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G = C4⋊Q8⋊29C4  order 128 = 27

24th semidirect product of C4⋊Q8 and C4 acting via C4/C2=C2

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

Aliases: C4⋊Q8⋊29C4, (C2×C42)⋊11C4, C23.9(C2×D4), C4○(C42⋊3C4), C4○(C42⋊C4), C42⋊C4⋊8C2, C42⋊3C4⋊8C2, (C2×D4).131D4, C4.4D4⋊21C4, C42.24(C2×C4), (C22×C4).94D4, C4.30(C23⋊C4), (C2×D4).20C23, C23⋊C4.12C22, C4⋊1D4.134C22, C23.36(C22⋊C4), C23.C23⋊14C2, C4.4D4.121C22, C22.26C24.23C2, (C2×C4○D4)⋊7C4, (C2×D4).37(C2×C4), C2.38(C2×C23⋊C4), (C22×C4).81(C2×C4), (C2×C4).95(C22×C4), (C2×Q8).106(C2×C4), (C2×C4).26(C22⋊C4), (C2×C4○D4).74C22, C22.62(C2×C22⋊C4), SmallGroup(128,858)

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

Derived series
C1 — C2×C4 — C4⋊Q8⋊29C4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C2×C4○D4 — C22.26C24 — C4⋊Q8⋊29C4
Lower central
C1 — C2 — C22 — C2×C4 — C4⋊Q8⋊29C4
Upper central
C1 — C4 — C2×C4 — C2×C4○D4 — C4⋊Q8⋊29C4
Jennings
C1 — C2 — C22 — C2×D4 — C4⋊Q8⋊29C4

Generators and relations for C4⋊Q8⋊29C4
 G = < a,b,c,d | a4=b4=d4=1, c2=b2, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a2b-1, cd=dc >

Subgroups: 324 in 129 conjugacy classes, 42 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C23⋊C4, C23⋊C4, C2×C42, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C42⋊C4, C42⋊3C4, C23.C23, C22.26C24, C4⋊Q8⋊29C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C4⋊Q8⋊29C4

Character table of C4⋊Q8⋊29C4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S
 size 11244481124444444888888888
ρ111111111111111111111111111    trivial
ρ2111-111-1-1-1-11-11-1-1-1111-11-11-11-1    linear of order 2
ρ3111111-1111-1111-1-1-1-11111-1-1-1-1    linear of order 2
ρ4111-1111-1-1-1-1-11-111-1-11-11-1-11-11    linear of order 2
ρ5111-1111-1-1-1-1-11-111-11-11-11-1-11-1    linear of order 2
ρ6111111-1111-1111-1-1-11-1-1-1-1-1111    linear of order 2
ρ7111-111-1-1-1-11-11-1-1-11-1-11-1111-11    linear of order 2
ρ811111111111111111-1-1-1-1-11-1-1-1    linear of order 2
ρ91111-1-1-11111-11-1111i-i-iii-1-i-ii    linear of order 4
ρ10111-1-1-11-1-1-11111-1-11i-iii-i-1i-i-i    linear of order 4
ρ11111-1-1-11-1-1-11111-1-11-ii-i-ii-1-iii    linear of order 4
ρ121111-1-1-11111-11-1111-iii-i-i-1ii-i    linear of order 4
ρ13111-1-1-1-1-1-1-1-111111-1ii-i-ii1i-i-i    linear of order 4
ρ141111-1-11111-1-11-1-1-1-1iii-i-i1-i-ii    linear of order 4
ρ151111-1-11111-1-11-1-1-1-1-i-i-iii1ii-i    linear of order 4
ρ16111-1-1-1-1-1-1-1-111111-1-i-iii-i1-iii    linear of order 4
ρ17222-22-202220-2-22000000000000    orthogonal lifted from D4
ρ18222-2-22022202-2-2000000000000    orthogonal lifted from D4
ρ1922222-20-2-2-202-2-2000000000000    orthogonal lifted from D4
ρ202222-220-2-2-20-2-22000000000000    orthogonal lifted from D4
ρ2144-4000044-40000000000000000    orthogonal lifted from C23⋊C4
ρ2244-40000-4-440000000000000000    orthogonal lifted from C23⋊C4
ρ234-400000-4i4i02i0002-2-2i000000000    complex faithful
ρ244-400000-4i4i0-2i000-222i000000000    complex faithful
ρ254-4000004i-4i0-2i0002-22i000000000    complex faithful
ρ264-4000004i-4i02i000-22-2i000000000    complex faithful

Permutation representations of C4⋊Q8⋊29C4
►On 16 points - transitive group 16T293
Generators in S16
(9 10 11 12)(13 14 15 16)

(1 2 3 5)(4 7 8 6)(9 10 11 12)(13 14 15 16)

(1 7 3 6)(2 4 5 8)(9 15 11 13)(10 14 12 16)

(1 11)(2 10 5 12)(3 9)(4 14 8 16)(6 15)(7 13)
 
G:=sub<Sym(16)| (9,10,11,12)(13,14,15,16), (1,2,3,5)(4,7,8,6)(9,10,11,12)(13,14,15,16), (1,7,3,6)(2,4,5,8)(9,15,11,13)(10,14,12,16), (1,11)(2,10,5,12)(3,9)(4,14,8,16)(6,15)(7,13)>;
 
G:=Group( (9,10,11,12)(13,14,15,16), (1,2,3,5)(4,7,8,6)(9,10,11,12)(13,14,15,16), (1,7,3,6)(2,4,5,8)(9,15,11,13)(10,14,12,16), (1,11)(2,10,5,12)(3,9)(4,14,8,16)(6,15)(7,13) );
 
G=PermutationGroup([[(9,10,11,12),(13,14,15,16)], [(1,2,3,5),(4,7,8,6),(9,10,11,12),(13,14,15,16)], [(1,7,3,6),(2,4,5,8),(9,15,11,13),(10,14,12,16)], [(1,11),(2,10,5,12),(3,9),(4,14,8,16),(6,15),(7,13)]])
 
G:=TransitiveGroup(16,293);
 

►On 16 points - transitive group 16T322
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

(1 3 2 4)(5 8 6 7)(9 12 11 10)(13 16 15 14)

(1 8 2 7)(3 5 4 6)(9 15 11 13)(10 14 12 16)

(1 15 6 12)(2 13 5 10)(3 16 8 11)(4 14 7 9)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,2,4)(5,8,6,7)(9,12,11,10)(13,16,15,14), (1,8,2,7)(3,5,4,6)(9,15,11,13)(10,14,12,16), (1,15,6,12)(2,13,5,10)(3,16,8,11)(4,14,7,9)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,2,4)(5,8,6,7)(9,12,11,10)(13,16,15,14), (1,8,2,7)(3,5,4,6)(9,15,11,13)(10,14,12,16), (1,15,6,12)(2,13,5,10)(3,16,8,11)(4,14,7,9) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,2,4),(5,8,6,7),(9,12,11,10),(13,16,15,14)], [(1,8,2,7),(3,5,4,6),(9,15,11,13),(10,14,12,16)], [(1,15,6,12),(2,13,5,10),(3,16,8,11),(4,14,7,9)]])
 
G:=TransitiveGroup(16,322);
 

Matrix representation of C4⋊Q8⋊29C4 ►in GL4(𝔽5) generated by

2000
0300
0010
0001
,
2000
0300
0030
0002
,
0300
3000
0003
0030
,
0010
0001
0100
1000
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[0,3,0,0,3,0,0,0,0,0,0,3,0,0,3,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
 

C4⋊Q8⋊29C4 in GAP, Magma, Sage, TeX 

C_4\rtimes Q_8\rtimes_{29}C_4
 
% in TeX
 
G:=Group("C4:Q8:29C4");
 
// GroupNames label
 
G:=SmallGroup(128,858);
 
// by ID
 
G=gap.SmallGroup(128,858);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,1018,248,1971,375,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,c*d=d*c>;
 
// generators/relations
 

Export

Character table of C4⋊Q8⋊29C4 in TeX

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