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## G = (C2×Q16)⋊S3order 192 = 26·3

### 2nd semidirect product of C2×Q16 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×Q16)⋊S3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — Q8×Dic3 — (C2×Q16)⋊S3
 Lower central C3 — C6 — C2×C12 — (C2×Q16)⋊S3
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for (C2×Q16)⋊S3
G = < a,b,c,d,e | a2=b8=d3=e2=1, c2=b4, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=ab-1, cd=dc, ede=d-1 >

Subgroups: 328 in 112 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, Q82S3, C2×C24, C3×Q16, C2×D12, C6×Q8, Q8.D4, Dic3⋊C8, C2.D24, Q82Dic3, C2×Q82S3, Q8×Dic3, C12.23D4, C6×Q16, (C2×Q16)⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C4○D8, C8.C22, S3×D4, D42S3, C2×C3⋊D4, Q8.D4, Q16⋊S3, D24⋊C2, C23.14D6, (C2×Q16)⋊S3

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 C4○D8 C8.C22 D4⋊2S3 S3×D4 Q16⋊S3 D24⋊C2 kernel (C2×Q16)⋊S3 Dic3⋊C8 C2.D24 Q8⋊2Dic3 C2×Q8⋊2S3 Q8×Dic3 C12.23D4 C6×Q16 C2×Q16 C2×Dic3 C3×Q8 C2×C8 C2×Q8 C12 Q8 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 2 2 4 4 1 1 1 2 2

Matrix representation of (C2×Q16)⋊S3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 43 13 0 0 60 30 0 0 0 0 16 57 0 0 16 16
,
 30 60 0 0 13 43 0 0 0 0 46 0 0 0 0 27
,
 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 1 1 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[43,60,0,0,13,30,0,0,0,0,16,16,0,0,57,16],[30,13,0,0,60,43,0,0,0,0,46,0,0,0,0,27],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[72,1,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;

(C2×Q16)⋊S3 in GAP, Magma, Sage, TeX

(C_2\times Q_{16})\rtimes S_3
% in TeX

G:=Group("(C2xQ16):S3");
// GroupNames label

G:=SmallGroup(192,744);
// by ID

G=gap.SmallGroup(192,744);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,1094,135,184,570,297,136,6278]);
// Polycyclic

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

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