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G = C42⋊C22order 64 = 26

1st semidirect product of C42 and C22 acting faithfully

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

Aliases: C421C22, C23.13D4, M4(2)⋊10C22, C4≀C25C2, C4○D44C4, (C2×Q8)⋊8C4, (C2×D4)⋊10C4, D4.7(C2×C4), (C2×C4).21D4, C4.70(C2×D4), Q8.7(C2×C4), C4.8(C22×C4), C42⋊C24C2, (C2×C4).66C23, C4○D4.6C22, C22.11(C2×D4), C4.25(C22⋊C4), (C2×M4(2))⋊13C2, C22.5(C22⋊C4), (C22×C4).37C22, (C2×C4).24(C2×C4), (C2×C4○D4).7C2, C2.24(C2×C22⋊C4), SmallGroup(64,102)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42⋊C22
C1C2C4C2×C4C22×C4C2×C4○D4 — C42⋊C22
C1C2C4 — C42⋊C22
C1C4C22×C4 — C42⋊C22
C1C2C2C2×C4 — C42⋊C22

Generators and relations for C42⋊C22
 G = < a,b,c,d | a4=b4=c2=d2=1, ab=ba, cac=a-1b, dad=ab2, bc=cb, bd=db, cd=dc >

Subgroups: 129 in 77 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C42⋊C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C42⋊C22

Character table of C42⋊C22

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 1122244112224444444444
ρ11111111111111111111111    trivial
ρ211111-1-111111-11111-1-1-1-1-1    linear of order 2
ρ3111-1-1-11111-1-1-1-1-1111-11-11    linear of order 2
ρ4111-1-11-1111-1-11-1-111-11-11-1    linear of order 2
ρ51111111111111-1-1-1-11-1-1-1-1    linear of order 2
ρ611111-1-111111-1-1-1-1-1-11111    linear of order 2
ρ7111-1-1-11111-1-1-111-1-111-11-1    linear of order 2
ρ8111-1-11-1111-1-1111-1-1-1-11-11    linear of order 2
ρ911-1-11-11-1-11-111i-i-ii-1ii-i-i    linear of order 4
ρ1011-1-111-1-1-11-11-1i-i-ii1-i-iii    linear of order 4
ρ1111-1-11-11-1-11-111-iii-i-1-i-iii    linear of order 4
ρ1211-1-111-1-1-11-11-1-iii-i1ii-i-i    linear of order 4
ρ1311-11-111-1-111-1-1-ii-ii-1-iii-i    linear of order 4
ρ1411-11-1-1-1-1-111-11-ii-ii1i-i-ii    linear of order 4
ρ1511-11-111-1-111-1-1i-ii-i-1i-i-ii    linear of order 4
ρ1611-11-1-1-1-1-111-11i-ii-i1-iii-i    linear of order 4
ρ1722-2-220022-22-20000000000    orthogonal lifted from D4
ρ1822-22-20022-2-220000000000    orthogonal lifted from D4
ρ19222-2-200-2-2-2220000000000    orthogonal lifted from D4
ρ202222200-2-2-2-2-20000000000    orthogonal lifted from D4
ρ214-400000-4i4i0000000000000    complex faithful
ρ224-4000004i-4i0000000000000    complex faithful

Permutation representations of C42⋊C22
On 16 points - transitive group 16T106
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7 4 5)(2 8 3 6)(9 16 11 14)(10 13 12 15)
(1 15)(2 9)(3 11)(4 13)(5 12)(6 14)(7 10)(8 16)
(1 4)(5 7)(10 12)(13 15)

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

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,7,4,5)(2,8,3,6)(9,16,11,14)(10,13,12,15), (1,15)(2,9)(3,11)(4,13)(5,12)(6,14)(7,10)(8,16), (1,4)(5,7)(10,12)(13,15) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,7,4,5),(2,8,3,6),(9,16,11,14),(10,13,12,15)], [(1,15),(2,9),(3,11),(4,13),(5,12),(6,14),(7,10),(8,16)], [(1,4),(5,7),(10,12),(13,15)])

G:=TransitiveGroup(16,106);

On 16 points - transitive group 16T107
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3 6 8)(2 4 5 7)(9 16 11 14)(10 13 12 15)
(1 11)(2 13)(3 14)(4 12)(5 15)(6 9)(7 10)(8 16)
(1 4)(2 8)(3 5)(6 7)(9 10)(11 12)(13 16)(14 15)

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

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,6,8)(2,4,5,7)(9,16,11,14)(10,13,12,15), (1,11)(2,13)(3,14)(4,12)(5,15)(6,9)(7,10)(8,16), (1,4)(2,8)(3,5)(6,7)(9,10)(11,12)(13,16)(14,15) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,6,8),(2,4,5,7),(9,16,11,14),(10,13,12,15)], [(1,11),(2,13),(3,14),(4,12),(5,15),(6,9),(7,10),(8,16)], [(1,4),(2,8),(3,5),(6,7),(9,10),(11,12),(13,16),(14,15)])

G:=TransitiveGroup(16,107);

On 16 points - transitive group 16T122
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 11 6)(2 13 12 7)(3 14 9 8)(4 15 10 5)
(2 15)(3 9)(4 7)(5 12)(8 14)(10 13)
(1 11)(3 9)(6 16)(8 14)

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

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,11,6)(2,13,12,7)(3,14,9,8)(4,15,10,5), (2,15)(3,9)(4,7)(5,12)(8,14)(10,13), (1,11)(3,9)(6,16)(8,14) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,11,6),(2,13,12,7),(3,14,9,8),(4,15,10,5)], [(2,15),(3,9),(4,7),(5,12),(8,14),(10,13)], [(1,11),(3,9),(6,16),(8,14)])

G:=TransitiveGroup(16,122);

C42⋊C22 is a maximal subgroup of
C42.5D4  C42.6D4  C42.7D4  C42.427D4  M4(2).30D4  M4(2).D4  C42.8D4  M4(2).8D4  M4(2).9D4  C42.9D4  C42.10D4  C42.283C23  M4(2)○D8  M4(2)⋊C23  C42.12C23  (C2×D4)⋊6F5  (C2×Q8)⋊6F5  D4⋊F5⋊C2
 C42⋊D2p: C42⋊D4  C422D4  C423D6  C426D6  C42⋊D10  C424D10  C42⋊D14  C424D14 ...
 M4(2)⋊D2p: M4(2)⋊19D4  M4(2)⋊5D4  M4(2)⋊24D6  C23.20D20  C23.20D28 ...
 C4○D4.D2p: 2+ 1+44C4  C4○D4.D4  (C22×Q8)⋊C4  C8○D4⋊C4  M4(2).41D4  M4(2).44D4  M4(2).46D4  M4(2).47D4 ...
C42⋊C22 is a maximal quotient of
C42.397D4  C42.398D4  C42.399D4  C42.45D4  C42.46D4  C42.373D4  D4⋊M4(2)  Q8⋊M4(2)  C42.374D4  C42.52D4  C42.53D4  C42.54D4  C24.56D4  C24.57D4  C42.58D4  C24.58D4  C42.59D4  C42.60D4  C24.59D4  C42.61D4  C42.62D4  C24.61D4  C42.63D4  C42.407D4  C42.408D4  C42.376D4  C42.67D4  C42.68D4  C42.69D4  C42.70D4  C42.71D4  C42.72D4  C42.73D4  C42.74D4  C24.63D4  D4.C42  C24.70D4  C24.72D4  M4(2)⋊13D4  M4(2)⋊7Q8  C42⋊Q8  (C2×D4)⋊6F5  (C2×Q8)⋊6F5  D4⋊F5⋊C2
 C42⋊D2p: C427D4  C428D4  C423D6  C426D6  C42⋊D10  C424D10  C42⋊D14  C424D14 ...
 C23.D4p: C24.60D4  M4(2)⋊24D6  C23.20D20  C23.20D28 ...
 C4○D4.D2p: C24.66D4  C42.102D4  (C6×D4)⋊9C4  (D4×C10)⋊21C4  (D4×C14)⋊9C4 ...

Matrix representation of C42⋊C22 in GL4(𝔽5) generated by

0010
3000
0004
0300
,
2000
0200
0020
0002
,
4000
0010
0100
0001
,
4000
0100
0010
0004
G:=sub<GL(4,GF(5))| [0,3,0,0,0,0,0,3,1,0,0,0,0,0,4,0],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4] >;

C42⋊C22 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_2^2
% in TeX

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

G:=SmallGroup(64,102);
// by ID

G=gap.SmallGroup(64,102);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,963,489,117,88]);
// Polycyclic

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

Export

Character table of C42⋊C22 in TeX

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