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## G = S3×C25order 192 = 26·3

### Direct product of C25 and S3

Aliases: S3×C25, C3⋊C26, C6⋊C25, (C24×C6)⋊5C2, (C2×C6)⋊4C24, (C23×C6)⋊20C22, (C22×C6)⋊10C23, SmallGroup(192,1542)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C25
 Chief series C1 — C3 — S3 — D6 — C22×S3 — S3×C23 — S3×C24 — S3×C25
 Lower central C3 — S3×C25
 Upper central C1 — C25

Generators and relations for S3×C25
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 10552 in 5650 conjugacy classes, 3199 normal (5 characteristic)
C1, C2 [×31], C2 [×32], C3, C22 [×155], C22 [×496], S3 [×32], C6 [×31], C23 [×155], C23 [×1240], D6 [×496], C2×C6 [×155], C24 [×31], C24 [×620], C22×S3 [×1240], C22×C6 [×155], C25, C25 [×62], S3×C23 [×620], C23×C6 [×31], C26, S3×C24 [×62], C24×C6, S3×C25
Quotients: C1, C2 [×63], C22 [×651], S3, C23 [×1395], D6 [×31], C24 [×651], C22×S3 [×155], C25 [×63], S3×C23 [×155], C26, S3×C24 [×31], S3×C25

Smallest permutation representation of S3×C25
On 96 points
Generators in S96
(1 94)(2 95)(3 96)(4 91)(5 92)(6 93)(7 88)(8 89)(9 90)(10 85)(11 86)(12 87)(13 82)(14 83)(15 84)(16 79)(17 80)(18 81)(19 76)(20 77)(21 78)(22 73)(23 74)(24 75)(25 70)(26 71)(27 72)(28 67)(29 68)(30 69)(31 64)(32 65)(33 66)(34 61)(35 62)(36 63)(37 58)(38 59)(39 60)(40 55)(41 56)(42 57)(43 52)(44 53)(45 54)(46 49)(47 50)(48 51)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(55 76)(56 77)(57 78)(58 73)(59 74)(60 75)(61 94)(62 95)(63 96)(64 91)(65 92)(66 93)(67 88)(68 89)(69 90)(70 85)(71 86)(72 87)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 64)(50 65)(51 66)(52 61)(53 62)(54 63)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(61 64)(62 65)(63 66)(67 70)(68 71)(69 72)(73 76)(74 77)(75 78)(79 82)(80 83)(81 84)(85 88)(86 89)(87 90)(91 94)(92 95)(93 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 67)(2 69)(3 68)(4 70)(5 72)(6 71)(7 61)(8 63)(9 62)(10 64)(11 66)(12 65)(13 55)(14 57)(15 56)(16 58)(17 60)(18 59)(19 49)(20 51)(21 50)(22 52)(23 54)(24 53)(25 91)(26 93)(27 92)(28 94)(29 96)(30 95)(31 85)(32 87)(33 86)(34 88)(35 90)(36 89)(37 79)(38 81)(39 80)(40 82)(41 84)(42 83)(43 73)(44 75)(45 74)(46 76)(47 78)(48 77)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2AE 2AF ··· 2BK 3 6A ··· 6AE order 1 2 ··· 2 2 ··· 2 3 6 ··· 6 size 1 1 ··· 1 3 ··· 3 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 S3 D6 kernel S3×C25 S3×C24 C24×C6 C25 C24 # reps 1 62 1 1 31

Matrix representation of S3×C25 in GL6(ℤ)

 1 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 0 0 0 0 0 0 1
,
 1 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 0 0 0 0 0 0 -1
,
 -1 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 0 0 0 0 0 0 1
,
 -1 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 0 0 0 0 0 0 -1
,
 -1 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 0 0 0 0 0 0 1
,
 1 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 0 -1 0 0 0 0 1 -1
,
 1 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 0 1 0 0 0 0 1 0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

S3×C25 in GAP, Magma, Sage, TeX

S_3\times C_2^5
% in TeX

G:=Group("S3xC2^5");
// GroupNames label

G:=SmallGroup(192,1542);
// by ID

G=gap.SmallGroup(192,1542);
# by ID

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

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

׿
×
𝔽