direct product, non-abelian, supersoluble, monomial
Aliases: C2×He3⋊5D4, C62.46D6, (C6×C12)⋊6S3, (C3×C12)⋊7D6, (C3×C6)⋊4D12, He3⋊9(C2×D4), (C2×He3)⋊5D4, C32⋊5(C2×D12), (C4×He3)⋊7C22, C6.21(C12⋊S3), (C2×He3).31C23, (C22×He3).33C22, (C2×C4×He3)⋊7C2, C12.83(C2×C3⋊S3), C4⋊2(C2×He3⋊C2), C3.2(C2×C12⋊S3), C6.63(C22×C3⋊S3), (C2×C12).23(C3⋊S3), (C2×C4)⋊2(He3⋊C2), (C3×C6).41(C22×S3), (C2×He3⋊C2)⋊5C22, (C22×He3⋊C2)⋊3C2, C2.4(C22×He3⋊C2), C22.10(C2×He3⋊C2), (C2×C6).58(C2×C3⋊S3), SmallGroup(432,386)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3⋊5D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >
Subgroups: 1313 in 297 conjugacy classes, 67 normal (13 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, He3, C3×C12, S3×C6, C62, C2×D12, C6×D4, He3⋊C2, C2×He3, C2×He3, C3×D12, C6×C12, S3×C2×C6, C4×He3, C2×He3⋊C2, C2×He3⋊C2, C22×He3, C6×D12, He3⋊5D4, C2×C4×He3, C22×He3⋊C2, C2×He3⋊5D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, C2×C3⋊S3, C2×D12, He3⋊C2, C12⋊S3, C22×C3⋊S3, C2×He3⋊C2, C2×C12⋊S3, He3⋊5D4, C22×He3⋊C2, C2×He3⋊5D4
►On 72 points
Generators in S72
(1 42)(2 43)(3 44)(4 41)(5 32)(6 29)(7 30)(8 31)(9 28)(10 25)(11 26)(12 27)(13 38)(14 39)(15 40)(16 37)(17 47)(18 48)(19 45)(20 46)(21 68)(22 65)(23 66)(24 67)(33 71)(34 72)(35 69)(36 70)(49 61)(50 62)(51 63)(52 64)(53 58)(54 59)(55 60)(56 57)
(1 23 32)(2 24 29)(3 21 30)(4 22 31)(5 42 66)(6 43 67)(7 44 68)(8 41 65)(9 19 54)(10 20 55)(11 17 56)(12 18 53)(13 52 36)(14 49 33)(15 50 34)(16 51 35)(25 46 60)(26 47 57)(27 48 58)(28 45 59)(37 63 69)(38 64 70)(39 61 71)(40 62 72)
(1 37 19)(2 38 20)(3 39 17)(4 40 18)(5 35 28)(6 36 25)(7 33 26)(8 34 27)(9 32 69)(10 29 70)(11 30 71)(12 31 72)(13 46 43)(14 47 44)(15 48 41)(16 45 42)(21 61 56)(22 62 53)(23 63 54)(24 64 55)(49 57 68)(50 58 65)(51 59 66)(52 60 67)
(1 69 63)(2 70 64)(3 71 61)(4 72 62)(5 66 45)(6 67 46)(7 68 47)(8 65 48)(9 54 37)(10 55 38)(11 56 39)(12 53 40)(13 25 60)(14 26 57)(15 27 58)(16 28 59)(17 30 21)(18 31 22)(19 32 23)(20 29 24)(33 49 44)(34 50 41)(35 51 42)(36 52 43)
(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 3)(5 68)(6 67)(7 66)(8 65)(9 56)(10 55)(11 54)(12 53)(14 16)(17 19)(21 32)(22 31)(23 30)(24 29)(25 60)(26 59)(27 58)(28 57)(33 51)(34 50)(35 49)(36 52)(37 39)(42 44)(45 47)(61 69)(62 72)(63 71)(64 70)
G:=sub<Sym(72)| (1,42)(2,43)(3,44)(4,41)(5,32)(6,29)(7,30)(8,31)(9,28)(10,25)(11,26)(12,27)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,68)(22,65)(23,66)(24,67)(33,71)(34,72)(35,69)(36,70)(49,61)(50,62)(51,63)(52,64)(53,58)(54,59)(55,60)(56,57), (1,23,32)(2,24,29)(3,21,30)(4,22,31)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,19,54)(10,20,55)(11,17,56)(12,18,53)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,63,69)(38,64,70)(39,61,71)(40,62,72), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,32,69)(10,29,70)(11,30,71)(12,31,72)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,61,56)(22,62,53)(23,63,54)(24,64,55)(49,57,68)(50,58,65)(51,59,66)(52,60,67), (1,69,63)(2,70,64)(3,71,61)(4,72,62)(5,66,45)(6,67,46)(7,68,47)(8,65,48)(9,54,37)(10,55,38)(11,56,39)(12,53,40)(13,25,60)(14,26,57)(15,27,58)(16,28,59)(17,30,21)(18,31,22)(19,32,23)(20,29,24)(33,49,44)(34,50,41)(35,51,42)(36,52,43), (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), (1,3)(5,68)(6,67)(7,66)(8,65)(9,56)(10,55)(11,54)(12,53)(14,16)(17,19)(21,32)(22,31)(23,30)(24,29)(25,60)(26,59)(27,58)(28,57)(33,51)(34,50)(35,49)(36,52)(37,39)(42,44)(45,47)(61,69)(62,72)(63,71)(64,70)>;
G:=Group( (1,42)(2,43)(3,44)(4,41)(5,32)(6,29)(7,30)(8,31)(9,28)(10,25)(11,26)(12,27)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,68)(22,65)(23,66)(24,67)(33,71)(34,72)(35,69)(36,70)(49,61)(50,62)(51,63)(52,64)(53,58)(54,59)(55,60)(56,57), (1,23,32)(2,24,29)(3,21,30)(4,22,31)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,19,54)(10,20,55)(11,17,56)(12,18,53)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,63,69)(38,64,70)(39,61,71)(40,62,72), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,32,69)(10,29,70)(11,30,71)(12,31,72)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,61,56)(22,62,53)(23,63,54)(24,64,55)(49,57,68)(50,58,65)(51,59,66)(52,60,67), (1,69,63)(2,70,64)(3,71,61)(4,72,62)(5,66,45)(6,67,46)(7,68,47)(8,65,48)(9,54,37)(10,55,38)(11,56,39)(12,53,40)(13,25,60)(14,26,57)(15,27,58)(16,28,59)(17,30,21)(18,31,22)(19,32,23)(20,29,24)(33,49,44)(34,50,41)(35,51,42)(36,52,43), (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), (1,3)(5,68)(6,67)(7,66)(8,65)(9,56)(10,55)(11,54)(12,53)(14,16)(17,19)(21,32)(22,31)(23,30)(24,29)(25,60)(26,59)(27,58)(28,57)(33,51)(34,50)(35,49)(36,52)(37,39)(42,44)(45,47)(61,69)(62,72)(63,71)(64,70) );
G=PermutationGroup([[(1,42),(2,43),(3,44),(4,41),(5,32),(6,29),(7,30),(8,31),(9,28),(10,25),(11,26),(12,27),(13,38),(14,39),(15,40),(16,37),(17,47),(18,48),(19,45),(20,46),(21,68),(22,65),(23,66),(24,67),(33,71),(34,72),(35,69),(36,70),(49,61),(50,62),(51,63),(52,64),(53,58),(54,59),(55,60),(56,57)], [(1,23,32),(2,24,29),(3,21,30),(4,22,31),(5,42,66),(6,43,67),(7,44,68),(8,41,65),(9,19,54),(10,20,55),(11,17,56),(12,18,53),(13,52,36),(14,49,33),(15,50,34),(16,51,35),(25,46,60),(26,47,57),(27,48,58),(28,45,59),(37,63,69),(38,64,70),(39,61,71),(40,62,72)], [(1,37,19),(2,38,20),(3,39,17),(4,40,18),(5,35,28),(6,36,25),(7,33,26),(8,34,27),(9,32,69),(10,29,70),(11,30,71),(12,31,72),(13,46,43),(14,47,44),(15,48,41),(16,45,42),(21,61,56),(22,62,53),(23,63,54),(24,64,55),(49,57,68),(50,58,65),(51,59,66),(52,60,67)], [(1,69,63),(2,70,64),(3,71,61),(4,72,62),(5,66,45),(6,67,46),(7,68,47),(8,65,48),(9,54,37),(10,55,38),(11,56,39),(12,53,40),(13,25,60),(14,26,57),(15,27,58),(16,28,59),(17,30,21),(18,31,22),(19,32,23),(20,29,24),(33,49,44),(34,50,41),(35,51,42),(36,52,43)], [(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)], [(1,3),(5,68),(6,67),(7,66),(8,65),(9,56),(10,55),(11,54),(12,53),(14,16),(17,19),(21,32),(22,31),(23,30),(24,29),(25,60),(26,59),(27,58),(28,57),(33,51),(34,50),(35,49),(36,52),(37,39),(42,44),(45,47),(61,69),(62,72),(63,71),(64,70)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6R | 6S | ··· | 6Z | 12A | 12B | 12C | 12D | 12E | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D12 | He3⋊C2 | C2×He3⋊C2 | C2×He3⋊C2 | He3⋊5D4 |
| kernel | C2×He3⋊5D4 | He3⋊5D4 | C2×C4×He3 | C22×He3⋊C2 | C6×C12 | C2×He3 | C3×C12 | C62 | C3×C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 8 | 4 | 16 | 4 | 8 | 4 | 4 |
Matrix representation of C2×He3⋊5D4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 12 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[12,1,0,0,0,12,0,0,0,0,0,0,0,0,3,0,0,12,0,0,0,0,0,4,0],[10,6,0,0,0,7,3,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,12,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12] >;
C2×He3⋊5D4 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes_5D_4 % in TeX
G:=Group("C2xHe3:5D4"); // GroupNames label
G:=SmallGroup(432,386);
// by ID
G=gap.SmallGroup(432,386);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,58,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations