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G = S3×C4.10D4order 192 = 26·3

Direct product of S3 and C4.10D4

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

Aliases: S3×C4.10D4, M4(2).20D6, (C4×S3).2D4, C12.95(C2×D4), C4.150(S3×D4), (C2×C12).7C23, (C2×Q8).118D6, (C2×Dic6).4C4, (C6×Q8).5C22, C12.10D43C2, (S3×M4(2)).1C2, C12.47D411C2, D6.17(C22⋊C4), C4.Dic3.4C22, Dic3.5(C22⋊C4), (C2×Dic6).46C22, (C3×M4(2)).20C22, (S3×C2×C4).2C4, (C2×S3×Q8).1C2, (C2×C4).6(C4×S3), (C2×C12).6(C2×C4), C31(C2×C4.10D4), (S3×C2×C4).3C22, C22.16(S3×C2×C4), C6.14(C2×C22⋊C4), C2.15(S3×C22⋊C4), (C2×C4).7(C22×S3), (C3×C4.10D4)⋊9C2, (C2×C6).10(C22×C4), (C2×Dic3).3(C2×C4), (C22×S3).56(C2×C4), SmallGroup(192,309)

Series: Derived Chief Lower central Upper central

C1C2×C6 — S3×C4.10D4
C1C3C6C12C2×C12S3×C2×C4C2×S3×Q8 — S3×C4.10D4
C3C6C2×C6 — S3×C4.10D4
C1C2C2×C4C4.10D4

Generators and relations for S3×C4.10D4
 G = < a,b,c,d,e | a3=b2=c4=1, d4=c2, e2=dcd-1=c-1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >

Subgroups: 368 in 146 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], S3, C6, C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], Q8 [×8], C23, Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4 [×3], C2×Q8, C2×Q8 [×7], C3⋊C8 [×2], C24 [×2], Dic6 [×6], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C4.10D4, C4.10D4 [×3], C2×M4(2) [×2], C22×Q8, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3 [×2], C3×M4(2) [×2], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], S3×Q8 [×4], C6×Q8, C2×C4.10D4, C12.47D4 [×2], C12.10D4, C3×C4.10D4, S3×M4(2) [×2], C2×S3×Q8, S3×C4.10D4
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], C22×S3, C4.10D4 [×2], C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], C2×C4.10D4, S3×C22⋊C4, S3×C4.10D4

Smallest permutation representation of S3×C4.10D4
On 48 points
Generators in S48
(1 28 35)(2 29 36)(3 30 37)(4 31 38)(5 32 39)(6 25 40)(7 26 33)(8 27 34)(9 48 19)(10 41 20)(11 42 21)(12 43 22)(13 44 23)(14 45 24)(15 46 17)(16 47 18)
(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 19 21 23)(18 24 22 20)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 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 16 7 10 5 12 3 14)(2 9 4 15 6 13 8 11)(17 40 23 34 21 36 19 38)(18 33 20 39 22 37 24 35)(25 44 27 42 29 48 31 46)(26 41 32 43 30 45 28 47)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D
order12222234444444466888888881212121224242424
size112336222446612122444441212121244888888

33 irreducible representations

dim1111111122222448
type++++++++++-+-
imageC1C2C2C2C2C2C4C4S3D4D6D6C4×S3C4.10D4S3×D4S3×C4.10D4
kernelS3×C4.10D4C12.47D4C12.10D4C3×C4.10D4S3×M4(2)C2×S3×Q8C2×Dic6S3×C2×C4C4.10D4C4×S3M4(2)C2×Q8C2×C4S3C4C1
# reps1211214414214221

Matrix representation of S3×C4.10D4 in GL6(𝔽73)

72720000
100000
001000
000100
000010
000001
,
100000
72720000
001000
000100
000010
000001
,
100000
010000
000100
0072000
0000072
000010
,
7200000
0720000
000010
000001
000100
0072000
,
7200000
0720000
00005316
00001620
00575300
00531600

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,57,53,0,0,0,0,53,16,0,0,53,16,0,0,0,0,16,20,0,0] >;

S3×C4.10D4 in GAP, Magma, Sage, TeX

S_3\times C_4._{10}D_4
% in TeX

G:=Group("S3xC4.10D4");
// GroupNames label

G:=SmallGroup(192,309);
// by ID

G=gap.SmallGroup(192,309);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,58,570,136,438,6278]);
// Polycyclic

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

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