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G = C62.C4order 144 = 24·32

2nd non-split extension by C62 of C4 acting faithfully

metabelian, soluble, monomial

Aliases: C62.2C4, C324M4(2), C322C84C2, C3⋊Dic3.6C4, C22.(C32⋊C4), C3⋊Dic3.10C22, (C3×C6).6(C2×C4), C2.6(C2×C32⋊C4), (C2×C3⋊Dic3).7C2, SmallGroup(144,135)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.C4
C1C32C3×C6C3⋊Dic3C322C8 — C62.C4
C32C3×C6 — C62.C4
C1C2C22

Generators and relations for C62.C4
 G = < a,b,c | a6=b6=1, c4=b3, ab=ba, cac-1=a-1b, cbc-1=a4b >

2C2
2C3
2C3
9C4
9C4
2C6
2C6
2C6
2C6
2C6
2C6
9C2×C4
9C8
9C8
2C2×C6
2C2×C6
6Dic3
6Dic3
6Dic3
6Dic3
2C3×C6
9M4(2)
6C2×Dic3
6C2×Dic3

Character table of C62.C4

 class 12A2B3A3B4A4B4C6A6B6C6D6E6F8A8B8C8D
 size 11244991844444418181818
ρ1111111111111111111    trivial
ρ211111111111111-1-1-1-1    linear of order 2
ρ311-11111-1-11-1-1-1111-1-1    linear of order 2
ρ411-11111-1-11-1-1-11-1-111    linear of order 2
ρ511-111-1-11-11-1-1-11-iii-i    linear of order 4
ρ611111-1-1-1111111i-ii-i    linear of order 4
ρ711-111-1-11-11-1-1-11i-i-ii    linear of order 4
ρ811111-1-1-1111111-ii-ii    linear of order 4
ρ92-20222i-2i00-2000-20000    complex lifted from M4(2)
ρ102-2022-2i2i00-2000-20000    complex lifted from M4(2)
ρ1144-4-2100021-1-12-20000    orthogonal lifted from C2×C32⋊C4
ρ12444-21000-2111-2-20000    orthogonal lifted from C32⋊C4
ρ134441-20001-2-2-2110000    orthogonal lifted from C32⋊C4
ρ1444-41-2000-1-222-110000    orthogonal lifted from C2×C32⋊C4
ρ154-401-2000-32003-10000    symplectic faithful, Schur index 2
ρ164-40-210000-1-33020000    symplectic faithful, Schur index 2
ρ174-401-20003200-3-10000    symplectic faithful, Schur index 2
ρ184-40-210000-13-3020000    symplectic faithful, Schur index 2

Permutation representations of C62.C4
On 24 points - transitive group 24T207
Generators in S24
(1 9 21)(2 6)(3 23 11)(4 8)(5 13 17)(7 19 15)(10 14)(12 16)(18 22)(20 24)
(1 13 21 5 9 17)(2 18 10 6 22 14)(3 19 11 7 23 15)(4 16 24 8 12 20)
(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)| (1,9,21)(2,6)(3,23,11)(4,8)(5,13,17)(7,19,15)(10,14)(12,16)(18,22)(20,24), (1,13,21,5,9,17)(2,18,10,6,22,14)(3,19,11,7,23,15)(4,16,24,8,12,20), (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( (1,9,21)(2,6)(3,23,11)(4,8)(5,13,17)(7,19,15)(10,14)(12,16)(18,22)(20,24), (1,13,21,5,9,17)(2,18,10,6,22,14)(3,19,11,7,23,15)(4,16,24,8,12,20), (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([(1,9,21),(2,6),(3,23,11),(4,8),(5,13,17),(7,19,15),(10,14),(12,16),(18,22),(20,24)], [(1,13,21,5,9,17),(2,18,10,6,22,14),(3,19,11,7,23,15),(4,16,24,8,12,20)], [(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(24,207);

C62.C4 is a maximal subgroup of
Dic3≀C2  C62.2Q8  C3⋊Dic3.D4  (C2×C62).C4  C62.12D4  C62.15D4  C3⋊S3⋊M4(2)  C62.(C2×C4)  C33⋊M4(2)  C3312M4(2)
C62.C4 is a maximal quotient of
C322C8⋊C4  C325(C4⋊C8)  C623C8  He34M4(2)  C33⋊M4(2)  C3312M4(2)

Matrix representation of C62.C4 in GL4(𝔽5) generated by

4001
0400
0040
4000
,
1004
0130
0300
1000
,
0300
1004
0002
0010
G:=sub<GL(4,GF(5))| [4,0,0,4,0,4,0,0,0,0,4,0,1,0,0,0],[1,0,0,1,0,1,3,0,0,3,0,0,4,0,0,0],[0,1,0,0,3,0,0,0,0,0,0,1,0,4,2,0] >;

C62.C4 in GAP, Magma, Sage, TeX

C_6^2.C_4
% in TeX

G:=Group("C6^2.C4");
// GroupNames label

G:=SmallGroup(144,135);
// by ID

G=gap.SmallGroup(144,135);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,121,50,3364,256,4613,881]);
// Polycyclic

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

Export

Subgroup lattice of C62.C4 in TeX
Character table of C62.C4 in TeX

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