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## G = C2×Q8⋊2S3order 96 = 25·3

### Direct product of C2 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Q8⋊2S3
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×Q8⋊2S3
 Lower central C3 — C6 — C12 — C2×Q8⋊2S3
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×Q82S3
G = < a,b,c,d,e | a2=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 178 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C8, SD16 [×4], C2×D4, C2×Q8, C3⋊C8 [×2], D12 [×2], D12, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C22×S3, C2×SD16, C2×C3⋊C8, Q82S3 [×4], C2×D12, C6×Q8, C2×Q82S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C3⋊D4 [×2], C22×S3, C2×SD16, Q82S3 [×2], C2×C3⋊D4, C2×Q82S3

Character table of C2×Q82S3

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 1 1 12 12 2 2 2 4 4 2 2 2 6 6 6 6 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 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 -1 2 2 -2 -2 -1 -1 -1 0 0 0 0 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ10 2 2 -2 -2 0 0 -1 -2 2 2 -2 1 -1 1 0 0 0 0 -1 1 -1 1 -1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 -1 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 -2 0 0 2 2 -2 0 0 -2 2 -2 0 0 0 0 -2 0 0 0 0 2 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 2 -2 -2 0 0 2 2 2 0 0 0 0 -2 0 0 0 0 -2 orthogonal lifted from D4 ρ14 2 2 -2 -2 0 0 -1 -2 2 -2 2 1 -1 1 0 0 0 0 -1 -1 1 -1 1 1 orthogonal lifted from D6 ρ15 2 2 -2 -2 0 0 -1 2 -2 0 0 1 -1 1 0 0 0 0 1 -√-3 -√-3 √-3 √-3 -1 complex lifted from C3⋊D4 ρ16 2 2 -2 -2 0 0 -1 2 -2 0 0 1 -1 1 0 0 0 0 1 √-3 √-3 -√-3 -√-3 -1 complex lifted from C3⋊D4 ρ17 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 √-3 -√-3 -√-3 √-3 1 complex lifted from C3⋊D4 ρ18 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 -√-3 √-3 √-3 -√-3 1 complex lifted from C3⋊D4 ρ19 2 -2 -2 2 0 0 2 0 0 0 0 2 -2 -2 √-2 √-2 -√-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 √-2 -√-2 √-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ21 2 -2 -2 2 0 0 2 0 0 0 0 2 -2 -2 -√-2 -√-2 √-2 √-2 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 -√-2 √-2 -√-2 √-2 0 0 0 0 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 orthogonal lifted from Q8⋊2S3 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 orthogonal lifted from Q8⋊2S3

Smallest permutation representation of C2×Q82S3
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 48 3 46)(2 47 4 45)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)(21 35 23 33)(22 34 24 36)(25 44 27 42)(26 43 28 41)(29 40 31 38)(30 39 32 37)
(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)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 29)(26 32)(27 31)(28 30)(33 35)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)

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

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

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

Matrix representation of C2×Q82S3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 1 48 0 0 3 72
,
 1 0 0 0 0 1 0 0 0 0 61 4 0 0 55 12
,
 0 1 0 0 72 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 72 72 0 0 0 0 1 0 0 0 3 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,1,3,0,0,48,72],[1,0,0,0,0,1,0,0,0,0,61,55,0,0,4,12],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,1,3,0,0,0,72] >;

C2×Q82S3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_2S_3
% in TeX

G:=Group("C2xQ8:2S3");
// GroupNames label

G:=SmallGroup(96,148);
// by ID

G=gap.SmallGroup(96,148);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,86,579,159,69,2309]);
// Polycyclic

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

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