p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42:1C22, C23.13D4, M4(2):10C22, C4wrC2:5C2, C4oD4:4C4, (C2xQ8):8C4, (C2xD4):10C4, D4.7(C2xC4), (C2xC4).21D4, C4.70(C2xD4), Q8.7(C2xC4), C4.8(C22xC4), C42:C2:4C2, (C2xC4).66C23, C4oD4.6C22, C22.11(C2xD4), C4.25(C22:C4), (C2xM4(2)):13C2, C22.5(C22:C4), (C22xC4).37C22, (C2xC4).24(C2xC4), (C2xC4oD4).7C2, C2.24(C2xC22:C4), SmallGroup(64,102)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
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, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C22:C4, C4:C4, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C4wrC2, C42:C2, C2xM4(2), C2xC4oD4, C42:C22
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C2xC22:C4, C42:C22
Character table of C42:C22
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | |
size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | -i | i | -1 | i | i | -i | -i | linear of order 4 |
ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -i | -i | i | i | linear of order 4 |
ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -i | -i | i | i | linear of order 4 |
ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | i | i | -i | 1 | i | i | -i | -i | linear of order 4 |
ρ13 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | -i | i | -1 | -i | i | i | -i | linear of order 4 |
ρ14 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | i | -i | i | 1 | i | -i | -i | i | linear of order 4 |
ρ15 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | i | -i | -1 | i | -i | -i | i | linear of order 4 |
ρ16 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | -i | i | -i | 1 | -i | i | i | -i | linear of order 4 |
ρ17 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
(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 12)(2 14)(3 16)(4 10)(5 13)(6 11)(7 15)(8 9)
(2 3)(6 8)(9 11)(14 16)
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), (1,12)(2,14)(3,16)(4,10)(5,13)(6,11)(7,15)(8,9), (2,3)(6,8)(9,11)(14,16)>;
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), (1,12)(2,14)(3,16)(4,10)(5,13)(6,11)(7,15)(8,9), (2,3)(6,8)(9,11)(14,16) );
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)], [(1,12),(2,14),(3,16),(4,10),(5,13),(6,11),(7,15),(8,9)], [(2,3),(6,8),(9,11),(14,16)]])
G:=TransitiveGroup(16,106);
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3 6 7)(2 4 5 8)(9 14 11 16)(10 15 12 13)
(1 9)(2 13)(3 14)(4 10)(5 15)(6 11)(7 16)(8 12)
(1 8)(2 3)(4 6)(5 7)(9 12)(10 11)(13 14)(15 16)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,6,7)(2,4,5,8)(9,14,11,16)(10,15,12,13), (1,9)(2,13)(3,14)(4,10)(5,15)(6,11)(7,16)(8,12), (1,8)(2,3)(4,6)(5,7)(9,12)(10,11)(13,14)(15,16)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,6,7)(2,4,5,8)(9,14,11,16)(10,15,12,13), (1,9)(2,13)(3,14)(4,10)(5,15)(6,11)(7,16)(8,12), (1,8)(2,3)(4,6)(5,7)(9,12)(10,11)(13,14)(15,16) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,6,7),(2,4,5,8),(9,14,11,16),(10,15,12,13)], [(1,9),(2,13),(3,14),(4,10),(5,15),(6,11),(7,16),(8,12)], [(1,8),(2,3),(4,6),(5,7),(9,12),(10,11),(13,14),(15,16)]])
G:=TransitiveGroup(16,107);
(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)oD8 M4(2):C23 C42.12C23 (C2xD4):6F5 (C2xQ8):6F5 D4:F5:C2
C42:D2p: C42:D4 C42:2D4 C42:3D6 C42:6D6 C42:D10 C42:4D10 C42:D14 C42:4D14 ...
M4(2):D2p: M4(2):19D4 M4(2):5D4 M4(2):24D6 C23.20D20 C23.20D28 ...
C4oD4.D2p: 2+ 1+4:4C4 C4oD4.D4 (C22xQ8):C4 C8oD4: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 (C2xD4):6F5 (C2xQ8):6F5 D4:F5:C2
C42:D2p: C42:7D4 C42:8D4 C42:3D6 C42:6D6 C42:D10 C42:4D10 C42:D14 C42:4D14 ...
C23.D4p: C24.60D4 M4(2):24D6 C23.20D20 C23.20D28 ...
C4oD4.D2p: C24.66D4 C42.102D4 (C6xD4):9C4 (D4xC10):21C4 (D4xC14):9C4 ...
Matrix representation of C42:C22 ►in GL4(F5) generated by
0 | 0 | 1 | 0 |
3 | 0 | 0 | 0 |
0 | 0 | 0 | 4 |
0 | 3 | 0 | 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 |
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
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