direct product, metabelian, nilpotent (class 2), monomial
Aliases: C2×D4×He3, C12.27C62, (C6×C12)⋊6C6, C4⋊(C22×He3), C62⋊3(C2×C6), (C2×C62)⋊1C6, (D4×C32)⋊5C6, C32⋊11(C6×D4), (C2×C6).8C62, C23⋊2(C2×He3), C6.21(C2×C62), (C23×He3)⋊1C2, (C6×D4).2C32, C6.22(D4×C32), C2.2(C23×He3), (C4×He3)⋊10C22, C22⋊2(C22×He3), (C2×He3).40C23, (C22×He3)⋊6C22, (D4×C3×C6)⋊C3, C3.2(D4×C3×C6), (C3×C6)⋊6(C3×D4), (C3×C12)⋊4(C2×C6), (C2×C4×He3)⋊10C2, (C2×C4)⋊2(C2×He3), (C2×C12).17(C3×C6), (C3×D4).10(C3×C6), (C3×C6).31(C22×C6), (C22×C6).17(C3×C6), SmallGroup(432,404)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4×He3
G = < a,b,c,d,e,f | a2=b4=c2=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >
Subgroups: 665 in 297 conjugacy classes, 133 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, D4, C23, C32, C12, C12, C2×C6, C2×C6, C2×C6, C2×D4, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, He3, C3×C12, C62, C62, C6×D4, C6×D4, C2×He3, C2×He3, C2×He3, C6×C12, D4×C32, C2×C62, C4×He3, C22×He3, C22×He3, C22×He3, D4×C3×C6, C2×C4×He3, D4×He3, C23×He3, C2×D4×He3
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, He3, C62, C6×D4, C2×He3, D4×C32, C2×C62, C22×He3, D4×C3×C6, D4×He3, C23×He3, C2×D4×He3
►On 72 points
Generators in S72
(1 15)(2 16)(3 13)(4 14)(5 65)(6 66)(7 67)(8 68)(9 31)(10 32)(11 29)(12 30)(17 25)(18 26)(19 27)(20 28)(21 59)(22 60)(23 57)(24 58)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 71)(50 72)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)
(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 16)(2 15)(3 14)(4 13)(5 66)(6 65)(7 68)(8 67)(9 32)(10 31)(11 30)(12 29)(17 26)(18 25)(19 28)(20 27)(21 60)(22 59)(23 58)(24 57)(33 42)(34 41)(35 44)(36 43)(37 46)(38 45)(39 48)(40 47)(49 72)(50 71)(51 70)(52 69)(53 62)(54 61)(55 64)(56 63)
(1 63 41)(2 64 42)(3 61 43)(4 62 44)(5 45 27)(6 46 28)(7 47 25)(8 48 26)(9 49 57)(10 50 58)(11 51 59)(12 52 60)(13 53 35)(14 54 36)(15 55 33)(16 56 34)(17 67 39)(18 68 40)(19 65 37)(20 66 38)(21 29 69)(22 30 70)(23 31 71)(24 32 72)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 67 29)(14 68 30)(15 65 31)(16 66 32)(17 21 35)(18 22 36)(19 23 33)(20 24 34)(25 59 43)(26 60 44)(27 57 41)(28 58 42)(37 71 55)(38 72 56)(39 69 53)(40 70 54)(45 49 63)(46 50 64)(47 51 61)(48 52 62)
(17 35 21)(18 36 22)(19 33 23)(20 34 24)(25 43 59)(26 44 60)(27 41 57)(28 42 58)(37 71 55)(38 72 56)(39 69 53)(40 70 54)(45 49 63)(46 50 64)(47 51 61)(48 52 62)
G:=sub<Sym(72)| (1,15)(2,16)(3,13)(4,14)(5,65)(6,66)(7,67)(8,68)(9,31)(10,32)(11,29)(12,30)(17,25)(18,26)(19,27)(20,28)(21,59)(22,60)(23,57)(24,58)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,71)(50,72)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64), (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), (1,16)(2,15)(3,14)(4,13)(5,66)(6,65)(7,68)(8,67)(9,32)(10,31)(11,30)(12,29)(17,26)(18,25)(19,28)(20,27)(21,60)(22,59)(23,58)(24,57)(33,42)(34,41)(35,44)(36,43)(37,46)(38,45)(39,48)(40,47)(49,72)(50,71)(51,70)(52,69)(53,62)(54,61)(55,64)(56,63), (1,63,41)(2,64,42)(3,61,43)(4,62,44)(5,45,27)(6,46,28)(7,47,25)(8,48,26)(9,49,57)(10,50,58)(11,51,59)(12,52,60)(13,53,35)(14,54,36)(15,55,33)(16,56,34)(17,67,39)(18,68,40)(19,65,37)(20,66,38)(21,29,69)(22,30,70)(23,31,71)(24,32,72), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,67,29)(14,68,30)(15,65,31)(16,66,32)(17,21,35)(18,22,36)(19,23,33)(20,24,34)(25,59,43)(26,60,44)(27,57,41)(28,58,42)(37,71,55)(38,72,56)(39,69,53)(40,70,54)(45,49,63)(46,50,64)(47,51,61)(48,52,62), (17,35,21)(18,36,22)(19,33,23)(20,34,24)(25,43,59)(26,44,60)(27,41,57)(28,42,58)(37,71,55)(38,72,56)(39,69,53)(40,70,54)(45,49,63)(46,50,64)(47,51,61)(48,52,62)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,65)(6,66)(7,67)(8,68)(9,31)(10,32)(11,29)(12,30)(17,25)(18,26)(19,27)(20,28)(21,59)(22,60)(23,57)(24,58)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,71)(50,72)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64), (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), (1,16)(2,15)(3,14)(4,13)(5,66)(6,65)(7,68)(8,67)(9,32)(10,31)(11,30)(12,29)(17,26)(18,25)(19,28)(20,27)(21,60)(22,59)(23,58)(24,57)(33,42)(34,41)(35,44)(36,43)(37,46)(38,45)(39,48)(40,47)(49,72)(50,71)(51,70)(52,69)(53,62)(54,61)(55,64)(56,63), (1,63,41)(2,64,42)(3,61,43)(4,62,44)(5,45,27)(6,46,28)(7,47,25)(8,48,26)(9,49,57)(10,50,58)(11,51,59)(12,52,60)(13,53,35)(14,54,36)(15,55,33)(16,56,34)(17,67,39)(18,68,40)(19,65,37)(20,66,38)(21,29,69)(22,30,70)(23,31,71)(24,32,72), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,67,29)(14,68,30)(15,65,31)(16,66,32)(17,21,35)(18,22,36)(19,23,33)(20,24,34)(25,59,43)(26,60,44)(27,57,41)(28,58,42)(37,71,55)(38,72,56)(39,69,53)(40,70,54)(45,49,63)(46,50,64)(47,51,61)(48,52,62), (17,35,21)(18,36,22)(19,33,23)(20,34,24)(25,43,59)(26,44,60)(27,41,57)(28,42,58)(37,71,55)(38,72,56)(39,69,53)(40,70,54)(45,49,63)(46,50,64)(47,51,61)(48,52,62) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,65),(6,66),(7,67),(8,68),(9,31),(10,32),(11,29),(12,30),(17,25),(18,26),(19,27),(20,28),(21,59),(22,60),(23,57),(24,58),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,71),(50,72),(51,69),(52,70),(53,61),(54,62),(55,63),(56,64)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,66),(6,65),(7,68),(8,67),(9,32),(10,31),(11,30),(12,29),(17,26),(18,25),(19,28),(20,27),(21,60),(22,59),(23,58),(24,57),(33,42),(34,41),(35,44),(36,43),(37,46),(38,45),(39,48),(40,47),(49,72),(50,71),(51,70),(52,69),(53,62),(54,61),(55,64),(56,63)], [(1,63,41),(2,64,42),(3,61,43),(4,62,44),(5,45,27),(6,46,28),(7,47,25),(8,48,26),(9,49,57),(10,50,58),(11,51,59),(12,52,60),(13,53,35),(14,54,36),(15,55,33),(16,56,34),(17,67,39),(18,68,40),(19,65,37),(20,66,38),(21,29,69),(22,30,70),(23,31,71),(24,32,72)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,67,29),(14,68,30),(15,65,31),(16,66,32),(17,21,35),(18,22,36),(19,23,33),(20,24,34),(25,59,43),(26,60,44),(27,57,41),(28,58,42),(37,71,55),(38,72,56),(39,69,53),(40,70,54),(45,49,63),(46,50,64),(47,51,61),(48,52,62)], [(17,35,21),(18,36,22),(19,33,23),(20,34,24),(25,43,59),(26,44,60),(27,41,57),(28,42,58),(37,71,55),(38,72,56),(39,69,53),(40,70,54),(45,49,63),(46,50,64),(47,51,61),(48,52,62)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3J | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6N | 6O | ··· | 6AL | 6AM | ··· | 6BR | 12A | 12B | 12C | 12D | 12E | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 | He3 | C2×He3 | C2×He3 | C2×He3 | D4×He3 |
| kernel | C2×D4×He3 | C2×C4×He3 | D4×He3 | C23×He3 | D4×C3×C6 | C6×C12 | D4×C32 | C2×C62 | C2×He3 | C3×C6 | C2×D4 | C2×C4 | D4 | C23 | C2 |
| # reps | 1 | 1 | 4 | 2 | 8 | 8 | 32 | 16 | 2 | 16 | 2 | 2 | 8 | 4 | 4 |
Matrix representation of C2×D4×He3 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 8 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 8 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,3,0,0,0,8,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,8,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3] >;
C2×D4×He3 in GAP, Magma, Sage, TeX
C_2\times D_4\times {\rm He}_3 % in TeX
G:=Group("C2xD4xHe3"); // GroupNames label
G:=SmallGroup(432,404);
// by ID
G=gap.SmallGroup(432,404);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,760]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=c^2=d^3=e^3=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d*e^-1,e*f=f*e>;
// generators/relations