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G = (C22×C8)⋊C4order 128 = 27

4th semidirect product of C22×C8 and C4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C22×C8)⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — (C22×C8)⋊C2 — (C22×C8)⋊C4
 Lower central C1 — C2 — C22 — C2×C4 — (C22×C8)⋊C4
 Upper central C1 — C4 — C2×C4 — C2×C4○D4 — (C22×C8)⋊C4
 Jennings C1 — C2 — C22 — C2×C4○D4 — (C22×C8)⋊C4

Generators and relations for (C22×C8)⋊C4
G = < a,b,c,d | a2=b2=c8=d4=1, dad-1=ab=ba, ac=ca, bc=cb, dbd-1=bc4, dcd-1=abc3 >

Subgroups: 192 in 81 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C23⋊C4, C23⋊C4, C4.D4, C4.10D4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×C4○D4, (C22×C8)⋊C2, C23.C23, M4(2).8C22, (C22×C8)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C23⋊C4, C23.9D4, (C22×C8)⋊C4

Character table of (C22×C8)⋊C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 2 4 4 4 1 1 2 4 4 4 8 8 8 8 4 4 4 4 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 -i i i -i -i -i i i -i i linear of order 4 ρ6 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 i -i -i i i -i i i -i -i linear of order 4 ρ7 1 1 1 -1 1 -1 1 1 1 -1 1 -1 i i -i -i -1 -1 -1 -1 1 -i -i i i 1 linear of order 4 ρ8 1 1 1 -1 1 -1 1 1 1 -1 1 -1 i i -i -i 1 1 1 1 -1 i i -i -i -1 linear of order 4 ρ9 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 i -i -i i i -i -i i -i -1 1 -1 1 i linear of order 4 ρ10 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -i i i -i -i i i -i i -1 1 -1 1 -i linear of order 4 ρ11 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -i i i -i i -i -i i -i 1 -1 1 -1 i linear of order 4 ρ12 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 i -i -i i -i i i -i i 1 -1 1 -1 -i linear of order 4 ρ13 1 1 1 -1 1 -1 1 1 1 -1 1 -1 -i -i i i 1 1 1 1 -1 -i -i i i -1 linear of order 4 ρ14 1 1 1 -1 1 -1 1 1 1 -1 1 -1 -i -i i i -1 -1 -1 -1 1 i i -i -i 1 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 -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 i -i -i i i i -i -i i -i linear of order 4 ρ17 2 2 2 -2 2 2 -2 -2 -2 2 -2 -2 0 0 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 -2 0 0 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 2 0 0 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 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 4 -4 0 0 0 4 4 -4 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 0 0 0 -4 -4 4 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 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 2ζ8 2ζ83 2ζ87 2ζ85 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 2ζ83 2ζ8 2ζ85 2ζ87 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 2ζ85 2ζ87 2ζ83 2ζ8 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 2ζ87 2ζ85 2ζ8 2ζ83 0 0 0 0 0 0 complex faithful

Smallest permutation representation of (C22×C8)⋊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 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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)
(2 16 28 18)(3 7)(4 14 30 24)(6 12 32 22)(8 10 26 20)(9 23 13 19)(11 21 15 17)(27 31)

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

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

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

Matrix representation of (C22×C8)⋊C4 in GL4(𝔽17) generated by

 0 0 13 8 0 0 0 4 4 9 0 0 0 13 0 0
,
 1 15 0 0 0 16 0 0 0 0 1 15 0 0 0 16
,
 0 8 9 8 0 8 0 0 8 9 0 8 0 0 0 8
,
 1 0 0 0 1 16 0 0 0 0 16 2 0 0 16 1
G:=sub<GL(4,GF(17))| [0,0,4,0,0,0,9,13,13,0,0,0,8,4,0,0],[1,0,0,0,15,16,0,0,0,0,1,0,0,0,15,16],[0,0,8,0,8,8,9,0,9,0,0,0,8,0,8,8],[1,1,0,0,0,16,0,0,0,0,16,16,0,0,2,1] >;

(C22×C8)⋊C4 in GAP, Magma, Sage, TeX

(C_2^2\times C_8)\rtimes C_4
% in TeX

G:=Group("(C2^2xC8):C4");
// GroupNames label

G:=SmallGroup(128,127);
// by ID

G=gap.SmallGroup(128,127);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,2804,172,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=d^4=1,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d^-1=b*c^4,d*c*d^-1=a*b*c^3>;
// generators/relations

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