Copied to
clipboard

## G = (C2×C62).C4order 288 = 25·32

### 5th non-split extension by C2×C62 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — (C2×C62).C4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C62.C4 — (C2×C62).C4
 Lower central C32 — C3×C6 — C62 — (C2×C62).C4
 Upper central C1 — C2 — C22 — C23

Generators and relations for (C2×C62).C4
G = < a,b,c,d | a2=b6=c6=1, d4=c3, ab=ba, ac=ca, dad-1=ab3c3, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 584 in 100 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, C23, C23, C32, Dic3, D6, C2×C6, M4(2), C2×D4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C4.D4, C3⋊Dic3, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C322C8, C2×C3⋊Dic3, C327D4, C22×C3⋊S3, C2×C62, C62.C4, C2×C327D4, (C2×C62).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C4.D4, C32⋊C4, C2×C32⋊C4, C62⋊C4, (C2×C62).C4

Character table of (C2×C62).C4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 8A 8B 8C 8D size 1 1 2 4 36 4 4 18 18 4 4 4 4 4 4 4 4 4 4 4 4 4 4 36 36 36 36 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 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 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 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 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 -i i -i i linear of order 4 ρ6 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -i i i -i linear of order 4 ρ7 1 1 1 1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i -i i -i linear of order 4 ρ8 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 i -i -i i linear of order 4 ρ9 2 2 -2 0 0 2 2 2 -2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 0 2 2 -2 2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 1 -2 0 0 -2 2 1 2 -1 0 0 3 -3 0 3 0 -3 -1 0 0 0 0 orthogonal lifted from C62⋊C4 ρ12 4 4 4 4 0 -2 1 0 0 1 1 -2 1 -2 1 1 -2 -2 1 -2 1 -2 -2 0 0 0 0 orthogonal lifted from C32⋊C4 ρ13 4 4 -4 0 0 -2 1 0 0 1 -1 -2 -1 2 -3 -3 0 0 3 0 3 0 2 0 0 0 0 orthogonal lifted from C62⋊C4 ρ14 4 4 -4 0 0 1 -2 0 0 -2 2 1 2 -1 0 0 -3 3 0 -3 0 3 -1 0 0 0 0 orthogonal lifted from C62⋊C4 ρ15 4 4 4 -4 0 1 -2 0 0 -2 -2 1 -2 1 2 2 -1 -1 2 -1 2 -1 1 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ16 4 4 4 4 0 1 -2 0 0 -2 -2 1 -2 1 -2 -2 1 1 -2 1 -2 1 1 0 0 0 0 orthogonal lifted from C32⋊C4 ρ17 4 4 4 -4 0 -2 1 0 0 1 1 -2 1 -2 -1 -1 2 2 -1 2 -1 2 -2 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ18 4 4 -4 0 0 -2 1 0 0 1 -1 -2 -1 2 3 3 0 0 -3 0 -3 0 2 0 0 0 0 orthogonal lifted from C62⋊C4 ρ19 4 -4 0 0 0 4 4 0 0 -4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ20 4 -4 0 0 0 1 -2 0 0 2 0 -1 0 -3 0 0 √-3 √-3 -2√-3 -√-3 2√-3 -√-3 3 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 -2 1 0 0 -1 3 2 -3 0 -√-3 √-3 -2√-3 0 -√-3 2√-3 √-3 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 1 -2 0 0 2 0 -1 0 3 -2√-3 2√-3 √-3 -√-3 0 -√-3 0 √-3 -3 0 0 0 0 complex faithful ρ23 4 -4 0 0 0 -2 1 0 0 -1 -3 2 3 0 -√-3 √-3 0 2√-3 √-3 0 -√-3 -2√-3 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 1 -2 0 0 2 0 -1 0 3 2√-3 -2√-3 -√-3 √-3 0 √-3 0 -√-3 -3 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 1 -2 0 0 2 0 -1 0 -3 0 0 -√-3 -√-3 2√-3 √-3 -2√-3 √-3 3 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 -2 1 0 0 -1 -3 2 3 0 √-3 -√-3 0 -2√-3 -√-3 0 √-3 2√-3 0 0 0 0 0 complex faithful ρ27 4 -4 0 0 0 -2 1 0 0 -1 3 2 -3 0 √-3 -√-3 2√-3 0 √-3 -2√-3 -√-3 0 0 0 0 0 0 complex faithful

Permutation representations of (C2×C62).C4
On 24 points - transitive group 24T585
Generators in S24
(2 6)(3 7)(9 13)(12 16)(18 22)(19 23)
(1 21 15)(2 18 16 6 22 12)(3 9 23)(4 14 24 8 10 20)(5 17 11)(7 13 19)
(1 11 21 5 15 17)(2 6)(3 19 9 7 23 13)(4 8)(10 14)(12 16)(18 22)(20 24)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,585);

On 24 points - transitive group 24T625
Generators in S24
(1 7)(2 4)(3 5)(6 8)(9 23)(10 20)(11 21)(12 18)(13 19)(14 24)(15 17)(16 22)
(1 21 13)(2 18 14 6 22 10)(3 15 23)(4 12 24 8 16 20)(5 17 9)(7 11 19)
(1 9 21 5 13 17)(2 6)(3 19 15 7 23 11)(4 8)(10 14)(12 16)(18 22)(20 24)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,625);

Matrix representation of (C2×C62).C4 in GL4(𝔽7) generated by

 4 5 2 6 0 6 0 2 3 3 3 1 0 0 0 1
,
 3 5 3 2 3 1 2 6 4 4 0 6 0 0 0 3
,
 4 5 3 4 3 2 2 1 4 4 1 6 0 0 0 6
,
 2 1 5 6 0 2 1 5 3 4 2 4 3 3 2 1
G:=sub<GL(4,GF(7))| [4,0,3,0,5,6,3,0,2,0,3,0,6,2,1,1],[3,3,4,0,5,1,4,0,3,2,0,0,2,6,6,3],[4,3,4,0,5,2,4,0,3,2,1,0,4,1,6,6],[2,0,3,3,1,2,4,3,5,1,2,2,6,5,4,1] >;

(C2×C62).C4 in GAP, Magma, Sage, TeX

(C_2\times C_6^2).C_4
% in TeX

G:=Group("(C2xC6^2).C4");
// GroupNames label

G:=SmallGroup(288,436);
// by ID

G=gap.SmallGroup(288,436);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,219,100,675,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=1,d^4=c^3,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^3*c^3,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations

Export

׿
×
𝔽