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## G = C2×D4.S3order 96 = 25·3

### Direct product of C2 and D4.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D4.S3
 Chief series C1 — C3 — C6 — C12 — Dic6 — C2×Dic6 — C2×D4.S3
 Lower central C3 — C6 — C12 — C2×D4.S3
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4.S3
G = < a,b,c,d,e | a2=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 146 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×4], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C2×SD16, C2×C3⋊C8, D4.S3 [×4], C2×Dic6, C6×D4, C2×D4.S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C3⋊D4 [×2], C22×S3, C2×SD16, D4.S3 [×2], C2×C3⋊D4, C2×D4.S3

Character table of C2×D4.S3

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B size 1 1 1 1 4 4 2 2 2 12 12 2 2 2 4 4 4 4 6 6 6 6 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 0 0 2 -2 -2 0 0 2 2 2 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 -2 2 -1 2 -2 0 0 1 -1 1 -1 -1 1 1 0 0 0 0 -1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 0 2 -2 2 0 0 -2 2 -2 0 0 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 -1 2 2 0 0 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ14 2 2 -2 -2 2 -2 -1 2 -2 0 0 1 -1 1 1 1 -1 -1 0 0 0 0 -1 1 orthogonal lifted from D6 ρ15 2 2 -2 -2 0 0 -1 -2 2 0 0 1 -1 1 -√-3 √-3 √-3 -√-3 0 0 0 0 1 -1 complex lifted from C3⋊D4 ρ16 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ17 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 0 0 -1 -2 2 0 0 1 -1 1 √-3 -√-3 -√-3 √-3 0 0 0 0 1 -1 complex lifted from C3⋊D4 ρ19 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 0 0 0 0 -√-2 -√-2 √-2 √-2 0 0 complex lifted from SD16 ρ20 2 -2 -2 2 0 0 2 0 0 0 0 2 -2 -2 0 0 0 0 -√-2 √-2 -√-2 √-2 0 0 complex lifted from SD16 ρ21 2 -2 -2 2 0 0 2 0 0 0 0 2 -2 -2 0 0 0 0 √-2 -√-2 √-2 -√-2 0 0 complex lifted from SD16 ρ22 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 0 0 0 0 √-2 √-2 -√-2 -√-2 0 0 complex lifted from SD16 ρ23 4 -4 -4 4 0 0 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ24 4 -4 4 -4 0 0 -2 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2

Smallest permutation representation of C2×D4.S3
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 21)(5 39)(6 40)(7 37)(8 38)(9 43)(10 44)(11 41)(12 42)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 45)(34 46)(35 47)(36 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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 26)(27 28)(29 32)(30 31)(34 36)(38 40)(41 43)(46 48)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 48 3 46)(2 47 4 45)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)(21 33 23 35)(22 36 24 34)(25 44 27 42)(26 43 28 41)(29 40 31 38)(30 39 32 37)

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

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

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

Matrix representation of C2×D4.S3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 72 0
,
 72 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 64 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 67 6 0 0 6 6
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,67,6,0,0,6,6] >;

C2×D4.S3 in GAP, Magma, Sage, TeX

C_2\times D_4.S_3
% in TeX

G:=Group("C2xD4.S3");
// GroupNames label

G:=SmallGroup(96,140);
// by ID

G=gap.SmallGroup(96,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,579,159,69,2309]);
// Polycyclic

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

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