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G = C2xD4.S3  order 96 = 25·3

Direct product of C2 and D4.S3

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

Aliases: C2xD4.S3, D4.7D6, C6:2SD16, C12.16D4, C12.13C23, Dic6:6C22, C3:C8:8C22, C3:3(C2xSD16), (C2xD4).4S3, (C6xD4).3C2, (C2xC6).40D4, (C2xC4).48D6, C6.46(C2xD4), (C2xDic6):9C2, C4.6(C3:D4), C4.13(C22xS3), (C3xD4).7C22, (C2xC12).31C22, C22.22(C3:D4), (C2xC3:C8):5C2, C2.10(C2xC3:D4), SmallGroup(96,140)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C2xD4.S3
Chief series
C1 — C3 — C6 — C12 — Dic6 — C2xDic6 — C2xD4.S3
Lower central
C3 — C6 — C12 — C2xD4.S3
Upper central
C1 — C22 — C2xC4 — C2xD4

Generators and relations for C2xD4.S3
 G = < a,b,c,d,e | a2=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 146 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, C12, C2xC6, C2xC6, C2xC8, SD16, C2xD4, C2xQ8, C3:C8, Dic6, Dic6, C2xDic3, C2xC12, C3xD4, C3xD4, C22xC6, C2xSD16, C2xC3:C8, D4.S3, C2xDic6, C6xD4, C2xD4.S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C3:D4, C22xS3, C2xSD16, D4.S3, C2xC3:D4, C2xD4.S3

Character table of C2xD4.S3

 class 12A2B2C2D2E34A4B4C4D6A6B6C6D6E6F6G8A8B8C8D12A12B
 size 11114422212122224444666644
ρ1111111111111111111111111    trivial
ρ211-1-11-111-11-1-11-1-1-111-111-11-1    linear of order 2
ρ311-1-11-111-1-11-11-1-1-1111-1-111-1    linear of order 2
ρ4111111111-1-11111111-1-1-1-111    linear of order 2
ρ511-1-1-1111-11-1-11-111-1-11-1-111-1    linear of order 2
ρ61111-1-111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ71111-1-1111-1-1111-1-1-1-1111111    linear of order 2
ρ811-1-1-1111-1-11-11-111-1-1-111-11-1    linear of order 2
ρ92222002-2-20022200000000-2-2    orthogonal lifted from D4
ρ10222222-12200-1-1-1-1-1-1-10000-1-1    orthogonal lifted from S3
ρ1122-2-2-22-12-2001-11-1-1110000-11    orthogonal lifted from D6
ρ1222-2-2002-2200-22-200000000-22    orthogonal lifted from D4
ρ132222-2-2-12200-1-1-111110000-1-1    orthogonal lifted from D6
ρ1422-2-22-2-12-2001-1111-1-10000-11    orthogonal lifted from D6
ρ1522-2-200-1-22001-11-√-3√-3√-3-√-300001-1    complex lifted from C3:D4
ρ16222200-1-2-200-1-1-1-√-3√-3-√-3√-3000011    complex lifted from C3:D4
ρ17222200-1-2-200-1-1-1√-3-√-3√-3-√-3000011    complex lifted from C3:D4
ρ1822-2-200-1-22001-11√-3-√-3-√-3√-300001-1    complex lifted from C3:D4
ρ192-22-20020000-2-220000-√-2-√-2√-2√-200    complex lifted from SD16
ρ202-2-2200200002-2-20000-√-2√-2-√-2√-200    complex lifted from SD16
ρ212-2-2200200002-2-20000√-2-√-2√-2-√-200    complex lifted from SD16
ρ222-22-20020000-2-220000√-2√-2-√-2-√-200    complex lifted from SD16
ρ234-4-4400-20000-2220000000000    symplectic lifted from D4.S3, Schur index 2
ρ244-44-400-2000022-20000000000    symplectic lifted from D4.S3, Schur index 2

Smallest permutation representation of C2xD4.S3
►On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 21)(5 39)(6 40)(7 37)(8 38)(9 43)(10 44)(11 41)(12 42)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 45)(34 46)(35 47)(36 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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 26)(27 28)(29 32)(30 31)(34 36)(38 40)(41 43)(46 48)

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

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

C2xD4.S3 is a maximal subgroup of
 D12.2D4  D4.S3:C4  Dic3:6SD16  Dic6:2D4  Dic6.D4  D6:5SD16  D6:SD16  C3:C8:1D4  D4.D12  D4.1D12  C42.51D6  D4.2D12  D12:17D4  Dic6:17D4  C3:C8:23D4  C3:C8:5D4  C42.61D6  C42.214D6  C42.65D6  C42.74D6  Dic6:9D4  C12:4SD16  (C6xD8).C2  C24:11D4  C24.22D4  Dic6:D4  Dic3:3SD16  C24.31D4  D6:8SD16  C24:15D4  M4(2).13D6  (C3xD4).31D4  (C3xD4).32D4  C2xS3xSD16  D8:6D6  D12.33C23
C2xD4.S3 is a maximal quotient of
 C4:C4.231D6  C12.38SD16  D4.2D12  C4:D4.S3  Dic6:17D4  C3:C8:23D4  C12.16D8  Dic6:9D4  C12:4SD16  C12.SD16  C12.Q16  Dic6:6Q8  (C3xD4).31D4

Matrix representation of C2xD4.S3 ►in GL4(F73) generated by

72000
07200
00720
00072
,
72000
07200
0001
00720
,
72000
0100
0001
0010
,
64000
0800
0010
0001
,
07200
72000
00676
0066
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,67,6,0,0,6,6] >;
 

C2xD4.S3 in GAP, Magma, Sage, TeX 

C_2\times D_4.S_3
 
% in TeX
 
G:=Group("C2xD4.S3");
 
// GroupNames label
 
G:=SmallGroup(96,140);
 
// by ID
 
G=gap.SmallGroup(96,140);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,579,159,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of C2xD4.S3 in TeX

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