p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24:3C4, C25.1C2, C23.33D4, C23.55C23, C24.23C22, C2.1C22wrC2, C23.24(C2xC4), (C22xC4):1C22, C22.29(C2xD4), C22:2(C22:C4), C22.28(C22xC4), (C2xC22:C4):1C2, C2.4(C2xC22:C4), SmallGroup(64,60)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24:3C4
G = < a,b,c,d,e | a2=b2=c2=d2=e4=1, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, cd=dc, ce=ec, de=ed >
Subgroups: 449 in 253 conjugacy classes, 65 normal (5 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2xC4, C23, C23, C23, C22:C4, C22xC4, C24, C24, C2xC22:C4, C25, C24:3C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C2xC22:C4, C22wrC2, C24:3C4
Character table of C24:3C4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 2P | 2Q | 2R | 2S | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | |
size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | -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 | -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 | 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 | -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 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | i | -i | linear of order 4 |
ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | linear of order 4 |
ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | i | -i | -i | i | -i | -i | i | i | linear of order 4 |
ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -i | -i | i | linear of order 4 |
ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | linear of order 4 |
ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -i | i | i | -i | i | i | -i | -i | linear of order 4 |
ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
ρ17 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ22 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ23 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ24 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ25 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ26 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ27 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ28 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
(1 9)(2 16)(3 11)(4 14)(5 12)(6 15)(7 10)(8 13)
(1 6)(2 10)(3 8)(4 12)(5 14)(7 16)(9 15)(11 13)
(1 6)(2 7)(3 8)(4 5)(9 15)(10 16)(11 13)(12 14)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,9)(2,16)(3,11)(4,14)(5,12)(6,15)(7,10)(8,13), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,9)(2,16)(3,11)(4,14)(5,12)(6,15)(7,10)(8,13), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,9),(2,16),(3,11),(4,14),(5,12),(6,15),(7,10),(8,13)], [(1,6),(2,10),(3,8),(4,12),(5,14),(7,16),(9,15),(11,13)], [(1,6),(2,7),(3,8),(4,5),(9,15),(10,16),(11,13),(12,14)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,79);
C24:3C4 is a maximal subgroup of
C24:C8 C24:D4 C24:2Q8 C25.85C22 C4xC22wrC2 C23.191C24 C23.194C24 C23.203C24 D4xC22:C4 C23.224C24 C23.257C24 C24:7D4 C23.308C24 C24:8D4 C23.311C24 C23.318C24 C23.335C24 C24:4Q8 C23.372C24 C23.380C24 C23.382C24 C23.434C24 C23.461C24 C24:9D4 C24:5Q8 C23.568C24 C23.570C24 C23.578C24 C23.584C24 C23.585C24 C23.597C24 C23.635C24 C23.636C24 C24:11D4 C24:6Q8 C24.15Q8 C23wrC2 C24:8Q8 C24:C12 C24:4Dic3 C24:4F5 C25.D5
C24.D2p: C24.4Q8 C25:C4 C24.68D4 C25.C22 C24.78D4 C24.90D4 C24.94D4 C24.96D4 ...
C24:3C4 is a maximal quotient of
C24.636C23 C24:3C8 C24.51(C2xC4) C24.165C23 C25.C4 C4.C22wrC2 (C23xC4).C4 2+ 1+4.2C4 2- 1+4:2C4 2+ 1+4:4C4 C4oD4.D4 (C22xQ8):C4 C24:4F5
C24.D2p: C24.17Q8 C24.50D4 C25:C4 C23.35D8 C24.155D4 C24.65D4 C24.66D4 2+ 1+4:2C4 ...
Matrix representation of C24:3C4 ►in GL5(F5)
4 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 4 |
1 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 4 | 4 |
1 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 4 |
3 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 3 |
0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,4,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[3,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,3,1] >;
C24:3C4 in GAP, Magma, Sage, TeX
C_2^4\rtimes_3C_4
% in TeX
G:=Group("C2^4:3C4");
// GroupNames label
G:=SmallGroup(64,60);
// by ID
G=gap.SmallGroup(64,60);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,362]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^4=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations
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