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## G = C2×C42⋊4S3order 192 = 26·3

### Direct product of C2 and C42⋊4S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C42⋊4S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4○D12 — C2×C4○D12 — C2×C42⋊4S3
 Lower central C3 — C6 — C12 — C2×C42⋊4S3
 Upper central C1 — C2×C4 — C22×C4 — C2×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 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E ··· 4N 4O 4P 6A ··· 6G 8A 8B 8C 8D 12A ··· 12X order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 12 12 2 1 1 1 1 2 ··· 2 12 12 2 ··· 2 12 12 12 12 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 C3⋊D4 C4≀C2 C42⋊4S3 kernel C2×C42⋊4S3 C42⋊4S3 C2×C4.Dic3 C2×C4×C12 C2×C4○D12 C2×Dic6 C2×D12 C4○D12 C2×C42 C2×C12 C22×C6 C42 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 2 2 4 1 3 1 2 1 4 4 2 2 8 16

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

 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
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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