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

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

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

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

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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