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## G = C42.Dic3order 192 = 26·3

### 2nd non-split extension by C42 of Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×Q8 — C12.10D4 — C42.Dic3
 Lower central C3 — C6 — C2×C6 — C2×C12 — C42.Dic3
 Upper central C1 — C2 — C22 — C2×Q8 — C4.4D4

Generators and relations for C42.Dic3
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b2c3, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >

Subgroups: 176 in 64 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.10D4, C4.4D4, C4.Dic3, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C42.C4, C12.10D4, C3×C4.4D4, C42.Dic3
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, C42.Dic3

Character table of C42.Dic3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 2 8 2 4 4 4 4 4 2 2 2 8 8 24 24 24 24 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 1 1 1 -1 -1 -i i i -i 1 1 1 1 1 1 -1 -1 linear of order 4 ρ6 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 i i -i -i -1 1 -1 -1 1 -1 -1 -1 linear of order 4 ρ7 1 1 1 -1 1 1 -1 1 1 -1 1 1 1 -1 -1 i -i -i i 1 1 1 1 1 1 -1 -1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -i -i i i -1 1 -1 -1 1 -1 -1 -1 linear of order 4 ρ9 2 2 2 0 2 0 2 -2 0 -2 2 2 2 0 0 0 0 0 0 0 -2 0 0 -2 0 2 -2 orthogonal lifted from D4 ρ10 2 2 2 0 2 0 -2 -2 0 2 2 2 2 0 0 0 0 0 0 0 -2 0 0 -2 0 -2 2 orthogonal lifted from D4 ρ11 2 2 2 -2 -1 -2 2 2 -2 2 -1 -1 -1 1 1 0 0 0 0 1 -1 1 1 -1 1 -1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 2 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 2 -1 -2 -2 2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 1 -1 1 1 -1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 2 -2 -1 2 -2 2 2 -2 -1 -1 -1 1 1 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 0 -1 0 2 -2 0 -2 -1 -1 -1 -√-3 √-3 0 0 0 0 -√-3 1 √-3 -√-3 1 √-3 -1 1 complex lifted from C3⋊D4 ρ16 2 2 2 0 -1 0 -2 -2 0 2 -1 -1 -1 -√-3 √-3 0 0 0 0 √-3 1 -√-3 √-3 1 -√-3 1 -1 complex lifted from C3⋊D4 ρ17 2 2 2 0 -1 0 -2 -2 0 2 -1 -1 -1 √-3 -√-3 0 0 0 0 -√-3 1 √-3 -√-3 1 √-3 1 -1 complex lifted from C3⋊D4 ρ18 2 2 2 0 -1 0 2 -2 0 -2 -1 -1 -1 √-3 -√-3 0 0 0 0 √-3 1 -√-3 √-3 1 -√-3 -1 1 complex lifted from C3⋊D4 ρ19 4 4 -4 0 4 0 0 0 0 0 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 4 -4 0 -2 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 -2√-3 0 0 2√-3 0 0 0 complex lifted from C23.7D6 ρ21 4 4 -4 0 -2 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 2√-3 0 0 -2√-3 0 0 0 complex lifted from C23.7D6 ρ22 4 -4 0 0 4 2i 0 0 -2i 0 -4 0 0 0 0 0 0 0 0 -2i 0 2i 2i 0 -2i 0 0 complex lifted from C42.C4 ρ23 4 -4 0 0 4 -2i 0 0 2i 0 -4 0 0 0 0 0 0 0 0 2i 0 -2i -2i 0 2i 0 0 complex lifted from C42.C4 ρ24 4 -4 0 0 -2 -2i 0 0 2i 0 2 2√-3 -2√-3 0 0 0 0 0 0 2ζ4ζ32 0 2ζ43ζ3 2ζ43ζ32 0 2ζ4ζ3 0 0 complex faithful ρ25 4 -4 0 0 -2 2i 0 0 -2i 0 2 2√-3 -2√-3 0 0 0 0 0 0 2ζ43ζ32 0 2ζ4ζ3 2ζ4ζ32 0 2ζ43ζ3 0 0 complex faithful ρ26 4 -4 0 0 -2 -2i 0 0 2i 0 2 -2√-3 2√-3 0 0 0 0 0 0 2ζ4ζ3 0 2ζ43ζ32 2ζ43ζ3 0 2ζ4ζ32 0 0 complex faithful ρ27 4 -4 0 0 -2 2i 0 0 -2i 0 2 -2√-3 2√-3 0 0 0 0 0 0 2ζ43ζ3 0 2ζ4ζ32 2ζ4ζ3 0 2ζ43ζ32 0 0 complex faithful

Smallest permutation representation of C42.Dic3
On 48 points
Generators in S48
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 43)(14 38)(15 45)(16 40)(17 47)(18 42)(19 37)(20 44)(21 39)(22 46)(23 41)(24 48)(25 28 31 34)(26 29 32 35)(27 30 33 36)
(1 33 7 27)(2 28 8 34)(3 35 9 29)(4 30 10 36)(5 25 11 31)(6 32 12 26)(13 46 19 40)(14 41 20 47)(15 48 21 42)(16 43 22 37)(17 38 23 44)(18 45 24 39)
(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)
(1 40 10 37 7 46 4 43)(2 45 11 42 8 39 5 48)(3 38 12 47 9 44 6 41)(13 30 22 27 19 36 16 33)(14 35 23 32 20 29 17 26)(15 28 24 25 21 34 18 31)

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

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

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

Matrix representation of C42.Dic3 in GL4(𝔽73) generated by

 46 0 0 0 0 46 0 0 0 0 46 19 0 0 27 27
,
 72 71 0 0 1 1 0 0 0 0 1 2 0 0 72 72
,
 3 0 0 0 70 70 0 0 0 0 24 0 0 0 49 49
,
 0 0 1 0 0 0 0 1 46 0 0 0 27 27 0 0
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,46,27,0,0,19,27],[72,1,0,0,71,1,0,0,0,0,1,72,0,0,2,72],[3,70,0,0,0,70,0,0,0,0,24,49,0,0,0,49],[0,0,46,27,0,0,0,27,1,0,0,0,0,1,0,0] >;

C42.Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,101);
// by ID

G=gap.SmallGroup(192,101);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,184,1571,570,297,136,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=b^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^5>;
// generators/relations

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