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## G = C2×S3×Q16order 192 = 26·3

### Direct product of C2, S3 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×S3×Q16
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×Q8 — C2×S3×Q16
 Lower central C3 — C6 — C12 — C2×S3×Q16
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C2×S3×Q16
G = < a,b,c,d,e | a2=b3=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 600 in 258 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C8, C2×C8, Q16, Q16, C22×C4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×C8, C2×Q16, C2×Q16, C22×Q8, S3×C8, Dic12, C2×C3⋊C8, C3⋊Q16, C2×C24, C3×Q16, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, S3×Q8, C6×Q8, C22×Q16, S3×C2×C8, C2×Dic12, S3×Q16, C2×C3⋊Q16, C6×Q16, C2×S3×Q8, C2×S3×Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C24, C22×S3, C2×Q16, C22×D4, S3×D4, S3×C23, C22×Q16, S3×Q16, C2×S3×D4, C2×S3×Q16

Smallest permutation representation of C2×S3×Q16
On 96 points
Generators in S96
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 89)
(1 91 82)(2 92 83)(3 93 84)(4 94 85)(5 95 86)(6 96 87)(7 89 88)(8 90 81)(9 56 46)(10 49 47)(11 50 48)(12 51 41)(13 52 42)(14 53 43)(15 54 44)(16 55 45)(17 57 75)(18 58 76)(19 59 77)(20 60 78)(21 61 79)(22 62 80)(23 63 73)(24 64 74)(25 66 38)(26 67 39)(27 68 40)(28 69 33)(29 70 34)(30 71 35)(31 72 36)(32 65 37)
(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(33 69)(34 70)(35 71)(36 72)(37 65)(38 66)(39 67)(40 68)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 73)(64 74)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)
(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 5 21)(2 24 6 20)(3 23 7 19)(4 22 8 18)(9 65 13 69)(10 72 14 68)(11 71 15 67)(12 70 16 66)(25 41 29 45)(26 48 30 44)(27 47 31 43)(28 46 32 42)(33 56 37 52)(34 55 38 51)(35 54 39 50)(36 53 40 49)(57 95 61 91)(58 94 62 90)(59 93 63 89)(60 92 64 96)(73 88 77 84)(74 87 78 83)(75 86 79 82)(76 85 80 81)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 3 3 3 3 2 2 2 4 4 4 4 6 6 12 12 12 12 2 2 2 2 2 2 2 6 6 6 6 4 4 8 8 8 8 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 Q16 S3×D4 S3×D4 S3×Q16 kernel C2×S3×Q16 S3×C2×C8 C2×Dic12 S3×Q16 C2×C3⋊Q16 C6×Q16 C2×S3×Q8 C2×Q16 C4×S3 C2×Dic3 C22×S3 C2×C8 Q16 C2×Q8 D6 C4 C22 C2 # reps 1 1 1 8 2 1 2 1 2 1 1 1 4 2 8 1 1 4

Matrix representation of C2×S3×Q16 in GL5(𝔽73)

 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 72 72 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 41 25 0 0 0 35 0
,
 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 49 23 0 0 0 13 24

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,72,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,41,35,0,0,0,25,0],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,49,13,0,0,0,23,24] >;

C2×S3×Q16 in GAP, Magma, Sage, TeX

C_2\times S_3\times Q_{16}
% in TeX

G:=Group("C2xS3xQ16");
// GroupNames label

G:=SmallGroup(192,1322);
// by ID

G=gap.SmallGroup(192,1322);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,185,136,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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