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## G = C2×D4⋊2S3order 96 = 25·3

### Direct product of C2 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D4⋊2S3
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — C2×D4⋊2S3
 Lower central C3 — C6 — C2×D4⋊2S3
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D42S3
G = < a,b,c,d,e | a2=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 306 in 164 conjugacy classes, 89 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], S3 [×2], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, Dic3 [×6], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C2×C4○D4, C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C2×D42S3
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, D42S3 [×2], S3×C23, C2×D42S3

Character table of C2×D42S3

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 1 1 2 2 2 2 6 6 2 2 2 3 3 3 3 6 6 6 6 2 2 2 4 4 4 4 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 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 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 -2 -2 2 -2 -2 2 0 0 -1 -2 2 0 0 0 0 0 0 0 0 1 1 -1 1 1 -1 -1 1 -1 orthogonal lifted from D6 ρ18 2 2 2 2 -2 2 -2 2 0 0 -1 -2 -2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 1 -1 1 1 1 orthogonal lifted from D6 ρ19 2 2 2 2 2 2 2 2 0 0 -1 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ20 2 2 -2 -2 -2 -2 2 2 0 0 -1 2 -2 0 0 0 0 0 0 0 0 1 1 -1 1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ21 2 2 2 2 -2 -2 -2 -2 0 0 -1 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ22 2 2 -2 -2 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 0 0 1 1 -1 -1 1 1 -1 -1 1 orthogonal lifted from D6 ρ23 2 2 -2 -2 -2 2 2 -2 0 0 -1 -2 2 0 0 0 0 0 0 0 0 1 1 -1 -1 -1 1 1 1 -1 orthogonal lifted from D6 ρ24 2 2 2 2 2 -2 2 -2 0 0 -1 -2 -2 0 0 0 0 0 0 0 0 -1 -1 -1 1 -1 1 -1 1 1 orthogonal lifted from D6 ρ25 2 -2 -2 2 0 0 0 0 0 0 2 0 0 2i 2i -2i -2i 0 0 0 0 -2 2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 -2 2 0 0 0 0 0 0 2 0 0 -2i -2i 2i 2i 0 0 0 0 -2 2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 2 -2 0 0 0 0 0 0 2 0 0 2i -2i -2i 2i 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 -2 2 -2 0 0 0 0 0 0 2 0 0 -2i 2i 2i -2i 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 4 -4 4 -4 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ30 4 -4 -4 4 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of C2×D42S3
On 48 points
Generators in S48
(1 35)(2 36)(3 33)(4 34)(5 29)(6 30)(7 31)(8 32)(9 38)(10 39)(11 40)(12 37)(13 25)(14 26)(15 27)(16 28)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 20)(2 19)(3 18)(4 17)(5 26)(6 25)(7 28)(8 27)(9 24)(10 23)(11 22)(12 21)(13 30)(14 29)(15 32)(16 31)(33 42)(34 41)(35 44)(36 43)(37 45)(38 48)(39 47)(40 46)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(29 38 35)(30 39 36)(31 40 33)(32 37 34)
(5 9)(6 10)(7 11)(8 12)(13 45)(14 46)(15 47)(16 48)(17 19)(18 20)(21 25)(22 26)(23 27)(24 28)(29 38)(30 39)(31 40)(32 37)(41 43)(42 44)

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

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

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

Matrix representation of C2×D42S3 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 5 0 0 0 0 0 8
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 5 0
,
 1 0 0 0 0 0 12 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,5,0,0,0,0,0,8],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,8,0],[1,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12] >;

C2×D42S3 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2S_3
% in TeX

G:=Group("C2xD4:2S3");
// GroupNames label

G:=SmallGroup(96,210);
// by ID

G=gap.SmallGroup(96,210);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,579,159,2309]);
// Polycyclic

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

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