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G = C2xC4oD12  order 96 = 25·3

Direct product of C2 and C4oD12

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

Aliases: C2xC4oD12, C6.4C24, D6.1C23, C23.31D6, D12:12C22, C12.43C23, Dic6:11C22, Dic3.2C23, (C2xC4)oD12, C4o(C2xD12), (C2xC4):10D6, C4o(C2xDic6), (C2xC4)oDic6, C4o(C4oD12), C6:1(C4oD4), (C22xC4):8S3, (C2xD12):14C2, (C4xS3):6C22, (C22xC12):8C2, C3:D4:6C22, C2.5(S3xC23), (C2xC12):13C22, (C2xDic6):15C2, (C2xC6).65C23, C4.43(C22xS3), C22.5(C22xS3), (C22xC6).46C22, (C22xS3).28C22, (C2xDic3).43C22, C4o(C2xC3:D4), C3:1(C2xC4oD4), (S3xC2xC4):15C2, (C2xC4)o(C2xD12), (C2xC4)o(C3:D4), (C2xC4)o(C2xDic6), (C2xC3:D4):12C2, (C2xC4)o(C2xC3:D4), SmallGroup(96,208)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2xC4oD12
Chief series
C1 — C3 — C6 — D6 — C22xS3 — S3xC2xC4 — C2xC4oD12
Lower central
C3 — C6 — C2xC4oD12
Upper central
C1 — C2xC4 — C22xC4

Generators and relations for C2xC4oD12
 G = < a,b,c,d | a2=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 322 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xC6, C2xC4oD4, C2xDic6, S3xC2xC4, C2xD12, C4oD12, C2xC3:D4, C22xC12, C2xC4oD12
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, C4oD12, S3xC23, C2xC4oD12

Smallest permutation representation of C2xC4oD12
►On 48 points
Generators in S48
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)

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

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

C2xC4oD12 is a maximal subgroup of
 D6:C8:C2  D12.32D4  D12:14D4  C4oD12:C4  C4.(C2xD12)  C42:6D6  (C2xD12):13C4  D12:17D4  D12.37D4  (C22xC8):7S3  C23.28D12  D6:C8:40C2  M4(2).31D6  C23.54D12  M4(2):24D6  C42.276D6  C24.38D6  C6.82+ 1+4  C6.2- 1+4  C6.2+ 1+4  C42.188D6  C42.91D6  C42:10D6  C42:11D6  C42.92D6  C42:14D6  C42.228D6  D12:23D4  D12:24D4  Dic6:23D4  Dic6:24D4  Dic6:20D4  C6.382+ 1+4  C6.722- 1+4  D12:20D4  C6.162- 1+4  C6.172- 1+4  D12:22D4  Dic6:22D4  C6.1212+ 1+4  C6.822- 1+4  M4(2):26D6  C24.9C23  C24.83D6  C24.52D6  C6.442- 1+4  C12.C24  (C2xC12):17D4  C6.1082- 1+4  C2xS3xC4oD4  C6.C25
C2xC4oD12 is a maximal quotient of
 C2xC4xDic6  C42.274D6  C2xC4xD12  C42.276D6  C42.277D6  C24.38D6  C24.41D6  C24.42D6  C6.2- 1+4  C6.102+ 1+4  C6.52- 1+4  C6.112+ 1+4  C6.62- 1+4  C42.89D6  C42:12D6  C42.93D6  C42.94D6  C42.95D6  C42.96D6  C42.97D6  C42.98D6  C42.99D6  C42.100D6  C42.102D6  C42.104D6  C42.105D6  C42.106D6  C42:14D6  C42.228D6  D12:23D4  D12:24D4  Dic6:23D4  Dic6:24D4  C42:18D6  C42.229D6  C42.113D6  C42.114D6  C42:19D6  C42.115D6  C42.116D6  C42.117D6  C42.118D6  C42.119D6  Dic6:10Q8  C42.122D6  C42.232D6  D12:10Q8  C42.131D6  C42.132D6  C42.133D6  C42.134D6  C42.135D6  C42.136D6  C2xC4xC3:D4  C24.83D6

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J6A···6G12A···12H
order1222222222344444444446···612···12
size1111226666211112266662···22···2

36 irreducible representations

dim111111122222
type++++++++++
imageC1C2C2C2C2C2C2S3D6D6C4oD4C4oD12
kernelC2xC4oD12C2xDic6S3xC2xC4C2xD12C4oD12C2xC3:D4C22xC12C22xC4C2xC4C23C6C2
# reps112182116148

Matrix representation of C2xC4oD12 ►in GL3(F13) generated by

1200
010
001
,
100
080
008
,
100
0710
0310
,
100
0710
036
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,8,0,0,0,8],[1,0,0,0,7,3,0,10,10],[1,0,0,0,7,3,0,10,6] >;
 

C2xC4oD12 in GAP, Magma, Sage, TeX 

C_2\times C_4\circ D_{12}
 
% in TeX
 
G:=Group("C2xC4oD12");
 
// GroupNames label
 
G:=SmallGroup(96,208);
 
// by ID
 
G=gap.SmallGroup(96,208);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,579,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
 
// generators/relations
 

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