8 10 26 21)

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

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

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

Matrix representation of C2×C4.10D4 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
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 2 0 0 0 0 16 1 0 0 0 0 16 1 0 1 0 0 0 1 16 0
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 2 0 0 0 0 0 1 16 0 0 0 16 1 0 0 0 16 0 1 0
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 8 0 3 15 0 0 7 0 11 9 0 0 7 10 10 16 0 0 8 16 10 16

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,16,0,0,0,2,1,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,16,0,0,0,2,1,1,1,0,0,0,16,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,8,7,7,8,0,0,0,0,10,16,0,0,3,11,10,10,0,0,15,9,16,16] >;

C2×C4.10D4 in GAP, Magma, Sage, TeX

C_2\times C_4._{10}D_4
% in TeX

G:=Group("C2xC4.10D4");
// GroupNames label

G:=SmallGroup(64,93);
// by ID

G=gap.SmallGroup(64,93);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,963,730,88]);
// Polycyclic

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

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