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G = C2×C4○D12order 96 = 25·3

Direct product of C2 and C4○D12

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

Aliases: C2×C4○D12, C6.4C24, D6.1C23, C23.31D6, D1212C22, C12.43C23, Dic611C22, Dic3.2C23, (C2×C4)D12, C4(C2×D12), (C2×C4)⋊10D6, C4(C2×Dic6), (C2×C4)Dic6, C4(C4○D12), C61(C4○D4), (C22×C4)⋊8S3, (C2×D12)⋊14C2, (C4×S3)⋊6C22, (C22×C12)⋊8C2, C3⋊D46C22, C2.5(S3×C23), (C2×C12)⋊13C22, (C2×Dic6)⋊15C2, (C2×C6).65C23, C4.43(C22×S3), C22.5(C22×S3), (C22×C6).46C22, (C22×S3).28C22, (C2×Dic3).43C22, C4(C2×C3⋊D4), C31(C2×C4○D4), (S3×C2×C4)⋊15C2, (C2×C4)(C2×D12), (C2×C4)(C3⋊D4), (C2×C4)(C2×Dic6), (C2×C3⋊D4)⋊12C2, (C2×C4)(C2×C3⋊D4), SmallGroup(96,208)

Series: Derived Chief Lower central Upper central

C1C6 — C2×C4○D12
C1C3C6D6C22×S3S3×C2×C4 — C2×C4○D12
C3C6 — C2×C4○D12
C1C2×C4C22×C4

Generators and relations for C2×C4○D12
 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 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C2×C4○D4, C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C2×C4○D12
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, C4○D12 [×2], S3×C23, C2×C4○D12

Smallest permutation representation of C2×C4○D12
On 48 points
Generators in S48
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 47 19 41)(14 48 20 42)(15 37 21 43)(16 38 22 44)(17 39 23 45)(18 40 24 46)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

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

C2×C4○D12 is a maximal subgroup of
D6⋊C8⋊C2  D12.32D4  D1214D4  C4○D12⋊C4  C4.(C2×D12)  C426D6  (C2×D12)⋊13C4  D1217D4  D12.37D4  (C22×C8)⋊7S3  C23.28D12  D6⋊C840C2  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  C4210D6  C4211D6  C42.92D6  C4214D6  C42.228D6  D1223D4  D1224D4  Dic623D4  Dic624D4  Dic620D4  C6.382+ 1+4  C6.722- 1+4  D1220D4  C6.162- 1+4  C6.172- 1+4  D1222D4  Dic622D4  C6.1212+ 1+4  C6.822- 1+4  M4(2)⋊26D6  C24.9C23  C24.83D6  C24.52D6  C6.442- 1+4  C12.C24  (C2×C12)⋊17D4  C6.1082- 1+4  C2×S3×C4○D4  C6.C25
C2×C4○D12 is a maximal quotient of
C2×C4×Dic6  C42.274D6  C2×C4×D12  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  C4212D6  C42.93D6  C42.94D6  C42.95D6  C42.96D6  C42.97D6  C42.98D6  C42.99D6  C42.100D6  C42.102D6  C42.104D6  C42.105D6  C42.106D6  C4214D6  C42.228D6  D1223D4  D1224D4  Dic623D4  Dic624D4  C4218D6  C42.229D6  C42.113D6  C42.114D6  C4219D6  C42.115D6  C42.116D6  C42.117D6  C42.118D6  C42.119D6  Dic610Q8  C42.122D6  C42.232D6  D1210Q8  C42.131D6  C42.132D6  C42.133D6  C42.134D6  C42.135D6  C42.136D6  C2×C4×C3⋊D4  C24.83D6

36 conjugacy classes

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

36 irreducible representations

dim111111122222
type++++++++++
imageC1C2C2C2C2C2C2S3D6D6C4○D4C4○D12
kernelC2×C4○D12C2×Dic6S3×C2×C4C2×D12C4○D12C2×C3⋊D4C22×C12C22×C4C2×C4C23C6C2
# reps112182116148

Matrix representation of C2×C4○D12 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] >;

C2×C4○D12 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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