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## G = C2×C23.D4order 128 = 27

### Direct product of C2 and C23.D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C23.D4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C22×D4 — C2×C22.D4 — C2×C23.D4
 Lower central C1 — C2 — C22 — C23 — C2×C23.D4
 Upper central C1 — C22 — C23 — C22×D4 — C2×C23.D4
 Jennings C1 — C2 — C22 — C2×D4 — C2×C23.D4

Generators and relations for C2×C23.D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be3 >

Subgroups: 356 in 139 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C23⋊C4, C23⋊C4, C4.D4, C4.D4, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C22.D4, C2×M4(2), C23×C4, C22×D4, C23.D4, C2×C23⋊C4, C2×C4.D4, C2×C22.D4, C2×C23.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C23.D4, C2×C23⋊C4, C2×C23.D4

Character table of C2×C23.D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 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 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 -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 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 -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 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 -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 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 -1 1 -1 linear of order 2 ρ9 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 i -i -i i -1 1 i -i -i i linear of order 4 ρ10 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 i -i -i i 1 -1 -i i i -i linear of order 4 ρ11 1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 i i -i -i 1 1 -i -i i i linear of order 4 ρ12 1 1 1 1 1 1 -1 1 -1 1 1 1 1 -1 1 -1 i i -i -i -1 -1 i i -i -i linear of order 4 ρ13 1 1 1 1 1 1 -1 1 -1 1 1 1 1 -1 1 -1 -i -i i i -1 -1 -i -i i i linear of order 4 ρ14 1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 -i -i i i 1 1 i i -i -i linear of order 4 ρ15 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 -i i i -i 1 -1 i -i -i i linear of order 4 ρ16 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -i i i -i -1 1 -i i i -i linear of order 4 ρ17 2 2 2 2 2 2 2 -2 2 -2 0 0 0 -2 0 -2 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 2 0 2 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 2 0 -2 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 -2 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 4 -4 -4 0 0 0 0 0 0 -2i -2i 2i 0 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4 ρ24 4 -4 -4 4 0 0 0 0 0 0 -2i 2i 2i 0 -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4 ρ25 4 4 -4 -4 0 0 0 0 0 0 2i 2i -2i 0 -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4 ρ26 4 -4 -4 4 0 0 0 0 0 0 2i -2i -2i 0 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C23.D4

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

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

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

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

Matrix representation of C2×C23.D4 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
,
 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 1 0 0 0 0 0 0 16
,
 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 0 0 1 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 0 0 16 0 0 0 0 0 0 16
,
 0 1 0 0 0 0 16 0 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 4 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 0 0 0 0 0 1 0 0

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,13,0,0,0,0,13,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0] >;

C2×C23.D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3.D_4
% in TeX

G:=Group("C2xC2^3.D4");
// GroupNames label

G:=SmallGroup(128,851);
// by ID

G=gap.SmallGroup(128,851);
# by ID

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

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

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