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## G = S3×C23×C4order 192 = 26·3

### Direct product of C23×C4 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C23×C4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C24 — S3×C23×C4
 Lower central C3 — S3×C23×C4
 Upper central C1 — C23×C4

Generators and relations for S3×C23×C4
G = < a,b,c,d,e,f | a2=b2=c2=d4=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2488 in 1362 conjugacy classes, 799 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C22×C4, C22×C4, C24, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C23×C4, C23×C4, C25, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C23×C6, C24×C4, S3×C22×C4, C23×Dic3, C23×C12, S3×C24, S3×C23×C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, C25, S3×C2×C4, S3×C23, C24×C4, S3×C22×C4, S3×C24, S3×C23×C4

Smallest permutation representation of S3×C23×C4
On 96 points
Generators in S96
(1 60)(2 57)(3 58)(4 59)(5 51)(6 52)(7 49)(8 50)(9 55)(10 56)(11 53)(12 54)(13 63)(14 64)(15 61)(16 62)(17 67)(18 68)(19 65)(20 66)(21 71)(22 72)(23 69)(24 70)(25 75)(26 76)(27 73)(28 74)(29 79)(30 80)(31 77)(32 78)(33 83)(34 84)(35 81)(36 82)(37 87)(38 88)(39 85)(40 86)(41 91)(42 92)(43 89)(44 90)(45 95)(46 96)(47 93)(48 94)
(1 48)(2 45)(3 46)(4 47)(5 63)(6 64)(7 61)(8 62)(9 67)(10 68)(11 65)(12 66)(13 51)(14 52)(15 49)(16 50)(17 55)(18 56)(19 53)(20 54)(21 35)(22 36)(23 33)(24 34)(25 39)(26 40)(27 37)(28 38)(29 43)(30 44)(31 41)(32 42)(57 95)(58 96)(59 93)(60 94)(69 83)(70 84)(71 81)(72 82)(73 87)(74 88)(75 85)(76 86)(77 91)(78 92)(79 89)(80 90)
(1 22)(2 23)(3 24)(4 21)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(81 93)(82 94)(83 95)(84 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 17 16)(2 18 13)(3 19 14)(4 20 15)(5 95 10)(6 96 11)(7 93 12)(8 94 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 44 39)(34 41 40)(35 42 37)(36 43 38)(45 56 51)(46 53 52)(47 54 49)(48 55 50)(57 68 63)(58 65 64)(59 66 61)(60 67 62)(69 80 75)(70 77 76)(71 78 73)(72 79 74)(81 92 87)(82 89 88)(83 90 85)(84 91 86)
(1 46)(2 47)(3 48)(4 45)(5 66)(6 67)(7 68)(8 65)(9 64)(10 61)(11 62)(12 63)(13 54)(14 55)(15 56)(16 53)(17 52)(18 49)(19 50)(20 51)(21 33)(22 34)(23 35)(24 36)(25 42)(26 43)(27 44)(28 41)(29 40)(30 37)(31 38)(32 39)(57 93)(58 94)(59 95)(60 96)(69 81)(70 82)(71 83)(72 84)(73 90)(74 91)(75 92)(76 89)(77 88)(78 85)(79 86)(80 87)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 3 4A ··· 4P 4Q ··· 4AF 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 3 ··· 3 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4×S3 kernel S3×C23×C4 S3×C22×C4 C23×Dic3 C23×C12 S3×C24 S3×C23 C23×C4 C22×C4 C24 C23 # reps 1 28 1 1 1 32 1 14 1 16

Matrix representation of S3×C23×C4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 1 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,12],[1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

S3×C23×C4 in GAP, Magma, Sage, TeX

S_3\times C_2^3\times C_4
% in TeX

G:=Group("S3xC2^3xC4");
// GroupNames label

G:=SmallGroup(192,1511);
// by ID

G=gap.SmallGroup(192,1511);
# by ID

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

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

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