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G = C2×C42.C4order 128 = 27

Direct product of C2 and C42.C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C42.C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C22×Q8 — C2×C4.4D4 — C2×C42.C4
 Lower central C1 — C2 — C22 — C2×C4 — C2×C42.C4
 Upper central C1 — C22 — C23 — C22×Q8 — C2×C42.C4
 Jennings C1 — C2 — C22 — C2×Q8 — C2×C42.C4

Generators and relations for C2×C42.C4
G = < a,b,c,d | a2=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >

Subgroups: 324 in 130 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C24, C4.10D4, C4.10D4, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C2×M4(2), C22×D4, C22×Q8, C42.C4, C2×C4.10D4, C2×C4.4D4, C2×C42.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C42.C4, C2×C23⋊C4, C2×C42.C4

Character table of C2×C42.C4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 8 8 4 4 4 4 4 4 4 4 4 4 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 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 -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 -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 -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 -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 -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 -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 -1 -1 -1 1 i i -i -i -i -i i i linear of order 4 ρ17 2 2 2 2 2 2 0 0 0 2 0 0 -2 -2 2 -2 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 2 -2 0 0 0 -2 0 0 -2 -2 2 2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 0 0 0 2 0 0 2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 0 0 0 -2 0 0 2 -2 -2 2 0 -2 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 2i 0 2i -2i 0 0 0 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ24 4 4 -4 -4 0 0 0 0 -2i 0 -2i 2i 0 0 0 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ25 4 -4 -4 4 0 0 0 0 2i 0 -2i -2i 0 0 0 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ26 4 -4 -4 4 0 0 0 0 -2i 0 2i 2i 0 0 0 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4

Smallest permutation representation of C2×C42.C4
On 32 points
Generators in S32
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 11 9 3)(2 18 14 26)(4 32 16 24)(5 15 13 7)(6 22 10 30)(8 28 12 20)(17 19 29 31)(21 23 25 27)
(1 23 5 19)(2 32 6 28)(3 21 7 17)(4 30 8 26)(9 27 13 31)(10 20 14 24)(11 25 15 29)(12 18 16 22)
(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)

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

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

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

Matrix representation of C2×C42.C4 in GL6(𝔽17)

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

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C42.C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2.C_4
% in TeX

G:=Group("C2xC4^2.C4");
// GroupNames label

G:=SmallGroup(128,862);
// by ID

G=gap.SmallGroup(128,862);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1018,248,1971,375,172,4037]);
// Polycyclic

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

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