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## G = (C2×C4).5D8order 128 = 27

### 5th non-split extension by C2×C4 of D8 acting via D8/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4).5D8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.31C24 — (C2×C4).5D8
 Lower central C1 — C22 — C22×C4 — (C2×C4).5D8
 Upper central C1 — C22 — C22×C4 — (C2×C4).5D8
 Jennings C1 — C2 — C22 — C22×C4 — (C2×C4).5D8

Generators and relations for (C2×C4).5D8
G = < a,b,c,d | a2=b4=c8=1, d2=b2, cbc-1=dbd-1=ab=ba, cac-1=ab2, ad=da, dcd-1=b2c-1 >

Subgroups: 324 in 127 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2.C42, C2.C42, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, C22.M4(2), C22.SD16, C23.81C23, C22.D8, C22.31C24, (C2×C4).5D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4.10D4, C23.7D4, (C2×C4).5D8

Character table of (C2×C4).5D8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D size 1 1 1 1 2 2 8 8 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 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 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 -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 -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 -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 -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 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 1 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 0 -2 0 0 2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 2 2 0 0 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 -2 2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 0 0 -2 2 2 -2 0 0 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 2 0 -2 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 -2 2 0 0 0 -2 2 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ16 2 2 -2 -2 -2 2 0 0 0 -2 2 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ17 2 2 -2 -2 -2 2 0 0 0 2 -2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ18 2 2 -2 -2 -2 2 0 0 0 2 -2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ19 4 4 -4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 -4 4 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ21 4 -4 -4 4 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ22 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 0 0 0 0 complex lifted from C23.7D4 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 0 0 0 0 complex lifted from C23.7D4

Smallest permutation representation of (C2×C4).5D8
On 32 points
Generators in S32
(2 29)(4 31)(6 25)(8 27)(9 24)(11 18)(13 20)(15 22)
(1 21 28 14)(2 22 29 15)(3 16 30 23)(4 9 31 24)(5 17 32 10)(6 18 25 11)(7 12 26 19)(8 13 27 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)
(1 14 28 21)(2 20 29 13)(3 12 30 19)(4 18 31 11)(5 10 32 17)(6 24 25 9)(7 16 26 23)(8 22 27 15)

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

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

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

Matrix representation of (C2×C4).5D8 in GL6(𝔽17)

 1 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 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13
,
 3 14 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 13 0 0 0 0 13 0 0 0 13 0 0 0 0 0 0 13 0 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 13 0 0 0 0 13 0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,13,0,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,13,0] >;

(C2×C4).5D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)._5D_8
% in TeX

G:=Group("(C2xC4).5D8");
// GroupNames label

G:=SmallGroup(128,342);
// by ID

G=gap.SmallGroup(128,342);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,352,1123,570,521,136,1411]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2,c*b*c^-1=d*b*d^-1=a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,d*c*d^-1=b^2*c^-1>;
// generators/relations

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