direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22⋊C16, C22⋊2C48, C24.106D4, C23.3C24, C6.8M5(2), C12.37M4(2), (C2×C16)⋊1C6, (C2×C48)⋊5C2, (C2×C6)⋊1C16, C2.1(C2×C48), (C2×C4).3C24, (C2×C12).7C8, (C2×C8).6C12, C8.26(C3×D4), C6.11(C2×C16), (C2×C24).16C4, (C22×C8).6C6, (C22×C6).3C8, (C22×C24).7C2, C22.9(C2×C24), C2.2(C3×M5(2)), C6.25(C22⋊C8), (C22×C12).15C4, (C22×C4).11C12, C4.10(C3×M4(2)), (C2×C24).451C22, C12.111(C22⋊C4), (C2×C6).40(C2×C8), C2.2(C3×C22⋊C8), (C2×C4).83(C2×C12), (C2×C8).105(C2×C6), C4.28(C3×C22⋊C4), (C2×C12).345(C2×C4), SmallGroup(192,154)
Series: Derived ►Chief ►Lower central ►Upper central
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
►On 96 points
(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