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## G = C42⋊7S3order 96 = 25·3

### 6th semidirect product of C42 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42⋊7S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — D6⋊C4 — C42⋊7S3
 Lower central C3 — C2×C6 — C42⋊7S3
 Upper central C1 — C22 — C42

Generators and relations for C427S3
G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 202 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.4D4, D6⋊C4, C4×C12, C2×Dic6, C2×D12, C427S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C2×D12, C4○D12, C427S3

Character table of C427S3

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L size 1 1 1 1 12 12 2 2 2 2 2 2 2 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 -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 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 -1 -1 1 linear of order 2 ρ5 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 1 1 1 linear of order 2 ρ6 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 -1 -1 -1 linear of order 2 ρ7 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 1 1 -1 linear of order 2 ρ8 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 -1 -1 1 linear of order 2 ρ9 2 2 2 2 0 0 -1 2 -2 -2 -2 2 -2 0 0 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 2 2 0 0 0 -2 0 0 0 -2 2 -2 0 0 0 -2 -2 0 2 2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -1 -2 -2 2 2 -2 -2 0 0 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ12 2 2 2 2 0 0 -1 -2 2 -2 -2 -2 2 0 0 -1 -1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ13 2 2 2 2 0 0 -1 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 -2 -2 2 0 0 2 -2 0 0 0 2 0 0 0 -2 2 -2 0 0 0 2 2 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 0 0 -1 2 0 0 0 -2 0 0 0 1 -1 1 -√3 √3 -√3 1 1 √3 -1 -1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ16 2 -2 -2 2 0 0 -1 2 0 0 0 -2 0 0 0 1 -1 1 √3 -√3 √3 1 1 -√3 -1 -1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ17 2 -2 -2 2 0 0 -1 -2 0 0 0 2 0 0 0 1 -1 1 -√3 √3 √3 -1 -1 -√3 1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ18 2 -2 -2 2 0 0 -1 -2 0 0 0 2 0 0 0 1 -1 1 √3 -√3 -√3 -1 -1 √3 1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ19 2 2 -2 -2 0 0 2 0 0 -2i 2i 0 0 0 0 2 -2 -2 2i 2i 0 0 0 0 0 0 0 -2i -2i 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 0 0 2 0 -2i 0 0 0 2i 0 0 -2 -2 2 0 0 -2i 0 0 -2i 0 0 2i 0 0 2i complex lifted from C4○D4 ρ21 2 2 -2 -2 0 0 2 0 0 2i -2i 0 0 0 0 2 -2 -2 -2i -2i 0 0 0 0 0 0 0 2i 2i 0 complex lifted from C4○D4 ρ22 2 -2 2 -2 0 0 2 0 2i 0 0 0 -2i 0 0 -2 -2 2 0 0 2i 0 0 2i 0 0 -2i 0 0 -2i complex lifted from C4○D4 ρ23 2 -2 2 -2 0 0 -1 0 2i 0 0 0 -2i 0 0 1 1 -1 √-3 -√-3 -i √3 -√3 -i -√3 √3 i √-3 -√-3 i complex lifted from C4○D12 ρ24 2 2 -2 -2 0 0 -1 0 0 -2i 2i 0 0 0 0 -1 1 1 -i -i -√-3 -√3 √3 √-3 -√3 √3 √-3 i i -√-3 complex lifted from C4○D12 ρ25 2 -2 2 -2 0 0 -1 0 2i 0 0 0 -2i 0 0 1 1 -1 -√-3 √-3 -i -√3 √3 -i √3 -√3 i -√-3 √-3 i complex lifted from C4○D12 ρ26 2 -2 2 -2 0 0 -1 0 -2i 0 0 0 2i 0 0 1 1 -1 √-3 -√-3 i -√3 √3 i √3 -√3 -i √-3 -√-3 -i complex lifted from C4○D12 ρ27 2 -2 2 -2 0 0 -1 0 -2i 0 0 0 2i 0 0 1 1 -1 -√-3 √-3 i √3 -√3 i -√3 √3 -i -√-3 √-3 -i complex lifted from C4○D12 ρ28 2 2 -2 -2 0 0 -1 0 0 -2i 2i 0 0 0 0 -1 1 1 -i -i √-3 √3 -√3 -√-3 √3 -√3 -√-3 i i √-3 complex lifted from C4○D12 ρ29 2 2 -2 -2 0 0 -1 0 0 2i -2i 0 0 0 0 -1 1 1 i i -√-3 √3 -√3 √-3 √3 -√3 √-3 -i -i -√-3 complex lifted from C4○D12 ρ30 2 2 -2 -2 0 0 -1 0 0 2i -2i 0 0 0 0 -1 1 1 i i √-3 -√3 √3 -√-3 -√3 √3 -√-3 -i -i √-3 complex lifted from C4○D12

Smallest permutation representation of C427S3
On 48 points
Generators in S48
(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 26 43 17)(2 27 44 18)(3 28 41 19)(4 25 42 20)(5 22 34 16)(6 23 35 13)(7 24 36 14)(8 21 33 15)(9 30 47 39)(10 31 48 40)(11 32 45 37)(12 29 46 38)
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 38 20)(6 39 17)(7 40 18)(8 37 19)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(25 34 29)(26 35 30)(27 36 31)(28 33 32)
(2 44)(4 42)(5 31)(6 37)(7 29)(8 39)(9 23)(10 14)(11 21)(12 16)(13 47)(15 45)(17 19)(18 25)(20 27)(22 46)(24 48)(26 28)(30 33)(32 35)(34 40)(36 38)

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

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

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

Matrix representation of C427S3 in GL4(𝔽13) generated by

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

C427S3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_7S_3
% in TeX

G:=Group("C4^2:7S3");
// GroupNames label

G:=SmallGroup(96,82);
// by ID

G=gap.SmallGroup(96,82);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,218,86,2309]);
// Polycyclic

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

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