direct product, metabelian, nilpotent (class 2), monomial
Aliases: C2xC4xHe3, C62.4C6, C6.6C62, (C6xC12):C3, (C3xC12):4C6, (C3xC6):3C12, C3.1(C6xC12), C6.4(C3xC12), C22.(C2xHe3), C12.12(C3xC6), C32:6(C2xC12), (C2xC12).1C32, C2.1(C22xHe3), (C22xHe3).3C2, (C2xHe3).16C22, (C3xC6).11(C2xC6), (C2xC6).10(C3xC6), SmallGroup(216,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC4xHe3
G = < a,b,c,d,e | a2=b4=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 152 in 88 conjugacy classes, 56 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2xC4, C32, C12, C12, C2xC6, C2xC6, C3xC6, C2xC12, C2xC12, He3, C3xC12, C62, C2xHe3, C2xHe3, C6xC12, C4xHe3, C22xHe3, C2xC4xHe3
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C32, C12, C2xC6, C3xC6, C2xC12, He3, C3xC12, C62, C2xHe3, C6xC12, C4xHe3, C22xHe3, C2xC4xHe3
►On 72 points
Generators in S72
(1 8)(2 5)(3 6)(4 7)(9 22)(10 23)(11 24)(12 21)(13 26)(14 27)(15 28)(16 25)(17 48)(18 45)(19 46)(20 47)(29 69)(30 70)(31 71)(32 72)(33 53)(34 54)(35 55)(36 56)(37 64)(38 61)(39 62)(40 63)(41 68)(42 65)(43 66)(44 67)(49 58)(50 59)(51 60)(52 57)
(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)
(1 42 71)(2 43 72)(3 44 69)(4 41 70)(5 66 32)(6 67 29)(7 68 30)(8 65 31)(9 25 36)(10 26 33)(11 27 34)(12 28 35)(13 53 23)(14 54 24)(15 55 21)(16 56 22)(17 57 38)(18 58 39)(19 59 40)(20 60 37)(45 49 62)(46 50 63)(47 51 64)(48 52 61)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 45 26)(6 46 27)(7 47 28)(8 48 25)(9 31 61)(10 32 62)(11 29 63)(12 30 64)(21 70 37)(22 71 38)(23 72 39)(24 69 40)(33 66 49)(34 67 50)(35 68 51)(36 65 52)(41 60 55)(42 57 56)(43 58 53)(44 59 54)
(1 42 38)(2 43 39)(3 44 40)(4 41 37)(5 66 62)(6 67 63)(7 68 64)(8 65 61)(9 48 52)(10 45 49)(11 46 50)(12 47 51)(13 53 72)(14 54 69)(15 55 70)(16 56 71)(17 57 22)(18 58 23)(19 59 24)(20 60 21)(25 36 31)(26 33 32)(27 34 29)(28 35 30)
G:=sub<Sym(72)| (1,8)(2,5)(3,6)(4,7)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25)(17,48)(18,45)(19,46)(20,47)(29,69)(30,70)(31,71)(32,72)(33,53)(34,54)(35,55)(36,56)(37,64)(38,61)(39,62)(40,63)(41,68)(42,65)(43,66)(44,67)(49,58)(50,59)(51,60)(52,57), (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), (1,42,71)(2,43,72)(3,44,69)(4,41,70)(5,66,32)(6,67,29)(7,68,30)(8,65,31)(9,25,36)(10,26,33)(11,27,34)(12,28,35)(13,53,23)(14,54,24)(15,55,21)(16,56,22)(17,57,38)(18,58,39)(19,59,40)(20,60,37)(45,49,62)(46,50,63)(47,51,64)(48,52,61), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,45,26)(6,46,27)(7,47,28)(8,48,25)(9,31,61)(10,32,62)(11,29,63)(12,30,64)(21,70,37)(22,71,38)(23,72,39)(24,69,40)(33,66,49)(34,67,50)(35,68,51)(36,65,52)(41,60,55)(42,57,56)(43,58,53)(44,59,54), (1,42,38)(2,43,39)(3,44,40)(4,41,37)(5,66,62)(6,67,63)(7,68,64)(8,65,61)(9,48,52)(10,45,49)(11,46,50)(12,47,51)(13,53,72)(14,54,69)(15,55,70)(16,56,71)(17,57,22)(18,58,23)(19,59,24)(20,60,21)(25,36,31)(26,33,32)(27,34,29)(28,35,30)>;
G:=Group( (1,8)(2,5)(3,6)(4,7)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25)(17,48)(18,45)(19,46)(20,47)(29,69)(30,70)(31,71)(32,72)(33,53)(34,54)(35,55)(36,56)(37,64)(38,61)(39,62)(40,63)(41,68)(42,65)(43,66)(44,67)(49,58)(50,59)(51,60)(52,57), (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), (1,42,71)(2,43,72)(3,44,69)(4,41,70)(5,66,32)(6,67,29)(7,68,30)(8,65,31)(9,25,36)(10,26,33)(11,27,34)(12,28,35)(13,53,23)(14,54,24)(15,55,21)(16,56,22)(17,57,38)(18,58,39)(19,59,40)(20,60,37)(45,49,62)(46,50,63)(47,51,64)(48,52,61), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,45,26)(6,46,27)(7,47,28)(8,48,25)(9,31,61)(10,32,62)(11,29,63)(12,30,64)(21,70,37)(22,71,38)(23,72,39)(24,69,40)(33,66,49)(34,67,50)(35,68,51)(36,65,52)(41,60,55)(42,57,56)(43,58,53)(44,59,54), (1,42,38)(2,43,39)(3,44,40)(4,41,37)(5,66,62)(6,67,63)(7,68,64)(8,65,61)(9,48,52)(10,45,49)(11,46,50)(12,47,51)(13,53,72)(14,54,69)(15,55,70)(16,56,71)(17,57,22)(18,58,23)(19,59,24)(20,60,21)(25,36,31)(26,33,32)(27,34,29)(28,35,30) );
G=PermutationGroup([[(1,8),(2,5),(3,6),(4,7),(9,22),(10,23),(11,24),(12,21),(13,26),(14,27),(15,28),(16,25),(17,48),(18,45),(19,46),(20,47),(29,69),(30,70),(31,71),(32,72),(33,53),(34,54),(35,55),(36,56),(37,64),(38,61),(39,62),(40,63),(41,68),(42,65),(43,66),(44,67),(49,58),(50,59),(51,60),(52,57)], [(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)], [(1,42,71),(2,43,72),(3,44,69),(4,41,70),(5,66,32),(6,67,29),(7,68,30),(8,65,31),(9,25,36),(10,26,33),(11,27,34),(12,28,35),(13,53,23),(14,54,24),(15,55,21),(16,56,22),(17,57,38),(18,58,39),(19,59,40),(20,60,37),(45,49,62),(46,50,63),(47,51,64),(48,52,61)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,45,26),(6,46,27),(7,47,28),(8,48,25),(9,31,61),(10,32,62),(11,29,63),(12,30,64),(21,70,37),(22,71,38),(23,72,39),(24,69,40),(33,66,49),(34,67,50),(35,68,51),(36,65,52),(41,60,55),(42,57,56),(43,58,53),(44,59,54)], [(1,42,38),(2,43,39),(3,44,40),(4,41,37),(5,66,62),(6,67,63),(7,68,64),(8,65,61),(9,48,52),(10,45,49),(11,46,50),(12,47,51),(13,53,72),(14,54,69),(15,55,70),(16,56,71),(17,57,22),(18,58,23),(19,59,24),(20,60,21),(25,36,31),(26,33,32),(27,34,29),(28,35,30)]])
C2xC4xHe3 is a maximal subgroup of
He3:7M4(2) C62.19D6 C62.20D6 C62.21D6 He3:8M4(2) C62.29D6 C62.30D6 C62.31D6 C62.36D6 C62.47D6
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3J | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6AD | 12A | ··· | 12H | 12I | ··· | 12AN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | He3 | C2xHe3 | C2xHe3 | C4xHe3 |
| kernel | C2xC4xHe3 | C4xHe3 | C22xHe3 | C6xC12 | C2xHe3 | C3xC12 | C62 | C3xC6 | C2xC4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 2 | 4 | 2 | 8 |
Matrix representation of C2xC4xHe3 ►in GL4(F13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,0,0,1,0,9,0,0,0,0,3,0] >;
C2xC4xHe3 in GAP, Magma, Sage, TeX
C_2\times C_4\times {\rm He}_3 % in TeX
G:=Group("C2xC4xHe3"); // GroupNames label
G:=SmallGroup(216,74);
// by ID
G=gap.SmallGroup(216,74);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,519]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations