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G = C2×Q82S3order 96 = 25·3

Direct product of C2 and Q82S3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q82S3, Q84D6, C63SD16, C12.18D4, C12.14C23, D12.9C22, C3⋊C89C22, (C2×Q8)⋊3S3, (C6×Q8)⋊1C2, C34(C2×SD16), C6.53(C2×D4), (C2×C6).41D4, (C2×C4).53D6, (C2×D12).8C2, C4.8(C3⋊D4), (C3×Q8)⋊3C22, C4.14(C22×S3), (C2×C12).36C22, C22.23(C3⋊D4), (C2×C3⋊C8)⋊6C2, C2.17(C2×C3⋊D4), SmallGroup(96,148)

Series: Derived Chief Lower central Upper central

C1C12 — C2×Q82S3
C1C3C6C12D12C2×D12 — C2×Q82S3
C3C6C12 — C2×Q82S3
C1C22C2×C4C2×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 12A2B2C2D2E34A4B4C4D6A6B6C8A8B8C8D12A12B12C12D12E12F
 size 11111212222442226666444444
ρ1111111111111111111111111    trivial
ρ211-1-1-111-11-11-11-1-111-111-11-1-1    linear of order 2
ρ3111111111-1-1111-1-1-1-11-1-1-1-11    linear of order 2
ρ411-1-1-111-111-1-11-11-1-111-11-11-1    linear of order 2
ρ51111-1-111111111-1-1-1-1111111    linear of order 2
ρ611-1-11-11-11-11-11-11-1-1111-11-1-1    linear of order 2
ρ71111-1-1111-1-111111111-1-1-1-11    linear of order 2
ρ811-1-11-11-111-1-11-1-111-11-11-11-1    linear of order 2
ρ9222200-122-2-2-1-1-10000-11111-1    orthogonal lifted from D6
ρ1022-2-200-1-222-21-110000-11-11-11    orthogonal lifted from D6
ρ11222200-12222-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-2-20022-200-22-20000-200002    orthogonal lifted from D4
ρ132222002-2-2002220000-20000-2    orthogonal lifted from D4
ρ1422-2-200-1-22-221-110000-1-11-111    orthogonal lifted from D6
ρ1522-2-200-12-2001-1100001--3--3-3-3-1    complex lifted from C3⋊D4
ρ1622-2-200-12-2001-1100001-3-3--3--3-1    complex lifted from C3⋊D4
ρ17222200-1-2-200-1-1-100001-3--3--3-31    complex lifted from C3⋊D4
ρ18222200-1-2-200-1-1-100001--3-3-3--31    complex lifted from C3⋊D4
ρ192-2-2200200002-2-2-2-2--2--2000000    complex lifted from SD16
ρ202-22-20020000-2-22-2--2-2--2000000    complex lifted from SD16
ρ212-2-2200200002-2-2--2--2-2-2000000    complex lifted from SD16
ρ222-22-20020000-2-22--2-2--2-2000000    complex lifted from SD16
ρ234-4-4400-20000-2220000000000    orthogonal lifted from Q82S3
ρ244-44-400-2000022-20000000000    orthogonal lifted from Q82S3

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)])

C2×Q82S3 is a maximal subgroup of
D12.6D4  Dic37SD16  Q83D12  D62SD16  Q84D12  C3⋊(C8⋊D4)  Q83(C4×S3)  Dic3⋊SD16  D12.12D4  C42.56D6  Q82D12  Q8.6D12  D12.36D4  D12.37D4  C3⋊C824D4  C3⋊C86D4  D12.23D4  C42.64D6  C42.214D6  C125SD16  C126SD16  C42.80D6  Dic35SD16  D66SD16  C2415D4  C249D4  (C2×Q16)⋊S3  D12.17D4  C24.37D4  C24.28D4  M4(2).15D6  (C3×Q8)⋊13D4  (C3×D4)⋊14D4  C2×S3×SD16  C24.C23  D12.34C23
C2×Q82S3 is a maximal quotient of
C4⋊C4.228D6  Q84Dic6  Q82D12  (C2×Q8).49D6  D12.36D4  C3⋊C824D4  C12.9Q16  C12.SD16  C125SD16  D125Q8  C126SD16  C12.D8  (C3×Q8)⋊13D4

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

72000
07200
00720
00072
,
1000
0100
00148
00372
,
1000
0100
00614
005512
,
0100
727200
0010
0001
,
1000
727200
0010
00372
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

Export

Character table of C2×Q82S3 in TeX

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