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## G = C3×C22⋊Q16order 192 = 26·3

### Direct product of C3 and C22⋊Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C22⋊Q16
G = < a,b,c,d,e | a3=b2=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 242 in 148 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×Q16, C22×C12, C22×C12, C6×Q8, C6×Q8, C6×Q8, C22⋊Q16, C3×C22⋊C8, C3×Q8⋊C4, C3×C22⋊Q8, C6×Q16, Q8×C2×C6, C3×C22⋊Q16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, Q16, C2×D4, C3×D4, C22×C6, C22≀C2, C2×Q16, C8.C22, C3×Q16, C6×D4, C22⋊Q16, C3×C22≀C2, C6×Q16, C3×C8.C22, C3×C22⋊Q16

Smallest permutation representation of C3×C22⋊Q16
On 96 points
Generators in S96
(1 56 32)(2 49 25)(3 50 26)(4 51 27)(5 52 28)(6 53 29)(7 54 30)(8 55 31)(9 65 81)(10 66 82)(11 67 83)(12 68 84)(13 69 85)(14 70 86)(15 71 87)(16 72 88)(17 78 89)(18 79 90)(19 80 91)(20 73 92)(21 74 93)(22 75 94)(23 76 95)(24 77 96)(33 43 63)(34 44 64)(35 45 57)(36 46 58)(37 47 59)(38 48 60)(39 41 61)(40 42 62)
(2 20)(4 22)(6 24)(8 18)(10 37)(12 39)(14 33)(16 35)(25 92)(27 94)(29 96)(31 90)(41 68)(43 70)(45 72)(47 66)(49 73)(51 75)(53 77)(55 79)(57 88)(59 82)(61 84)(63 86)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(25 92)(26 93)(27 94)(28 95)(29 96)(30 89)(31 90)(32 91)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 88)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)
(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 44 5 48)(2 43 6 47)(3 42 7 46)(4 41 8 45)(9 93 13 89)(10 92 14 96)(11 91 15 95)(12 90 16 94)(17 65 21 69)(18 72 22 68)(19 71 23 67)(20 70 24 66)(25 33 29 37)(26 40 30 36)(27 39 31 35)(28 38 32 34)(49 63 53 59)(50 62 54 58)(51 61 55 57)(52 60 56 64)(73 86 77 82)(74 85 78 81)(75 84 79 88)(76 83 80 87)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C ··· 4G 4H 4I 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12N 12O 12P 12Q 12R 24A ··· 24H order 1 2 2 2 2 2 3 3 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 1 1 2 2 4 ··· 4 8 8 1 ··· 1 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 8 8 8 8 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D4 Q16 C3×D4 C3×D4 C3×D4 C3×Q16 C8.C22 C3×C8.C22 kernel C3×C22⋊Q16 C3×C22⋊C8 C3×Q8⋊C4 C3×C22⋊Q8 C6×Q16 Q8×C2×C6 C22⋊Q16 C22⋊C8 Q8⋊C4 C22⋊Q8 C2×Q16 C22×Q8 C2×C12 C3×Q8 C22×C6 C2×C6 C2×C4 Q8 C23 C22 C6 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 1 4 1 4 2 8 2 8 1 2

Matrix representation of C3×C22⋊Q16 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 0 41 0 0 16 41
,
 72 0 0 0 0 72 0 0 0 0 60 24 0 0 72 13
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,0,16,0,0,41,41],[72,0,0,0,0,72,0,0,0,0,60,72,0,0,24,13] >;

C3×C22⋊Q16 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes Q_{16}
% in TeX

G:=Group("C3xC2^2:Q16");
// GroupNames label

G:=SmallGroup(192,884);
// by ID

G=gap.SmallGroup(192,884);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,680,1094,4204,2111,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=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,d*b*d^-1=b*c=c*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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