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G = C2×D42S3order 96 = 25·3

Direct product of C2 and D42S3

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

Aliases: C2×D42S3, D45D6, C6.6C24, D6.2C23, C23.24D6, C12.20C23, Dic67C22, Dic3.3C23, (C6×D4)⋊6C2, (C2×D4)⋊8S3, D4(C2×Dic3), Dic3(C2×D4), C62(C4○D4), (C2×C4).60D6, (C4×S3)⋊4C22, (C3×D4)⋊6C22, C3⋊D42C22, (C2×C6).1C23, C2.7(S3×C23), (C2×Dic6)⋊12C2, C4.20(C22×S3), (C2×C12).45C22, (C2×Dic3)⋊9C22, (C22×Dic3)⋊8C2, C22.1(C22×S3), (C22×C6).23C22, (C22×S3).29C22, (S3×C2×C4)⋊4C2, C32(C2×C4○D4), (C2×C3⋊D4)⋊10C2, SmallGroup(96,210)

Series: Derived Chief Lower central Upper central

C1C6 — C2×D42S3
C1C3C6D6C22×S3S3×C2×C4 — C2×D42S3
C3C6 — C2×D42S3
C1C22C2×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 12A2B2C2D2E2F2G2H2I34A4B4C4D4E4F4G4H4I4J6A6B6C6D6E6F6G12A12B
 size 111122226622233336666222444444
ρ1111111111111111111111111111111    trivial
ρ211111111-1-1111-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ311-1-1-111-11-11-11-11-11-111-1-1-1111-1-1-11    linear of order 2
ρ411-1-1-111-1-111-111-11-11-1-11-1-1111-1-1-11    linear of order 2
ρ511-1-11-1-111-11-11-11-111-1-11-1-11-1-111-11    linear of order 2
ρ611-1-11-1-11-111-111-11-1-111-1-1-11-1-111-11    linear of order 2
ρ71111-1-1-1-1111111111-1-1-1-1111-1-1-1-111    linear of order 2
ρ81111-1-1-1-1-1-1111-1-1-1-11111111-1-1-1-111    linear of order 2
ρ91111-11-11111-1-1-1-1-1-111-1-11111-11-1-1-1    linear of order 2
ρ101111-11-11-1-11-1-11111-1-1111111-11-1-1-1    linear of order 2
ρ1111-1-111-1-11-111-11-11-1-11-11-1-111-1-111-1    linear of order 2
ρ1211-1-111-1-1-1111-1-11-111-11-1-1-111-1-111-1    linear of order 2
ρ1311-1-1-1-1111-111-11-11-11-11-1-1-11-111-11-1    linear of order 2
ρ1411-1-1-1-111-1111-1-11-11-11-11-1-11-111-11-1    linear of order 2
ρ1511111-11-1111-1-1-1-1-1-1-1-111111-11-11-1-1    linear of order 2
ρ1611111-11-1-1-11-1-1111111-1-1111-11-11-1-1    linear of order 2
ρ1722-2-22-2-2200-1-220000000011-111-1-11-1    orthogonal lifted from D6
ρ182222-22-2200-1-2-200000000-1-1-1-11-1111    orthogonal lifted from D6
ρ192222222200-12200000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ2022-2-2-2-22200-12-20000000011-11-1-11-11    orthogonal lifted from D6
ρ212222-2-2-2-200-12200000000-1-1-11111-1-1    orthogonal lifted from D6
ρ2222-2-222-2-200-12-20000000011-1-111-1-11    orthogonal lifted from D6
ρ2322-2-2-222-200-1-220000000011-1-1-1111-1    orthogonal lifted from D6
ρ2422222-22-200-1-2-200000000-1-1-11-11-111    orthogonal lifted from D6
ρ252-2-220000002002i2i-2i-2i0000-22-2000000    complex lifted from C4○D4
ρ262-2-22000000200-2i-2i2i2i0000-22-2000000    complex lifted from C4○D4
ρ272-22-20000002002i-2i-2i2i00002-2-2000000    complex lifted from C4○D4
ρ282-22-2000000200-2i2i2i-2i00002-2-2000000    complex lifted from C4○D4
ρ294-44-4000000-20000000000-222000000    symplectic lifted from D42S3, Schur index 2
ρ304-4-44000000-200000000002-22000000    symplectic lifted from D42S3, 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)])

C2×D42S3 is a maximal subgroup of
C23⋊C45S3  M4(2).19D6  D4⋊(C4×S3)  D42S3⋊C4  D43D12  D4.D12  Dic6⋊D4  Dic6.16D4  C42.108D6  D45D12  D46D12  C24.67D6  C24.44D6  C24.45D6  C12⋊(C4○D4)  C6.322+ 1+4  Dic619D4  Dic620D4  C4⋊C421D6  C6.722- 1+4  C6.402+ 1+4  C6.732- 1+4  C6.792- 1+4  C6.822- 1+4  C4⋊C428D6  C42.233D6  C42.141D6  Dic610D4  C4228D6  C42.238D6  Dic611D4  D84D6  C24.53D6  C6.1042- 1+4  C2×S3×C4○D4  D6.C24
C2×D42S3 is a maximal quotient of
C24.42D6  C6.52- 1+4  C42.102D6  C42.105D6  C42.106D6  D46Dic6  D46D12  C42.229D6  C42.117D6  C42.119D6  C24.67D6  C24.43D6  C24.44D6  C24.46D6  C12⋊(C4○D4)  Dic619D4  C4⋊C4.178D6  C6.342+ 1+4  C6.702- 1+4  C6.712- 1+4  C4⋊C421D6  C6.732- 1+4  C6.432+ 1+4  C6.452+ 1+4  C6.462+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  (Q8×Dic3)⋊C2  C4⋊C4.187D6  C6.152- 1+4  C6.1182+ 1+4  C6.212- 1+4  C6.232- 1+4  C6.772- 1+4  C6.242- 1+4  C4⋊C4.197D6  C6.802- 1+4  C6.1222+ 1+4  C6.852- 1+4  C42.139D6  C42.234D6  C42.143D6  C42.144D6  C42.166D6  C42.238D6  Dic611D4  C42.168D6  Dic68Q8  C42.241D6  C42.176D6  C42.177D6  C2×D4×Dic3  C24.53D6

Matrix representation of C2×D42S3 in GL5(𝔽13)

120000
01000
00100
000120
000012
,
10000
012000
001200
00050
00008
,
10000
012000
001200
00008
00050
,
10000
012100
012000
00010
00001
,
120000
00100
01000
00010
000012

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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Character table of C2×D42S3 in TeX

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