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## G = C42⋊4Dic3order 192 = 26·3

### 2nd semidirect product of C42 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42⋊4Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23.7D6 — C42⋊4Dic3
 Lower central C3 — C6 — C2×C6 — C2×C12 — C42⋊4Dic3
 Upper central C1 — C2 — C22 — C2×D4 — C4.4D4

Generators and relations for C424Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 240 in 70 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C23⋊C4, C4.4D4, C6.D4, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C423C4, C23.7D6, C3×C4.4D4, C424Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C23⋊C4, C6.D4, C423C4, C23.7D6, C424Dic3

Character table of C424Dic3

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 2 4 4 2 4 4 4 8 24 24 24 24 2 2 2 8 8 4 4 4 4 4 4 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 -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 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -2 2 -2 0 0 0 0 0 0 0 2 2 2 -2 2 0 -2 0 0 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 -1 2 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 1 -1 1 1 -1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 -2 2 2 -2 0 0 0 0 0 0 0 2 2 2 2 -2 0 -2 0 0 -2 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 -1 2 -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 -1 2 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 -1 -2 0 0 0 0 0 0 0 -1 -1 -1 1 -1 -√-3 1 √-3 -√-3 1 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ16 2 2 2 -2 2 -1 -2 0 0 0 0 0 0 0 -1 -1 -1 -1 1 √-3 1 -√-3 √-3 1 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ17 2 2 2 -2 2 -1 -2 0 0 0 0 0 0 0 -1 -1 -1 -1 1 -√-3 1 √-3 -√-3 1 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 2 2 -2 -1 -2 0 0 0 0 0 0 0 -1 -1 -1 1 -1 √-3 1 -√-3 √-3 1 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ19 4 4 -4 0 0 4 0 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 ρ20 4 4 -4 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 -2√-3 0 0 2√-3 0 0 0 complex lifted from C23.7D6 ρ21 4 4 -4 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 2√-3 0 0 -2√-3 0 0 0 complex lifted from C23.7D6 ρ22 4 -4 0 0 0 4 0 -2i 2i 0 0 0 0 0 -4 0 0 0 0 -2i 0 2i 2i 0 -2i 0 0 complex lifted from C42⋊3C4 ρ23 4 -4 0 0 0 4 0 2i -2i 0 0 0 0 0 -4 0 0 0 0 2i 0 -2i -2i 0 2i 0 0 complex lifted from C42⋊3C4 ρ24 4 -4 0 0 0 -2 0 -2i 2i 0 0 0 0 0 2 2√-3 -2√-3 0 0 2ζ43ζ32 0 2ζ4ζ3 2ζ4ζ32 0 2ζ43ζ3 0 0 complex faithful ρ25 4 -4 0 0 0 -2 0 -2i 2i 0 0 0 0 0 2 -2√-3 2√-3 0 0 2ζ43ζ3 0 2ζ4ζ32 2ζ4ζ3 0 2ζ43ζ32 0 0 complex faithful ρ26 4 -4 0 0 0 -2 0 2i -2i 0 0 0 0 0 2 -2√-3 2√-3 0 0 2ζ4ζ3 0 2ζ43ζ32 2ζ43ζ3 0 2ζ4ζ32 0 0 complex faithful ρ27 4 -4 0 0 0 -2 0 2i -2i 0 0 0 0 0 2 2√-3 -2√-3 0 0 2ζ4ζ32 0 2ζ43ζ3 2ζ43ζ32 0 2ζ4ζ3 0 0 complex faithful

Smallest permutation representation of C424Dic3
On 48 points
Generators in S48
(1 32 11 29)(2 36 12 27)(3 34 10 25)(4 26 8 35)(5 30 9 33)(6 28 7 31)(13 48 20 45)(14 40 21 37)(15 44 22 47)(16 42 23 39)(17 46 24 43)(18 38 19 41)
(1 16 4 13)(2 14 5 17)(3 18 6 15)(7 22 10 19)(8 20 11 23)(9 24 12 21)(25 41 31 47)(26 48 32 42)(27 37 33 43)(28 44 34 38)(29 39 35 45)(30 46 36 40)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(2 3)(5 6)(7 12)(8 11)(9 10)(13 20 16 23)(14 19 17 22)(15 24 18 21)(25 37 28 40)(26 42 29 39)(27 41 30 38)(31 43 34 46)(32 48 35 45)(33 47 36 44)

G:=sub<Sym(48)| (1,32,11,29)(2,36,12,27)(3,34,10,25)(4,26,8,35)(5,30,9,33)(6,28,7,31)(13,48,20,45)(14,40,21,37)(15,44,22,47)(16,42,23,39)(17,46,24,43)(18,38,19,41), (1,16,4,13)(2,14,5,17)(3,18,6,15)(7,22,10,19)(8,20,11,23)(9,24,12,21)(25,41,31,47)(26,48,32,42)(27,37,33,43)(28,44,34,38)(29,39,35,45)(30,46,36,40), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (2,3)(5,6)(7,12)(8,11)(9,10)(13,20,16,23)(14,19,17,22)(15,24,18,21)(25,37,28,40)(26,42,29,39)(27,41,30,38)(31,43,34,46)(32,48,35,45)(33,47,36,44)>;

G:=Group( (1,32,11,29)(2,36,12,27)(3,34,10,25)(4,26,8,35)(5,30,9,33)(6,28,7,31)(13,48,20,45)(14,40,21,37)(15,44,22,47)(16,42,23,39)(17,46,24,43)(18,38,19,41), (1,16,4,13)(2,14,5,17)(3,18,6,15)(7,22,10,19)(8,20,11,23)(9,24,12,21)(25,41,31,47)(26,48,32,42)(27,37,33,43)(28,44,34,38)(29,39,35,45)(30,46,36,40), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (2,3)(5,6)(7,12)(8,11)(9,10)(13,20,16,23)(14,19,17,22)(15,24,18,21)(25,37,28,40)(26,42,29,39)(27,41,30,38)(31,43,34,46)(32,48,35,45)(33,47,36,44) );

G=PermutationGroup([[(1,32,11,29),(2,36,12,27),(3,34,10,25),(4,26,8,35),(5,30,9,33),(6,28,7,31),(13,48,20,45),(14,40,21,37),(15,44,22,47),(16,42,23,39),(17,46,24,43),(18,38,19,41)], [(1,16,4,13),(2,14,5,17),(3,18,6,15),(7,22,10,19),(8,20,11,23),(9,24,12,21),(25,41,31,47),(26,48,32,42),(27,37,33,43),(28,44,34,38),(29,39,35,45),(30,46,36,40)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(2,3),(5,6),(7,12),(8,11),(9,10),(13,20,16,23),(14,19,17,22),(15,24,18,21),(25,37,28,40),(26,42,29,39),(27,41,30,38),(31,43,34,46),(32,48,35,45),(33,47,36,44)]])

Matrix representation of C424Dic3 in GL4(𝔽13) generated by

 12 6 7 2 1 3 5 7 10 4 8 0 8 4 6 6
,
 0 0 12 1 12 12 11 12 5 5 1 0 4 5 1 0
,
 12 1 0 0 12 0 0 0 9 8 0 12 10 9 1 1
,
 0 1 0 0 1 0 0 0 6 6 2 4 4 4 2 11
G:=sub<GL(4,GF(13))| [12,1,10,8,6,3,4,4,7,5,8,6,2,7,0,6],[0,12,5,4,0,12,5,5,12,11,1,1,1,12,0,0],[12,12,9,10,1,0,8,9,0,0,0,1,0,0,12,1],[0,1,6,4,1,0,6,4,0,0,2,2,0,0,4,11] >;

C424Dic3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4{\rm Dic}_3
% in TeX

G:=Group("C4^2:4Dic3");
// GroupNames label

G:=SmallGroup(192,100);
// by ID

G=gap.SmallGroup(192,100);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,1571,570,297,136,1684,6278]);
// 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*b^2,d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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