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G = C42.6C22  order 64 = 26

6th non-split extension by C42 of C22 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.6C22, C4⋊C8⋊12C2, C4⋊C4.5C4, C4.73(C2×D4), C4.12(C4⋊C4), (C2×C4).17Q8, C4.22(C2×Q8), C2.5(C8○D4), (C2×C4).127D4, C22⋊C4.2C4, (C22×C8).7C2, C22.8(C4⋊C4), (C2×C8).46C22, C23.17(C2×C4), (C2×C4).149C23, C42⋊C2.5C2, (C2×M4(2)).14C2, C22.44(C22×C4), (C22×C4).112C22, C2.10(C2×C4⋊C4), (C2×C4).25(C2×C4), SmallGroup(64,105)

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

Derived series
C1 — C22 — C42.6C22
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C42.6C22
Lower central
C1 — C22 — C42.6C22
Upper central
C1 — C2×C4 — C42.6C22
Jennings
C1 — C2 — C2 — C2×C4 — C42.6C22

Generators and relations for C42.6C22
 G = < a,b,c,d | a4=b4=1, c2=b, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b2c >

Subgroups: 73 in 57 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C42.6C22
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C8○D4, C42.6C22

Character table of C42.6C22

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111221111224444222222224444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ31111-1-11111-1-111-1-1-11-1-111-11-11-11    linear of order 2
ρ41111-1-11111-1-1-1-111-11-1-111-111-11-1    linear of order 2
ρ5111111111111-1-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ61111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-11111-1-111-1-11-111-1-11-11-11-1    linear of order 2
ρ81111-1-11111-1-1-1-1111-111-1-11-1-11-11    linear of order 2
ρ9111111-1-1-1-1-1-1-11-11ii-i-i-i-iii-i-iii    linear of order 4
ρ10111111-1-1-1-1-1-1-11-11-i-iiiii-i-iii-i-i    linear of order 4
ρ11111111-1-1-1-1-1-11-11-1-i-iiiii-i-i-i-iii    linear of order 4
ρ12111111-1-1-1-1-1-11-11-1ii-i-i-i-iiiii-i-i    linear of order 4
ρ131111-1-1-1-1-1-1111-1-11i-i-i-iiii-ii-i-ii    linear of order 4
ρ141111-1-1-1-1-1-1111-1-11-iiii-i-i-ii-iii-i    linear of order 4
ρ151111-1-1-1-1-1-111-111-1-iiii-i-i-iii-i-ii    linear of order 4
ρ161111-1-1-1-1-1-111-111-1i-i-i-iiii-i-iii-i    linear of order 4
ρ172-2-22-22-22-22-220000000000000000    orthogonal lifted from D4
ρ182-2-222-2-22-222-20000000000000000    orthogonal lifted from D4
ρ192-2-22-222-22-22-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ202-2-222-22-22-2-220000000000000000    symplectic lifted from Q8, Schur index 2
ρ2122-2-2002i-2i-2i2i0000002ζ8502ζ872ζ83002ζ800000    complex lifted from C8○D4
ρ2222-2-2002i-2i-2i2i0000002ζ802ζ832ζ87002ζ8500000    complex lifted from C8○D4
ρ232-22-200-2i-2i2i2i00000002ζ85002ζ832ζ8702ζ80000    complex lifted from C8○D4
ρ2422-2-200-2i2i2i-2i0000002ζ8302ζ82ζ85002ζ8700000    complex lifted from C8○D4
ρ2522-2-200-2i2i2i-2i0000002ζ8702ζ852ζ8002ζ8300000    complex lifted from C8○D4
ρ262-22-200-2i-2i2i2i00000002ζ8002ζ872ζ8302ζ850000    complex lifted from C8○D4
ρ272-22-2002i2i-2i-2i00000002ζ83002ζ852ζ802ζ870000    complex lifted from C8○D4
ρ282-22-2002i2i-2i-2i00000002ζ87002ζ82ζ8502ζ830000    complex lifted from C8○D4

Smallest permutation representation of C42.6C22
►On 32 points
Generators in S32
(1 23 27 16)(2 13 28 20)(3 17 29 10)(4 15 30 22)(5 19 31 12)(6 9 32 24)(7 21 25 14)(8 11 26 18)

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

C42.6C22 is a maximal subgroup of
 C23.7M4(2)  C22⋊C4.C8  M4(2).42D4  (C2×D4).Q8  M4(2).3Q8  C4⋊C4.96D4  C4⋊C4.97D4  C4⋊C4.98D4  C22⋊C4.7D4  M4(2).12D4  M4(2).13D4  M4(2).Q8  M4(2).2Q8  C22⋊C4.Q8  C42.257C23  C42.674C23  C42.260C23  C42.261C23  C42.262C23  C42.264C23  C42.265C23  M4(2)⋊22D4  C42.286C23  C42.287C23  M4(2)⋊9Q8  C42.307C23  C42.308C23  C42.309C23  C42.310C23  C42.14C23  C42.15C23  C42.16C23  C42.17C23  C42.18C23  C42.19C23  C42.20C23  C42.21C23  C42.22C23  C42.23C23  C4.2- 1+4  C42.25C23  C42.26C23  C42.27C23  C42.28C23  C42.29C23  C42.30C23  C20⋊C8⋊C2
 C42.D2p: C42.428D4  C42.107D4  C42.9D4  C42.10D4  C42.30D6  C42.43D6  C42.30D10  C42.43D10 ...
 C4p.(C4⋊C4): C42.62Q8  C42.28Q8  Dic3⋊C8⋊C2  C12.88(C2×Q8)  C20.65(C4⋊C4)  C20.51(C4⋊C4)  C4⋊C4.9F5  Dic7⋊C8⋊C2 ...
C42.6C22 is a maximal quotient of
 C23.29C42  C20⋊C8⋊C2  C4⋊C4.9F5
 C42.D2p: C42.90D4  C42.91D4  C42.Q8  C42.92D4  C42.21Q8  C42.45Q8  C42.95D4  C42.23Q8 ...
 (C2×C8).D2p: C23.21M4(2)  (C2×C8).195D4  Dic3⋊C8⋊C2  C12.88(C2×Q8)  C20.65(C4⋊C4)  C20.51(C4⋊C4)  Dic7⋊C8⋊C2  C28.439(C2×D4) ...

Matrix representation of C42.6C22 ►in GL4(𝔽17) generated by

0100
16000
0001
0010
,
4000
0400
0040
0004
,
2000
0200
00150
0002
,
0100
1000
0001
00160
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,15,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,16,0,0,1,0] >;
 

C42.6C22 in GAP, Magma, Sage, TeX 

C_4^2._6C_2^2
 
% in TeX
 
G:=Group("C4^2.6C2^2");
 
// GroupNames label
 
G:=SmallGroup(64,105);
 
// by ID
 
G=gap.SmallGroup(64,105);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,332,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^2=b,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c>;
 
// generators/relations
 

Export

Character table of C42.6C22 in TeX

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