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G = C8⋊C4.C4  order 128 = 27

3rd non-split extension by C8⋊C4 of C4 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: (C2×Q8).7D4, C8⋊C4.3C4, C42.6(C2×C4), C4⋊Q8.4C22, C42.C2.4C4, C2.7(C42⋊3C4), C42.3C4.2C2, C22.23(C23⋊C4), C42.30C22.3C2, (C2×C4).39(C22⋊C4), SmallGroup(128,145)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — C8⋊C4.C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C4⋊Q8 — C42.30C22 — C8⋊C4.C4
Lower central
C1 — C2 — C22 — C2×C4 — C42 — C8⋊C4.C4
Upper central
C1 — C2 — C22 — C2×C4 — C4⋊Q8 — C8⋊C4.C4
Jennings
C1 — C2 — C2 — C22 — C2×C4 — C4⋊Q8 — C8⋊C4.C4

Generators and relations for C8⋊C4.C4
 G = < a,b,c | a8=b4=1, c4=a4, bab-1=a5, cac-1=ab, cbc-1=a2b >

C1
C2
2C2
C22
2C4
4C4
4C4
4C4
8C4
C2×C4
2C2×C4
2C2×C4
2C2×C4
4C2×C4
4Q8
4C8
4Q8
8C8
8C8
C2×Q8
C2×Q8
C42
2C2×C8
2C4⋊C4
4C4⋊C4
4M4(2)
4C4⋊C4
4M4(2)
C8⋊C4
C4⋊Q8
C42.C2
2C4.10D4
2C4.10D4
2Q8⋊C4
2Q8⋊C4
C42.30C22
C42.3C4
C42.3C4
C8⋊C4.C4

Character table of C8⋊C4.C4

 class 12A2B4A4B4C4D4E8A8B8C8D8E8F
 size 1124888168816161616
ρ111111111111111    trivial
ρ21111111-1-1-111-1-1    linear of order 2
ρ31111111-1-1-1-1-111    linear of order 2
ρ41111111111-1-1-1-1    linear of order 2
ρ51111-1-111-1-1i-ii-i    linear of order 4
ρ61111-1-11-111i-i-ii    linear of order 4
ρ71111-1-11-111-iii-i    linear of order 4
ρ81111-1-111-1-1-ii-ii    linear of order 4
ρ92222-22-20000000    orthogonal lifted from D4
ρ1022222-2-20000000    orthogonal lifted from D4
ρ11444-40000000000    orthogonal lifted from C23⋊C4
ρ1244-400000-2i2i0000    complex lifted from C42⋊3C4
ρ1344-4000002i-2i0000    complex lifted from C42⋊3C4
ρ148-8000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C8⋊C4.C4
►On 32 points
Generators in S32
(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 18 5 22)(2 23 6 19)(3 20 7 24)(4 17 8 21)(9 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)

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

Matrix representation of C8⋊C4.C4 ►in GL8(𝔽17)

1680150000
1381160000
131313140000
1124140000
0000160150
00001601616
000011610
00001010
,
80990000
1316090000
14540000
513450000
000016298
000012190
000003412
0000143513
,
00001000
00000100
00000010
00000001
1601500000
001610000
10100000
116100000

G:=sub<GL(8,GF(17))| [16,13,13,1,0,0,0,0,8,8,13,12,0,0,0,0,0,1,13,4,0,0,0,0,15,16,14,14,0,0,0,0,0,0,0,0,16,16,1,1,0,0,0,0,0,0,16,0,0,0,0,0,15,16,1,1,0,0,0,0,0,16,0,0],[8,13,1,5,0,0,0,0,0,16,4,13,0,0,0,0,9,0,5,4,0,0,0,0,9,9,4,5,0,0,0,0,0,0,0,0,16,12,0,14,0,0,0,0,2,1,3,3,0,0,0,0,9,9,4,5,0,0,0,0,8,0,12,13],[0,0,0,0,16,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

C8⋊C4.C4 in GAP, Magma, Sage, TeX 

C_8\rtimes C_4.C_4
 
% in TeX
 
G:=Group("C8:C4.C4");
 
// GroupNames label
 
G:=SmallGroup(128,145);
 
// by ID
 
G=gap.SmallGroup(128,145);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,232,422,387,520,794,745,248,1684,1411,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=1,c^4=a^4,b*a*b^-1=a^5,c*a*c^-1=a*b,c*b*c^-1=a^2*b>;
 
// generators/relations
 

Export

Subgroup lattice of C8⋊C4.C4 in TeX
Character table of C8⋊C4.C4 in TeX

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