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## G = C24⋊5Dic3order 192 = 26·3

### 1st semidirect product of C24 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24⋊5Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23.7D6 — C24⋊5Dic3
 Lower central C3 — C6 — C2×C6 — C22×C6 — C24⋊5Dic3
 Upper central C1 — C2 — C22 — C2×D4 — C22≀C2

Generators and relations for C245Dic3
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=e3, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 304 in 94 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22, C22 [×12], C6, C6 [×5], C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], Dic3, C12 [×2], C2×C6, C2×C6 [×12], C22⋊C4, C22⋊C4 [×2], M4(2), C2×D4, C2×D4, C24, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4 [×3], C22×C6 [×2], C22×C6 [×4], C23⋊C4, C4.D4, C22≀C2, C4.Dic3, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C6×D4, C23×C6, C2≀C4, C12.D4, C23.7D6, C3×C22≀C2, C245Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C23⋊C4, C6.D4, C2≀C4, C23.7D6, C245Dic3

Character table of C245Dic3

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 12A 12B 12C size 1 1 2 4 4 4 4 2 4 8 24 24 2 2 2 4 4 4 4 4 4 8 24 24 8 8 8 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 1 1 1 -1 1 -1 -1 1 -1 -1 i -i 1 1 -1 linear of order 4 ρ6 1 1 1 -1 1 -1 -1 1 -1 1 i -i 1 1 1 -1 1 -1 -1 1 -1 -1 -i i 1 1 -1 linear of order 4 ρ7 1 1 1 -1 1 1 1 1 -1 -1 -i i 1 1 1 1 1 1 1 1 1 -1 -i i -1 -1 -1 linear of order 4 ρ8 1 1 1 -1 1 1 1 1 -1 -1 i -i 1 1 1 1 1 1 1 1 1 -1 i -i -1 -1 -1 linear of order 4 ρ9 2 2 2 2 -2 0 0 2 -2 0 0 0 2 2 2 0 -2 0 0 -2 0 2 0 0 0 0 -2 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 -1 2 -2 0 0 -1 -1 -1 1 -1 1 1 -1 1 -1 0 0 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 -2 -2 0 0 2 2 0 0 0 2 2 2 0 -2 0 0 -2 0 -2 0 0 0 0 2 orthogonal lifted from D4 ρ12 2 2 2 2 2 2 2 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 -2 2 2 2 -1 -2 -2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 0 0 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 2 -2 2 -2 -2 -1 -2 2 0 0 -1 -1 -1 1 -1 1 1 -1 1 1 0 0 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 2 -2 0 0 -1 -2 0 0 0 -1 -1 -1 √-3 1 √-3 -√-3 1 -√-3 -1 0 0 √-3 -√-3 1 complex lifted from C3⋊D4 ρ16 2 2 2 -2 -2 0 0 -1 2 0 0 0 -1 -1 -1 √-3 1 √-3 -√-3 1 -√-3 1 0 0 -√-3 √-3 -1 complex lifted from C3⋊D4 ρ17 2 2 2 -2 -2 0 0 -1 2 0 0 0 -1 -1 -1 -√-3 1 -√-3 √-3 1 √-3 1 0 0 √-3 -√-3 -1 complex lifted from C3⋊D4 ρ18 2 2 2 2 -2 0 0 -1 -2 0 0 0 -1 -1 -1 -√-3 1 -√-3 √-3 1 √-3 -1 0 0 -√-3 √-3 1 complex lifted from C3⋊D4 ρ19 4 -4 0 0 0 -2 2 4 0 0 0 0 0 -4 0 2 0 -2 2 0 -2 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ20 4 -4 0 0 0 2 -2 4 0 0 0 0 0 -4 0 -2 0 2 -2 0 2 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ21 4 4 -4 0 0 0 0 4 0 0 0 0 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 0 0 0 0 -2 0 0 0 0 2 -2 2 0 -2√-3 0 0 2√-3 0 0 0 0 0 0 0 complex lifted from C23.7D6 ρ23 4 -4 0 0 0 -2 2 -2 0 0 0 0 -2√-3 2 2√-3 -1-√-3 0 1+√-3 -1+√-3 0 1-√-3 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 2 -2 -2 0 0 0 0 2√-3 2 -2√-3 1-√-3 0 -1+√-3 1+√-3 0 -1-√-3 0 0 0 0 0 0 complex faithful ρ25 4 4 -4 0 0 0 0 -2 0 0 0 0 2 -2 2 0 2√-3 0 0 -2√-3 0 0 0 0 0 0 0 complex lifted from C23.7D6 ρ26 4 -4 0 0 0 2 -2 -2 0 0 0 0 -2√-3 2 2√-3 1+√-3 0 -1-√-3 1-√-3 0 -1+√-3 0 0 0 0 0 0 complex faithful ρ27 4 -4 0 0 0 -2 2 -2 0 0 0 0 2√-3 2 -2√-3 -1+√-3 0 1-√-3 -1-√-3 0 1+√-3 0 0 0 0 0 0 complex faithful

Permutation representations of C245Dic3
On 24 points - transitive group 24T289
Generators in S24
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(8 22)(10 24)(12 20)
(2 16)(4 18)(6 14)(7 21)(9 23)(11 19)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 21)(8 22)(9 23)(10 24)(11 19)(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)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

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

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

G:=TransitiveGroup(24,289);

On 24 points - transitive group 24T356
Generators in S24
(1 17)(2 15)(3 13)(4 7)(5 8)(6 9)(10 18)(11 16)(12 14)(19 22)(20 23)(21 24)
(1 14)(2 18)(3 16)(4 19)(5 23)(6 21)(7 22)(8 20)(9 24)(10 15)(11 13)(12 17)
(1 12)(2 10)(3 11)(13 16)(14 17)(15 18)
(1 12)(2 10)(3 11)(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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)
(1 9)(2 8)(3 7)(4 11)(5 10)(6 12)(13 19 16 22)(14 24 17 21)(15 23 18 20)

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

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

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

G:=TransitiveGroup(24,356);

Matrix representation of C245Dic3 in GL4(𝔽7) generated by

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

C245Dic3 in GAP, Magma, Sage, TeX

C_2^4\rtimes_5{\rm Dic}_3
% in TeX

G:=Group("C2^4:5Dic3");
// GroupNames label

G:=SmallGroup(192,95);
// by ID

G=gap.SmallGroup(192,95);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,675,297,1684,6278]);
// Polycyclic

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

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