direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2xC3:C16, C6:C16, C12.3C8, C24.3C4, C8.21D6, C8.4Dic3, C24.25C22, C8o(C3:C16), C4o(C3:C16), C3:2(C2xC16), C4.3(C3:C8), (C2xC8).9S3, (C2xC6).2C8, C6.8(C2xC8), (C2xC24).12C2, C12.38(C2xC4), (C2xC12).11C4, C22.2(C3:C8), (C2xC4).8Dic3, C4.10(C2xDic3), C2.2(C2xC3:C8), SmallGroup(96,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C2xC3:C16 |
Generators and relations for C2xC3:C16
G = < a,b,c | a2=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, Dic3, D6, C16, C2xC8, C3:C8, C2xDic3, C2xC16, C3:C16, C2xC3:C8, C2xC3:C16
Smallest permutation representation of C2xC3:C16
►Regular action on 96 points
Generators in S96
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 95)(18 96)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 65)(46 66)(47 67)(48 68)
(1 32 38)(2 39 17)(3 18 40)(4 41 19)(5 20 42)(6 43 21)(7 22 44)(8 45 23)(9 24 46)(10 47 25)(11 26 48)(12 33 27)(13 28 34)(14 35 29)(15 30 36)(16 37 31)(49 94 74)(50 75 95)(51 96 76)(52 77 81)(53 82 78)(54 79 83)(55 84 80)(56 65 85)(57 86 66)(58 67 87)(59 88 68)(60 69 89)(61 90 70)(62 71 91)(63 92 72)(64 73 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)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,95)(18,96)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,65)(46,66)(47,67)(48,68), (1,32,38)(2,39,17)(3,18,40)(4,41,19)(5,20,42)(6,43,21)(7,22,44)(8,45,23)(9,24,46)(10,47,25)(11,26,48)(12,33,27)(13,28,34)(14,35,29)(15,30,36)(16,37,31)(49,94,74)(50,75,95)(51,96,76)(52,77,81)(53,82,78)(54,79,83)(55,84,80)(56,65,85)(57,86,66)(58,67,87)(59,88,68)(60,69,89)(61,90,70)(62,71,91)(63,92,72)(64,73,93), (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( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,95)(18,96)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,65)(46,66)(47,67)(48,68), (1,32,38)(2,39,17)(3,18,40)(4,41,19)(5,20,42)(6,43,21)(7,22,44)(8,45,23)(9,24,46)(10,47,25)(11,26,48)(12,33,27)(13,28,34)(14,35,29)(15,30,36)(16,37,31)(49,94,74)(50,75,95)(51,96,76)(52,77,81)(53,82,78)(54,79,83)(55,84,80)(56,65,85)(57,86,66)(58,67,87)(59,88,68)(60,69,89)(61,90,70)(62,71,91)(63,92,72)(64,73,93), (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([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,95),(18,96),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,65),(46,66),(47,67),(48,68)], [(1,32,38),(2,39,17),(3,18,40),(4,41,19),(5,20,42),(6,43,21),(7,22,44),(8,45,23),(9,24,46),(10,47,25),(11,26,48),(12,33,27),(13,28,34),(14,35,29),(15,30,36),(16,37,31),(49,94,74),(50,75,95),(51,96,76),(52,77,81),(53,82,78),(54,79,83),(55,84,80),(56,65,85),(57,86,66),(58,67,87),(59,88,68),(60,69,89),(61,90,70),(62,71,91),(63,92,72),(64,73,93)], [(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)]])
C2xC3:C16 is a maximal subgroup of
C24.C8 C12:C16 C6.6D16 C6.SD32 C6.D16 C6.Q32 C24.7Q8 Dic12.C4 Dic3xC16 Dic3:C16 C48:10C4 D6:C16 Dic6.C8 C24.98D4 C24.99D4 D8:1Dic3 C6.5Q32 C24.41D4 S3xC2xC16 C16.12D6 C24.78C23 Q16.D6 C24.F5
C2xC3:C16 is a maximal quotient of
C12:C16 C3:M6(2) C24.98D4 C24.F5
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 16A | ··· | 16P | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | Dic3 | D6 | Dic3 | C3:C8 | C3:C8 | C3:C16 |
| kernel | C2xC3:C16 | C3:C16 | C2xC24 | C24 | C2xC12 | C12 | C2xC6 | C6 | C2xC8 | C8 | C8 | C2xC4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 8 |
Matrix representation of C2xC3:C16 ►in GL4(F97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 96 | 96 |
| 0 | 0 | 1 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 13 | 78 |
| 0 | 0 | 65 | 84 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,96,1,0,0,96,0],[18,0,0,0,0,96,0,0,0,0,13,65,0,0,78,84] >;
C2xC3:C16 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes C_{16} % in TeX
G:=Group("C2xC3:C16"); // GroupNames label
G:=SmallGroup(96,18);
// by ID
G=gap.SmallGroup(96,18);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,50,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^2=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export