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G = C42⋊5Dic3order 192 = 26·3

3rd semidirect product of C42 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42⋊5Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23.7D6 — C42⋊5Dic3
 Lower central C3 — C6 — C2×C6 — C2×C12 — C42⋊5Dic3
 Upper central C1 — C2 — C22 — C2×D4 — C4⋊1D4

Generators and relations for C425Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 304 in 86 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×D4, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C23⋊C4, C41D4, C6.D4, C4×C12, C6×D4, C6×D4, C42⋊C4, C23.7D6, C3×C41D4, C425Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C23⋊C4, C6.D4, C42⋊C4, C23.7D6, C425Dic3

Character table of C425Dic3

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F size 1 1 2 4 4 8 2 4 4 4 24 24 24 24 2 2 2 8 8 8 8 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 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 i i -i -i 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 linear of order 4 ρ6 1 1 1 -1 -1 1 1 1 -1 -1 -i -i i i 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 1 1 1 1 i -i -i i 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 1 1 1 1 -i i i -i 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 2 -2 -1 2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ10 2 2 2 2 -2 0 2 -2 0 0 0 0 0 0 2 2 2 -2 2 0 0 0 0 -2 0 0 -2 orthogonal lifted from D4 ρ11 2 2 2 -2 2 0 2 -2 0 0 0 0 0 0 2 2 2 2 -2 0 0 0 0 -2 0 0 -2 orthogonal lifted from D4 ρ12 2 2 2 2 2 2 -1 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 -2 -2 -2 -1 2 2 2 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 2 -2 -2 2 -1 2 -2 -2 0 0 0 0 -1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 2 -2 0 -1 -2 0 0 0 0 0 0 -1 -1 -1 1 -1 -√-3 √-3 √-3 -√-3 1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ16 2 2 2 2 -2 0 -1 -2 0 0 0 0 0 0 -1 -1 -1 1 -1 √-3 -√-3 -√-3 √-3 1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ17 2 2 2 -2 2 0 -1 -2 0 0 0 0 0 0 -1 -1 -1 -1 1 -√-3 √-3 -√-3 √-3 1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ18 2 2 2 -2 2 0 -1 -2 0 0 0 0 0 0 -1 -1 -1 -1 1 √-3 -√-3 √-3 -√-3 1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 0 4 0 -2 2 0 0 0 0 -4 0 0 0 0 0 0 2 2 0 -2 -2 0 orthogonal lifted from C42⋊C4 ρ20 4 -4 0 0 0 0 4 0 2 -2 0 0 0 0 -4 0 0 0 0 0 0 -2 -2 0 2 2 0 orthogonal lifted from C42⋊C4 ρ21 4 4 -4 0 0 0 4 0 0 0 0 0 0 0 4 -4 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 -4 0 0 0 0 -2 0 2 -2 0 0 0 0 2 2√-3 -2√-3 0 0 0 0 1-√-3 1+√-3 0 -1+√-3 -1-√-3 0 complex faithful ρ23 4 -4 0 0 0 0 -2 0 2 -2 0 0 0 0 2 -2√-3 2√-3 0 0 0 0 1+√-3 1-√-3 0 -1-√-3 -1+√-3 0 complex faithful ρ24 4 4 -4 0 0 0 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 -2√-3 0 0 2√-3 complex lifted from C23.7D6 ρ25 4 -4 0 0 0 0 -2 0 -2 2 0 0 0 0 2 -2√-3 2√-3 0 0 0 0 -1-√-3 -1+√-3 0 1+√-3 1-√-3 0 complex faithful ρ26 4 -4 0 0 0 0 -2 0 -2 2 0 0 0 0 2 2√-3 -2√-3 0 0 0 0 -1+√-3 -1-√-3 0 1-√-3 1+√-3 0 complex faithful ρ27 4 4 -4 0 0 0 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 2√-3 0 0 -2√-3 complex lifted from C23.7D6

Permutation representations of C425Dic3
On 24 points - transitive group 24T354
Generators in S24
(1 5)(2 6)(3 4)(7 20 11 23)(8 24 12 21)(9 22 10 19)(13 16)(14 17)(15 18)
(1 17 5 14)(2 15 6 18)(3 13 4 16)(7 23 11 20)(8 21 12 24)(9 19 10 22)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 9)(2 8)(3 7)(4 11)(5 10)(6 12)(13 23 16 20)(14 22 17 19)(15 21 18 24)

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

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

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

G:=TransitiveGroup(24,354);

Matrix representation of C425Dic3 in GL4(𝔽7) generated by

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

C425Dic3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_5{\rm Dic}_3
% in TeX

G:=Group("C4^2:5Dic3");
// GroupNames label

G:=SmallGroup(192,104);
// by ID

G=gap.SmallGroup(192,104);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,1571,570,297,136,1684,6278]);
// Polycyclic

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

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