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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 — D12 — C2×D12 — 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=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 210 in 76 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, D4 [×2], D4 [×4], C23 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C2×C8, D8 [×4], C2×D4, C2×D4, C3⋊C8 [×2], D12 [×2], D12, C2×C12, C3×D4 [×2], C3×D4, C22×S3, C22×C6, C2×D8, C2×C3⋊C8, D4⋊S3 [×4], C2×D12, C6×D4, C2×D4⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, C3⋊D4 [×2], C22×S3, C2×D8, D4⋊S3 [×2], C2×C3⋊D4, C2×D4⋊S3

Character table of C2×D4⋊S3

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B size 1 1 1 1 4 4 12 12 2 2 2 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 2 -2 0 0 -1 -2 2 1 -1 1 -1 -1 1 1 0 0 0 0 -1 1 orthogonal lifted from D6 ρ10 2 2 2 2 2 2 0 0 -1 2 2 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 -2 -2 2 -2 2 0 0 -1 -2 2 1 -1 1 1 1 -1 -1 0 0 0 0 -1 1 orthogonal lifted from D6 ρ12 2 -2 -2 2 0 0 0 0 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 0 0 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 0 -1 2 2 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ15 2 -2 2 -2 0 0 0 0 2 0 0 2 -2 -2 0 0 0 0 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ16 2 2 -2 -2 0 0 0 0 2 0 0 -2 -2 2 0 0 0 0 √2 √2 -√2 -√2 0 0 orthogonal lifted from D8 ρ17 2 2 -2 -2 0 0 0 0 2 0 0 -2 -2 2 0 0 0 0 -√2 -√2 √2 √2 0 0 orthogonal lifted from D8 ρ18 2 -2 2 -2 0 0 0 0 2 0 0 2 -2 -2 0 0 0 0 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ19 2 2 2 2 0 0 0 0 -1 -2 -2 -1 -1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ20 2 2 2 2 0 0 0 0 -1 -2 -2 -1 -1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ21 2 -2 -2 2 0 0 0 0 -1 2 -2 1 -1 1 -√-3 √-3 √-3 -√-3 0 0 0 0 1 -1 complex lifted from C3⋊D4 ρ22 2 -2 -2 2 0 0 0 0 -1 2 -2 1 -1 1 √-3 -√-3 -√-3 √-3 0 0 0 0 1 -1 complex lifted from C3⋊D4 ρ23 4 4 -4 -4 0 0 0 0 -2 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 -4 4 -4 0 0 0 0 -2 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2

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

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

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

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

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

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 0 1 0 0 0 0 57 16 0 0 16 16
,
 72 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,57,16,0,0,16,16],[72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,1,0,0,0,0,72] >;

C2×D4⋊S3 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes S_3
% in TeX

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

G:=SmallGroup(96,138);
// by ID

G=gap.SmallGroup(96,138);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,579,159,69,2309]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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