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G = C2×C424S3order 192 = 26·3

Direct product of C2 and C424S3

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

Aliases: C2×C424S3, C4239D6, C61C4≀C2, C4○D121C4, (C2×D12)⋊6C4, (C2×C42)⋊6S3, D1216(C2×C4), (C2×Dic6)⋊6C4, C4.81(C2×D12), (C2×C4).88D12, C4.21(D6⋊C4), (C4×C12)⋊52C22, Dic615(C2×C4), (C2×C12).480D4, C12.301(C2×D4), (C22×C4).433D6, (C22×C6).179D4, C12.45(C22⋊C4), C12.109(C22×C4), (C2×C12).793C23, C22.43(D6⋊C4), C4○D12.37C22, C23.80(C3⋊D4), C4.Dic319C22, (C22×C12).536C22, C32(C2×C4≀C2), (C2×C4×C12)⋊10C2, C4.67(S3×C2×C4), C2.4(C2×D6⋊C4), (C2×C4).107(C4×S3), (C2×C4○D12).3C2, (C2×C6).422(C2×D4), C6.30(C2×C22⋊C4), (C2×C4.Dic3)⋊3C2, (C2×C12).222(C2×C4), C22.25(C2×C3⋊D4), (C2×C4).236(C3⋊D4), (C2×C6).55(C22⋊C4), (C2×C4).707(C22×S3), SmallGroup(192,486)

Series: Derived Chief Lower central Upper central

C1C12 — C2×C424S3
C1C3C6C2×C6C2×C12C4○D12C2×C4○D12 — C2×C424S3
C3C6C12 — C2×C424S3
C1C2×C4C22×C4C2×C42

Generators and relations for C2×C424S3
 G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, ebe=bc=cb, bd=db, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 440 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×6], C22 [×3], C22 [×6], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×7], Q8 [×3], C23, C23, Dic3 [×2], C12 [×4], C12 [×4], D6 [×4], C2×C6 [×3], C2×C6 [×2], C42 [×2], C42, C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], C3⋊C8 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×6], C22×S3, C22×C6, C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C4×C12 [×2], C4×C12, C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C22×C12, C2×C4≀C2, C424S3 [×4], C2×C4.Dic3, C2×C4×C12, C2×C4○D12, C2×C424S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C4≀C2 [×2], C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C4≀C2, C424S3 [×2], C2×D6⋊C4, C2×C424S3

Smallest permutation representation of C2×C424S3
On 48 points
Generators in S48
(1 7)(2 8)(3 5)(4 6)(9 19)(10 20)(11 14)(12 13)(15 21)(16 22)(17 23)(18 24)(25 42)(26 43)(27 44)(28 41)(29 35)(30 36)(31 33)(32 34)(37 48)(38 45)(39 46)(40 47)
(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 23 8 18)(2 24 7 17)(3 9 6 20)(4 10 5 19)(11 16 13 21)(12 15 14 22)(25 43 27 41)(26 44 28 42)(29 36 31 34)(30 33 32 35)(37 45 39 47)(38 46 40 48)
(1 16 5)(2 15 6)(3 7 22)(4 8 21)(9 17 12)(10 18 11)(13 19 23)(14 20 24)(25 34 38)(26 35 39)(27 36 40)(28 33 37)(29 46 43)(30 47 44)(31 48 41)(32 45 42)
(1 26)(2 41)(3 29)(4 33)(5 35)(6 31)(7 43)(8 28)(9 34)(10 30)(11 47)(12 38)(13 45)(14 40)(15 48)(16 39)(17 25)(18 44)(19 32)(20 36)(21 37)(22 46)(23 42)(24 27)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E···4N4O4P6A···6G8A8B8C8D12A···12X
order12222222344444···4446···6888812···12
size1111221212211112···212122···2121212122···2

60 irreducible representations

dim1111111122222222222
type+++++++++++
imageC1C2C2C2C2C4C4C4S3D4D4D6D6C4×S3D12C3⋊D4C3⋊D4C4≀C2C424S3
kernelC2×C424S3C424S3C2×C4.Dic3C2×C4×C12C2×C4○D12C2×Dic6C2×D12C4○D12C2×C42C2×C12C22×C6C42C22×C4C2×C4C2×C4C2×C4C23C6C2
# reps14111224131214422816

Matrix representation of C2×C424S3 in GL3(𝔽73) generated by

7200
0720
0072
,
4600
010
0027
,
100
0270
0046
,
100
080
0064
,
7200
0072
0720
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[46,0,0,0,1,0,0,0,27],[1,0,0,0,27,0,0,0,46],[1,0,0,0,8,0,0,0,64],[72,0,0,0,0,72,0,72,0] >;

C2×C424S3 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_4S_3
% in TeX

G:=Group("C2xC4^2:4S3");
// GroupNames label

G:=SmallGroup(192,486);
// by ID

G=gap.SmallGroup(192,486);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1123,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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