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G = C22.C42order 64 = 26

2nd non-split extension by C22 of C42 acting via C42/C2×C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: M4(2)⋊3C4, C22.2C42, C4.4(C4⋊C4), (C2×C4).2Q8, (C2×C4).113D4, (C22×C4).2C4, C22.5(C4⋊C4), C23.22(C2×C4), C4.21(C22⋊C4), C2.2(C4.D4), (C2×M4(2)).8C2, C2.2(C4.10D4), (C22×C4).20C22, C22.26(C22⋊C4), C2.8(C2.C42), (C2×C4⋊C4).3C2, (C2×C4).15(C2×C4), SmallGroup(64,24)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.C42
C1C2C22C2×C4C22×C4C2×C4⋊C4 — C22.C42
C1C2C22 — C22.C42
C1C22C22×C4 — C22.C42
C1C2C2C22×C4 — C22.C42

Generators and relations for C22.C42
 G = < a,b,c,d | a2=b2=d4=1, c4=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=abc >

2C2
2C2
2C22
2C22
4C4
4C4
2C2×C4
2C8
2C8
2C2×C4
2C8
2C8
4C2×C4
4C2×C4
2C4⋊C4
2C2×C8
2C4⋊C4
2M4(2)
2C2×C8
2M4(2)

Character table of C22.C42

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1111-1-1-11-1    linear of order 2
ρ41111111111-1-1-1-1-1-1-1111-11    linear of order 2
ρ5111111-1-1-1-1-1-111-iiii-i-i-ii    linear of order 4
ρ6111111-1-1-1-1-1-111i-i-i-iiii-i    linear of order 4
ρ7111111-1-1-1-111-1-1-iii-iii-i-i    linear of order 4
ρ81-11-1-11-11-11i-ii-i-11-1-ii-i1i    linear of order 4
ρ91-11-1-11-11-11-ii-ii-11-1i-ii1-i    linear of order 4
ρ101-11-1-111-11-1i-i-ii-i-ii11-1i-1    linear of order 4
ρ111-11-1-111-11-1-iii-i-i-ii-1-11i1    linear of order 4
ρ12111111-1-1-1-111-1-1i-i-ii-i-iii    linear of order 4
ρ131-11-1-11-11-11-ii-ii1-11-ii-i-1i    linear of order 4
ρ141-11-1-11-11-11i-ii-i1-11i-ii-1-i    linear of order 4
ρ151-11-1-111-11-1-iii-iii-i11-1-i-1    linear of order 4
ρ161-11-1-111-11-1i-i-iiii-i-1-11-i1    linear of order 4
ρ172222-2-22-2-22000000000000    orthogonal lifted from D4
ρ182-22-22-222-2-2000000000000    orthogonal lifted from D4
ρ192222-2-2-222-2000000000000    orthogonal lifted from D4
ρ202-22-22-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-44000000000000000000    orthogonal lifted from C4.D4
ρ2244-4-4000000000000000000    symplectic lifted from C4.10D4, Schur index 2

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

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

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

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

C22.C42 is a maximal subgroup of
C2.C2≀C4  (C2×C4).D8  (C2×C4).Q16  C2.7C2≀C4  C23.15C42  C4×C4.D4  C4×C4.10D4  C42.96D4  C42.97D4  C4○D4.4Q8  C4○D4.5Q8  (C22×C4).275D4  (C22×C4).276D4  C4≀C2⋊C4  C429(C2×C4)  C8.C22⋊C4  C8⋊C22⋊C4  M4(2)⋊20D4  M4(2).45D4  M4(2).48D4  M4(2).49D4  C4.10D43C4  C4.D43C4  M4(2)⋊8Q8  C42.128D4  M4(2)⋊D4  M4(2)⋊4D4  (C2×D4)⋊2Q8  (C2×Q8)⋊2Q8  C4211D4  C4212D4  M4(2).10D4  M4(2).11D4  M4(2)⋊Q8  C423Q8  M4(2).12D4  M4(2).13D4  (C2×C8).55D4  (C2×C8).165D4  M4(2).Q8  M4(2).2Q8  C22.F5⋊C4
 C4p.(C4⋊C4): M4(2).5Q8  M4(2).6Q8  C12.(C4⋊C4)  M4(2)⋊Dic3  (C2×C20).Q8  M4(2)⋊Dic5  M4(2)⋊F5  C28.(C4⋊C4) ...
C22.C42 is a maximal quotient of
C23.19C42  C42.3Q8  C42.4Q8  C42.25D4  C42.7Q8  C42.388D4  C23.C42  C42.30D4  C42.32D4  M4(2)⋊F5  C22.F5⋊C4
 (C2×C4).D4p: C42.27D4  C42.8Q8  C12.(C4⋊C4)  M4(2)⋊Dic3  (C2×C20).Q8  M4(2)⋊Dic5  C28.(C4⋊C4)  M4(2)⋊Dic7 ...

Matrix representation of C22.C42 in GL6(𝔽17)

1600000
0160000
001000
000100
00113160
00614016
,
100000
010000
0016000
0001600
0000160
0000016
,
010000
100000
00113150
000011
001091610
008917
,
400000
0130000
00101200
0013700
00112411
001331113

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,11,6,0,0,0,1,3,14,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,0,10,8,0,0,3,0,9,9,0,0,15,1,16,1,0,0,0,1,10,7],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,10,13,11,13,0,0,12,7,2,3,0,0,0,0,4,11,0,0,0,0,11,13] >;

C22.C42 in GAP, Magma, Sage, TeX

C_2^2.C_4^2
% in TeX

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

G:=SmallGroup(64,24);
// by ID

G=gap.SmallGroup(64,24);
# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489,117]);
// Polycyclic

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

Export

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

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