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G = S3×C22×C8  order 192 = 26·3

Direct product of C22×C8 and S3

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

Aliases: S3×C22×C8, C24⋊12C23, C12.65C24, C3⋊1(C23×C8), C3⋊C8⋊14C23, C6⋊1(C22×C8), (C22×C24)⋊20C2, (C2×C24)⋊49C22, C4.64(S3×C23), C6.28(C23×C4), C23.70(C4×S3), (C4×S3).39C23, (S3×C23).10C4, D6.25(C22×C4), (C22×C4).485D6, C12.144(C22×C4), (C2×C12).878C23, (C22×Dic3).20C4, Dic3.26(C22×C4), (C22×C12).566C22, (C2×C6)⋊6(C2×C8), (S3×C2×C4).25C4, C4.119(S3×C2×C4), C2.2(S3×C22×C4), (C22×C3⋊C8)⋊24C2, (C2×C3⋊C8)⋊49C22, C22.74(S3×C2×C4), (C4×S3).39(C2×C4), (C2×C4).186(C4×S3), (S3×C22×C4).25C2, (C2×C12).256(C2×C4), (S3×C2×C4).309C22, (C22×S3).75(C2×C4), (C2×C4).822(C22×S3), (C2×C6).154(C22×C4), (C22×C6).101(C2×C4), (C2×Dic3).112(C2×C4), SmallGroup(192,1295)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — S3×C22×C8
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C22×C4 — S3×C22×C8
Lower central
C3 — S3×C22×C8
Upper central
C1 — C22×C8

Generators and relations for S3×C22×C8
 G = < a,b,c,d,e | a2=b2=c8=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 600 in 338 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C8, C2×C8, C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C22×C8, C22×C8, C23×C4, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C23×C8, S3×C2×C8, C22×C3⋊C8, C22×C24, S3×C22×C4, S3×C22×C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C24, C4×S3, C22×S3, C22×C8, C23×C4, S3×C8, S3×C2×C4, S3×C23, C23×C8, S3×C2×C8, S3×C22×C4, S3×C22×C8

Smallest permutation representation of S3×C22×C8
►On 96 points
Generators in S96
(1 80)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 65)(32 66)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(41 93)(42 94)(43 95)(44 96)(45 89)(46 90)(47 91)(48 92)

(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 73)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)

(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)

(1 28 36)(2 29 37)(3 30 38)(4 31 39)(5 32 40)(6 25 33)(7 26 34)(8 27 35)(9 96 20)(10 89 21)(11 90 22)(12 91 23)(13 92 24)(14 93 17)(15 94 18)(16 95 19)(41 59 51)(42 60 52)(43 61 53)(44 62 54)(45 63 55)(46 64 56)(47 57 49)(48 58 50)(65 83 75)(66 84 76)(67 85 77)(68 86 78)(69 87 79)(70 88 80)(71 81 73)(72 82 74)

(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 73)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
 
G:=sub<Sym(96)| (1,80)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(33,85)(34,86)(35,87)(36,88)(37,81)(38,82)(39,83)(40,84)(41,93)(42,94)(43,95)(44,96)(45,89)(46,90)(47,91)(48,92), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,73)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,28,36)(2,29,37)(3,30,38)(4,31,39)(5,32,40)(6,25,33)(7,26,34)(8,27,35)(9,96,20)(10,89,21)(11,90,22)(12,91,23)(13,92,24)(14,93,17)(15,94,18)(16,95,19)(41,59,51)(42,60,52)(43,61,53)(44,62,54)(45,63,55)(46,64,56)(47,57,49)(48,58,50)(65,83,75)(66,84,76)(67,85,77)(68,86,78)(69,87,79)(70,88,80)(71,81,73)(72,82,74), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,73)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94)>;
 
G:=Group( (1,80)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(33,85)(34,86)(35,87)(36,88)(37,81)(38,82)(39,83)(40,84)(41,93)(42,94)(43,95)(44,96)(45,89)(46,90)(47,91)(48,92), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,73)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,28,36)(2,29,37)(3,30,38)(4,31,39)(5,32,40)(6,25,33)(7,26,34)(8,27,35)(9,96,20)(10,89,21)(11,90,22)(12,91,23)(13,92,24)(14,93,17)(15,94,18)(16,95,19)(41,59,51)(42,60,52)(43,61,53)(44,62,54)(45,63,55)(46,64,56)(47,57,49)(48,58,50)(65,83,75)(66,84,76)(67,85,77)(68,86,78)(69,87,79)(70,88,80)(71,81,73)(72,82,74), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,73)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94) );
 
G=PermutationGroup([[(1,80),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,54),(10,55),(11,56),(12,49),(13,50),(14,51),(15,52),(16,53),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,65),(32,66),(33,85),(34,86),(35,87),(36,88),(37,81),(38,82),(39,83),(40,84),(41,93),(42,94),(43,95),(44,96),(45,89),(46,90),(47,91),(48,92)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,73),(17,87),(18,88),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,28,36),(2,29,37),(3,30,38),(4,31,39),(5,32,40),(6,25,33),(7,26,34),(8,27,35),(9,96,20),(10,89,21),(11,90,22),(12,91,23),(13,92,24),(14,93,17),(15,94,18),(16,95,19),(41,59,51),(42,60,52),(43,61,53),(44,62,54),(45,63,55),(46,64,56),(47,57,49),(48,58,50),(65,83,75),(66,84,76),(67,85,77),(68,86,78),(69,87,79),(70,88,80),(71,81,73),(72,82,74)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,73),(17,69),(18,70),(19,71),(20,72),(21,65),(22,66),(23,67),(24,68),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(81,95),(82,96),(83,89),(84,90),(85,91),(86,92),(87,93),(88,94)]])
 

96 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P6A···6G8A···8P8Q···8AF12A···12H24A···24P
order12···22···234···44···46···68···88···812···1224···24
size11···13···321···13···32···21···13···32···22···2

96 irreducible representations

dim111111111222222
type++++++++
imageC1C2C2C2C2C4C4C4C8S3D6D6C4×S3C4×S3S3×C8
kernelS3×C22×C8S3×C2×C8C22×C3⋊C8C22×C24S3×C22×C4S3×C2×C4C22×Dic3S3×C23C22×S3C22×C8C2×C8C22×C4C2×C4C23C22
# reps1121111222321616216

Matrix representation of S3×C22×C8 ►in GL4(𝔽73) generated by

72000
0100
00720
00072
,
1000
07200
00720
00072
,
63000
04600
0010
0001
,
1000
0100
00072
00172
,
72000
0100
00072
00720
G:=sub<GL(4,GF(73))| [72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[63,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[72,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0] >;
 

S3×C22×C8 in GAP, Magma, Sage, TeX 

S_3\times C_2^2\times C_8
 
% in TeX
 
G:=Group("S3xC2^2xC8");
 
// GroupNames label
 
G:=SmallGroup(192,1295);
 
// by ID
 
G=gap.SmallGroup(192,1295);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,80,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^3=e^2=1,a*b=b*a,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*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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