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G = C2×D4⋊S3order 96 = 25·3

Direct product of C2 and D4⋊S3

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

Aliases: C2×D4⋊S3, C62D8, D43D6, C12.14D4, D125C22, C12.11C23, C33(C2×D8), (C6×D4)⋊1C2, (C2×D4)⋊1S3, C3⋊C87C22, (C2×D12)⋊8C2, C6.44(C2×D4), (C2×C6).38D4, (C2×C4).47D6, (C3×D4)⋊3C22, C4.5(C3⋊D4), C4.11(C22×S3), (C2×C12).29C22, C22.21(C3⋊D4), (C2×C3⋊C8)⋊4C2, C2.8(C2×C3⋊D4), SmallGroup(96,138)

Series: Derived Chief Lower central Upper central

C1C12 — C2×D4⋊S3
C1C3C6C12D12C2×D12 — C2×D4⋊S3
C3C6C12 — C2×D4⋊S3
C1C22C2×C4C2×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 12A2B2C2D2E2F2G34A4B6A6B6C6D6E6F6G8A8B8C8D12A12B
 size 11114412122222224444666644
ρ1111111111111111111111111    trivial
ρ21111-1-1-1-1111111-1-1-1-1111111    linear of order 2
ρ31-1-11-111-11-11-11-1-1-111-111-11-1    linear of order 2
ρ41-1-111-1-111-11-11-111-1-1-111-11-1    linear of order 2
ρ5111111-1-11111111111-1-1-1-111    linear of order 2
ρ61111-1-111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ71-1-11-11-111-11-11-1-1-1111-1-111-1    linear of order 2
ρ81-1-111-11-11-11-11-111-1-11-1-111-1    linear of order 2
ρ92-2-222-200-1-221-11-1-1110000-11    orthogonal lifted from D6
ρ1022222200-122-1-1-1-1-1-1-10000-1-1    orthogonal lifted from S3
ρ112-2-22-2200-1-221-1111-1-10000-11    orthogonal lifted from D6
ρ122-2-22000022-2-22-200000000-22    orthogonal lifted from D4
ρ13222200002-2-222200000000-2-2    orthogonal lifted from D4
ρ142222-2-200-122-1-1-111110000-1-1    orthogonal lifted from D6
ρ152-22-200002002-2-200002-22-200    orthogonal lifted from D8
ρ1622-2-20000200-2-22000022-2-200    orthogonal lifted from D8
ρ1722-2-20000200-2-220000-2-22200    orthogonal lifted from D8
ρ182-22-200002002-2-20000-22-2200    orthogonal lifted from D8
ρ1922220000-1-2-2-1-1-1--3-3--3-3000011    complex lifted from C3⋊D4
ρ2022220000-1-2-2-1-1-1-3--3-3--3000011    complex lifted from C3⋊D4
ρ212-2-220000-12-21-11--3-3-3--300001-1    complex lifted from C3⋊D4
ρ222-2-220000-12-21-11-3--3--3-300001-1    complex lifted from C3⋊D4
ρ2344-4-40000-20022-20000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ244-44-40000-200-2220000000000    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)])

C2×D4⋊S3 is a maximal subgroup of
D12.3D4  Dic34D8  D4⋊D12  D6⋊D8  D43D12  C3⋊C8⋊D4  D4⋊S3⋊C4  D123D4  D12.D4  C42.48D6  C127D8  D4.1D12  D1216D4  D1217D4  C3⋊C822D4  C4⋊D4⋊S3  D12.23D4  C42.64D6  C42.214D6  C122D8  C12⋊D8  C42.74D6  Dic3⋊D8  C245D4  C2411D4  D12⋊D4  (C3×D4).D4  C24.43D4  D127D4  C249D4  M4(2).D6  (C2×C6)⋊8D8  (C3×D4)⋊14D4  C2×S3×D8  D85D6  D12.32C23
C2×D4⋊S3 is a maximal quotient of
(C2×C6).40D8  C12.50D8  C127D8  (C2×C6).D8  D1216D4  C3⋊C822D4  C12.16D8  C122D8  C12⋊D8  C12.17D8  D126Q8  C12.D8  D8.D6  C24.27C23  Q16⋊D6  Q16.D6  D8.9D6  (C2×C6)⋊8D8

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

72000
07200
0010
0001
,
1000
0100
0001
00720
,
1000
0100
005716
001616
,
72100
72000
0010
0001
,
07200
72000
0010
00072
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

Export

Character table of C2×D4⋊S3 in TeX

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