Copied to
clipboard

G = C42.7C22order 64 = 26

7th non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.7C22, (C4×C8)⋊3C2, C4⋊C814C2, C4⋊C4.6C4, C8⋊C48C2, C2.6(C8○D4), C22⋊C8.8C2, C22⋊C4.3C4, C4.50(C4○D4), C23.10(C2×C4), (C2×C8).48C22, (C2×C4).152C23, C42⋊C2.8C2, (C22×C4).40C22, C22.46(C22×C4), C2.12(C42⋊C2), (C2×C4).28(C2×C4), SmallGroup(64,114)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.7C22
C1C2C4C2×C4C42C42⋊C2 — C42.7C22
C1C22 — C42.7C22
C1C2×C4 — C42.7C22
C1C2C2C2×C4 — C42.7C22

Generators and relations for C42.7C22
 G = < a,b,c,d | a4=b4=d2=1, c2=b, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, bd=db, dcd=a2b2c >

4C2
2C4
2C4
2C22
2C4
2C4
2C22
2C22
2C4
2C8
2C2×C4
2C8
2C8
2C8
2C2×C4

Character table of C42.7C22

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111411112222444222222224444
ρ11111111111111111111111111111    trivial
ρ21111-11111-1-1-1-1-111-11-1-111-111-11-1    linear of order 2
ρ31111-111111111-1-1-111111111-1-1-1-1    linear of order 2
ρ4111111111-1-1-1-11-1-1-11-1-111-11-11-11    linear of order 2
ρ51111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-11111-1-1-1-1-1111-111-1-11-1-11-11    linear of order 2
ρ71111-111111111-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ8111111111-1-1-1-11-1-11-111-1-11-11-11-1    linear of order 2
ρ91111-1-1-1-1-1-11-1111-1-iiii-i-i-ii-iii-i    linear of order 4
ρ101111-1-1-1-1-1-11-1111-1i-i-i-iiii-ii-i-ii    linear of order 4
ρ1111111-1-1-1-1-11-11-1-11-iiii-i-i-iii-i-ii    linear of order 4
ρ1211111-1-1-1-1-11-11-1-11i-i-i-iiii-i-iii-i    linear of order 4
ρ1311111-1-1-1-11-11-1-11-1-i-iiiii-i-iii-i-i    linear of order 4
ρ1411111-1-1-1-11-11-1-11-1ii-i-i-i-iii-i-iii    linear of order 4
ρ151111-1-1-1-1-11-11-11-11-i-iiiii-i-i-i-iii    linear of order 4
ρ161111-1-1-1-1-11-11-11-11ii-i-i-i-iiiii-i-i    linear of order 4
ρ172-2-220-2-222-2i-2i2i2i000000000000000    complex lifted from C4○D4
ρ182-2-22022-2-2-2i2i2i-2i000000000000000    complex lifted from C4○D4
ρ192-2-220-2-2222i2i-2i-2i000000000000000    complex lifted from C4○D4
ρ202-2-22022-2-22i-2i-2i2i000000000000000    complex lifted from C4○D4
ρ212-22-202i-2i-2i2i0000000080083870850000    complex lifted from C8○D4
ρ2222-2-202i-2i2i-2i0000000850878300800000    complex lifted from C8○D4
ρ2322-2-20-2i2i-2i2i0000000830885008700000    complex lifted from C8○D4
ρ242-22-20-2i2i2i-2i0000000083008850870000    complex lifted from C8○D4
ρ2522-2-202i-2i2i-2i0000000808387008500000    complex lifted from C8○D4
ρ262-22-20-2i2i2i-2i0000000087008580830000    complex lifted from C8○D4
ρ2722-2-20-2i2i-2i2i0000000870858008300000    complex lifted from C8○D4
ρ282-22-202i-2i-2i2i0000000085008783080000    complex lifted from C8○D4

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

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

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

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

C42.7C22 is a maximal subgroup of
C42.259C23  C42.261C23  C42.262C23  C42.292C23  C42.293C23  C42.297C23  C42.298C23  C42.299C23  C42.305C23  C42.307C23  C42.310C23  C42.352C23  C42.353C23  C42.354C23  C42.355C23  C42.356C23  C42.357C23  C42.358C23  C42.359C23  C42.360C23  C42.361C23  C42.406C23  C42.407C23  C42.408C23  C42.409C23  C42.410C23  C42.411C23  C42.423C23  C42.424C23  C42.425C23  C42.426C23
 C2p.(C8○D4): C42.260C23  C42.678C23  C42.291C23  C42.294C23  C42.696C23  C42.304C23  C42.308C23  C42.309C23 ...
C42.7C22 is a maximal quotient of
(C4×C8)⋊12C4  C24.53(C2×C4)  C424C4.C2  C23.(C2×F5)  C4⋊C4.7F5
 C42.D2p: C42.379D4  C42.45Q8  C42.95D4  C42.23Q8  C42.243D6  C42.185D6  C42.31D6  C42.187D6 ...
 (C2×C8).D2p: C4⋊C43C8  (C2×C8).Q8  C22⋊C44C8  C23.9M4(2)  C24⋊C4⋊C2  C408C4⋊C2  C56⋊C4⋊C2 ...

Matrix representation of C42.7C22 in GL4(𝔽17) generated by

01600
1000
00016
00160
,
4000
0400
0040
0004
,
15000
01500
0008
0090
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,0,16,0,0,16,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[15,0,0,0,0,15,0,0,0,0,0,9,0,0,8,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C42.7C22 in GAP, Magma, Sage, TeX

C_4^2._7C_2^2
% in TeX

G:=Group("C4^2.7C2^2");
// GroupNames label

G:=SmallGroup(64,114);
// by ID

G=gap.SmallGroup(64,114);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,50,88]);
// Polycyclic

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

Export

Subgroup lattice of C42.7C22 in TeX
Character table of C42.7C22 in TeX

׿
×
𝔽