Copied to
clipboard

## G = C3×C22⋊C16order 192 = 26·3

### Direct product of C3 and C22⋊C16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C22⋊C16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C24 — C2×C48 — C3×C22⋊C16
 Lower central C1 — C2 — C3×C22⋊C16
 Upper central C1 — C2×C24 — C3×C22⋊C16

Generators and relations for C3×C22⋊C16
G = < a,b,c,d | a3=b2=c2=d16=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 90 in 66 conjugacy classes, 42 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C16, C2×C8, C2×C8, C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, C2×C16, C22×C8, C48, C2×C24, C2×C24, C22×C12, C22⋊C16, C2×C48, C22×C24, C3×C22⋊C16
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C12, C2×C6, C16, C22⋊C4, C2×C8, M4(2), C24, C2×C12, C3×D4, C22⋊C8, C2×C16, M5(2), C48, C3×C22⋊C4, C2×C24, C3×M4(2), C22⋊C16, C3×C22⋊C8, C2×C48, C3×M5(2), C3×C22⋊C16

Smallest permutation representation of C3×C22⋊C16
On 96 points
Generators in S96
(1 69 55)(2 70 56)(3 71 57)(4 72 58)(5 73 59)(6 74 60)(7 75 61)(8 76 62)(9 77 63)(10 78 64)(11 79 49)(12 80 50)(13 65 51)(14 66 52)(15 67 53)(16 68 54)(17 36 94)(18 37 95)(19 38 96)(20 39 81)(21 40 82)(22 41 83)(23 42 84)(24 43 85)(25 44 86)(26 45 87)(27 46 88)(28 47 89)(29 48 90)(30 33 91)(31 34 92)(32 35 93)
(2 24)(4 26)(6 28)(8 30)(10 32)(12 18)(14 20)(16 22)(33 76)(35 78)(37 80)(39 66)(41 68)(43 70)(45 72)(47 74)(50 95)(52 81)(54 83)(56 85)(58 87)(60 89)(62 91)(64 93)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(33 76)(34 77)(35 78)(36 79)(37 80)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 94)(50 95)(51 96)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(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)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 8A ··· 8H 8I 8J 8K 8L 12A ··· 12H 12I 12J 12K 12L 16A ··· 16P 24A ··· 24P 24Q ··· 24X 48A ··· 48AF 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 8 8 12 ··· 12 12 12 12 12 16 ··· 16 24 ··· 24 24 ··· 24 48 ··· 48 size 1 1 1 1 2 2 1 1 1 1 1 1 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C16 C24 C24 C48 D4 M4(2) C3×D4 M5(2) C3×M4(2) C3×M5(2) kernel C3×C22⋊C16 C2×C48 C22×C24 C22⋊C16 C2×C24 C22×C12 C2×C16 C22×C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 C24 C12 C8 C6 C4 C2 # reps 1 2 1 2 2 2 4 2 4 4 4 4 16 8 8 32 2 2 4 4 4 8

Matrix representation of C3×C22⋊C16 in GL3(𝔽97) generated by

 1 0 0 0 35 0 0 0 35
,
 96 0 0 0 1 46 0 0 96
,
 1 0 0 0 96 0 0 0 96
,
 12 0 0 0 51 95 0 2 46
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,35],[96,0,0,0,1,0,0,46,96],[1,0,0,0,96,0,0,0,96],[12,0,0,0,51,2,0,95,46] >;

C3×C22⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC2^2:C16");
// GroupNames label

G:=SmallGroup(192,154);
// by ID

G=gap.SmallGroup(192,154);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

׿
×
𝔽