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## G = C23.(C2×D4)  order 128 = 27

### 6th non-split extension by C23 of C2×D4 acting via C2×D4/C2=D4

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.(C2×D4)
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C2×C4○D4 — C23.38C23 — C23.(C2×D4)
 Lower central C1 — C2 — C22 — C23 — C23.(C2×D4)
 Upper central C1 — C2 — C2×C4 — C2×C4○D4 — C23.(C2×D4)
 Jennings C1 — C2 — C22 — C2×D4 — C23.(C2×D4)

Generators and relations for C23.(C2×D4)
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e4=c, f2=ca=dad-1=ac, ab=ba, eae-1=abc, af=fa, ebe-1=fbf-1=bc=cb, ede-1=bd=db, cd=dc, ce=ec, cf=fc, fdf-1=bcd, fef-1=ace3 >

Subgroups: 268 in 117 conjugacy classes, 42 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C23⋊C4, C23⋊C4, C4.D4, C4.10D4, C42⋊C2, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C23.D4, C23.C23, M4(2).8C22, C23.38C23, C23.(C2×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C23.(C2×D4)

Character table of C23.(C2×D4)

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 8A 8B 8C 8D size 1 1 2 4 4 4 2 2 4 4 4 8 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 1 1 1 -1 1 1 -1 -1 -1 -1 1 -i -1 1 -1 -i i 1 i -i i i -i linear of order 4 ρ10 1 1 1 -1 1 -1 1 1 1 -1 -1 i 1 -1 -1 -i -i 1 i -i i -i i linear of order 4 ρ11 1 1 1 -1 1 1 -1 -1 -1 -1 1 i 1 -1 1 i -i -1 -i -i i i -i linear of order 4 ρ12 1 1 1 -1 1 -1 1 1 1 -1 -1 -i -1 1 1 i i -1 -i -i i -i i linear of order 4 ρ13 1 1 1 -1 1 -1 1 1 1 -1 -1 -i 1 -1 -1 i i 1 -i i -i i -i linear of order 4 ρ14 1 1 1 -1 1 1 -1 -1 -1 -1 1 i -1 1 -1 i -i 1 -i i -i -i i linear of order 4 ρ15 1 1 1 -1 1 -1 1 1 1 -1 -1 i -1 1 1 -i -i -1 i i -i i -i linear of order 4 ρ16 1 1 1 -1 1 1 -1 -1 -1 -1 1 -i 1 -1 1 -i i -1 i i -i -i i linear of order 4 ρ17 2 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 -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 -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 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C23.(C2×D4)
On 32 points
Generators in S32
(2 32)(3 7)(4 30)(6 28)(8 26)(9 13)(10 22)(12 20)(14 18)(16 24)(17 21)(25 29)
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 21 5 17)(2 10 6 14)(3 19 7 23)(4 16 8 12)(9 31 13 27)(11 29 15 25)(18 28 22 32)(20 26 24 30)
(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 23 5 19)(2 10 28 18)(3 13)(4 20 26 16)(6 14 32 22)(7 9)(8 24 30 12)(11 27 15 31)(17 25)(21 29)

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

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

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

Matrix representation of C23.(C2×D4) in GL8(𝔽17)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 16 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 16 0 0 1 0
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 0 0 16 0 0 0 15 0 0 0 0 0 0 0 1 1 16 0 0 0 15 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 16 16 0 0 16 0 1 0 0 0 1 0 0 0 16 16 0 0 16 0 0 0
,
 13 0 0 0 13 4 4 4 4 0 0 0 4 13 0 0 0 0 13 0 13 13 13 4 0 0 4 0 0 0 4 13 0 15 0 15 2 15 15 15 13 15 0 2 15 2 2 2 0 2 0 15 2 2 2 15 0 15 13 15 15 15 15 2
,
 16 0 0 0 15 0 0 0 1 0 0 0 1 16 0 0 0 0 16 0 0 0 0 2 0 0 1 0 0 0 1 16 1 0 0 0 1 0 0 0 0 1 0 0 16 0 0 0 0 0 1 1 0 0 0 16 0 0 0 0 0 0 0 1

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,15,1,0,0,1,16,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,1,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,16,0,0,0,1,16,0,0,0,0,0,0,0,16,16,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,15,1,0,0,1,16,0,0,0,1,0,0,0,0,15,1,0,0,1,16,0,0,0,1,0,0,0,0,0,0],[13,4,0,0,0,13,0,0,0,0,0,0,15,15,2,15,0,0,13,4,0,0,0,13,0,0,0,0,15,2,15,15,13,4,13,0,2,15,2,15,4,13,13,0,15,2,2,15,4,0,13,4,15,2,2,15,4,0,4,13,15,2,15,2],[16,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,1,0,0,1,0,0,0,0,0,0,0,1,0,15,1,0,0,1,16,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,16,0,0,16,1] >;

C23.(C2×D4) in GAP, Magma, Sage, TeX

C_2^3.(C_2\times D_4)
% in TeX

G:=Group("C2^3.(C2xD4)");
// GroupNames label

G:=SmallGroup(128,855);
// by ID

G=gap.SmallGroup(128,855);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,851,375,4037]);
// Polycyclic

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

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