direct product, non-abelian, soluble
Aliases: C22×He3⋊C4, He3⋊2(C22×C4), (C22×He3)⋊2C4, He3⋊C2.3C23, C3.(C22×C32⋊C4), (C2×He3)⋊1(C2×C4), C6.18(C2×C32⋊C4), (C2×He3⋊C2)⋊5C4, He3⋊C2⋊2(C2×C4), (C2×C6).7(C32⋊C4), (C22×He3⋊C2).5C2, (C2×He3⋊C2).20C22, SmallGroup(432,543)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C22×He3⋊C4 |
Generators and relations for C22×He3⋊C4
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf-1=cde, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 777 in 145 conjugacy classes, 37 normal (9 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3×C6, C2×C12, C22×S3, C22×C6, He3, S3×C6, C62, C22×C12, He3⋊C2, He3⋊C2, C2×He3, S3×C2×C6, He3⋊C4, C2×He3⋊C2, C22×He3, C2×He3⋊C4, C22×He3⋊C2, C22×He3⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C32⋊C4, C2×C32⋊C4, He3⋊C4, C22×C32⋊C4, C2×He3⋊C4, C22×He3⋊C4
►On 72 points
Generators in S72
(1 3)(2 4)(5 7)(6 8)(9 71)(10 72)(11 69)(12 70)(13 46)(14 47)(15 48)(16 45)(17 60)(18 57)(19 58)(20 59)(21 51)(22 52)(23 49)(24 50)(25 32)(26 29)(27 30)(28 31)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(53 64)(54 61)(55 62)(56 63)(65 67)(66 68)
(1 35)(2 36)(3 33)(4 34)(5 41)(6 42)(7 43)(8 44)(9 13)(10 14)(11 15)(12 16)(17 31)(18 32)(19 29)(20 30)(21 55)(22 56)(23 53)(24 54)(25 57)(26 58)(27 59)(28 60)(37 65)(38 66)(39 67)(40 68)(45 70)(46 71)(47 72)(48 69)(49 64)(50 61)(51 62)(52 63)
(1 11 26)(2 32 62)(3 69 29)(4 25 55)(5 53 72)(6 28 54)(7 64 10)(8 31 61)(9 39 63)(12 27 38)(13 67 52)(14 43 49)(15 58 35)(16 59 66)(17 50 44)(18 51 36)(19 33 48)(20 68 45)(21 34 57)(22 46 65)(23 47 41)(24 42 60)(30 40 70)(37 56 71)
(1 37 8)(2 38 5)(3 39 6)(4 40 7)(9 54 29)(10 55 30)(11 56 31)(12 53 32)(13 24 19)(14 21 20)(15 22 17)(16 23 18)(25 70 64)(26 71 61)(27 72 62)(28 69 63)(33 67 42)(34 68 43)(35 65 44)(36 66 41)(45 49 57)(46 50 58)(47 51 59)(48 52 60)
(2 32 27)(4 25 30)(5 53 62)(7 64 55)(9 29 54)(10 40 70)(11 56 31)(12 72 38)(13 19 24)(14 68 45)(15 22 17)(16 47 66)(18 59 36)(20 34 57)(21 43 49)(23 51 41)(26 61 71)(28 69 63)(46 58 50)(48 52 60)
(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)
G:=sub<Sym(72)| (1,3)(2,4)(5,7)(6,8)(9,71)(10,72)(11,69)(12,70)(13,46)(14,47)(15,48)(16,45)(17,60)(18,57)(19,58)(20,59)(21,51)(22,52)(23,49)(24,50)(25,32)(26,29)(27,30)(28,31)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(53,64)(54,61)(55,62)(56,63)(65,67)(66,68), (1,35)(2,36)(3,33)(4,34)(5,41)(6,42)(7,43)(8,44)(9,13)(10,14)(11,15)(12,16)(17,31)(18,32)(19,29)(20,30)(21,55)(22,56)(23,53)(24,54)(25,57)(26,58)(27,59)(28,60)(37,65)(38,66)(39,67)(40,68)(45,70)(46,71)(47,72)(48,69)(49,64)(50,61)(51,62)(52,63), (1,11,26)(2,32,62)(3,69,29)(4,25,55)(5,53,72)(6,28,54)(7,64,10)(8,31,61)(9,39,63)(12,27,38)(13,67,52)(14,43,49)(15,58,35)(16,59,66)(17,50,44)(18,51,36)(19,33,48)(20,68,45)(21,34,57)(22,46,65)(23,47,41)(24,42,60)(30,40,70)(37,56,71), (1,37,8)(2,38,5)(3,39,6)(4,40,7)(9,54,29)(10,55,30)(11,56,31)(12,53,32)(13,24,19)(14,21,20)(15,22,17)(16,23,18)(25,70,64)(26,71,61)(27,72,62)(28,69,63)(33,67,42)(34,68,43)(35,65,44)(36,66,41)(45,49,57)(46,50,58)(47,51,59)(48,52,60), (2,32,27)(4,25,30)(5,53,62)(7,64,55)(9,29,54)(10,40,70)(11,56,31)(12,72,38)(13,19,24)(14,68,45)(15,22,17)(16,47,66)(18,59,36)(20,34,57)(21,43,49)(23,51,41)(26,61,71)(28,69,63)(46,58,50)(48,52,60), (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)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,71)(10,72)(11,69)(12,70)(13,46)(14,47)(15,48)(16,45)(17,60)(18,57)(19,58)(20,59)(21,51)(22,52)(23,49)(24,50)(25,32)(26,29)(27,30)(28,31)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(53,64)(54,61)(55,62)(56,63)(65,67)(66,68), (1,35)(2,36)(3,33)(4,34)(5,41)(6,42)(7,43)(8,44)(9,13)(10,14)(11,15)(12,16)(17,31)(18,32)(19,29)(20,30)(21,55)(22,56)(23,53)(24,54)(25,57)(26,58)(27,59)(28,60)(37,65)(38,66)(39,67)(40,68)(45,70)(46,71)(47,72)(48,69)(49,64)(50,61)(51,62)(52,63), (1,11,26)(2,32,62)(3,69,29)(4,25,55)(5,53,72)(6,28,54)(7,64,10)(8,31,61)(9,39,63)(12,27,38)(13,67,52)(14,43,49)(15,58,35)(16,59,66)(17,50,44)(18,51,36)(19,33,48)(20,68,45)(21,34,57)(22,46,65)(23,47,41)(24,42,60)(30,40,70)(37,56,71), (1,37,8)(2,38,5)(3,39,6)(4,40,7)(9,54,29)(10,55,30)(11,56,31)(12,53,32)(13,24,19)(14,21,20)(15,22,17)(16,23,18)(25,70,64)(26,71,61)(27,72,62)(28,69,63)(33,67,42)(34,68,43)(35,65,44)(36,66,41)(45,49,57)(46,50,58)(47,51,59)(48,52,60), (2,32,27)(4,25,30)(5,53,62)(7,64,55)(9,29,54)(10,40,70)(11,56,31)(12,72,38)(13,19,24)(14,68,45)(15,22,17)(16,47,66)(18,59,36)(20,34,57)(21,43,49)(23,51,41)(26,61,71)(28,69,63)(46,58,50)(48,52,60), (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) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,71),(10,72),(11,69),(12,70),(13,46),(14,47),(15,48),(16,45),(17,60),(18,57),(19,58),(20,59),(21,51),(22,52),(23,49),(24,50),(25,32),(26,29),(27,30),(28,31),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(53,64),(54,61),(55,62),(56,63),(65,67),(66,68)], [(1,35),(2,36),(3,33),(4,34),(5,41),(6,42),(7,43),(8,44),(9,13),(10,14),(11,15),(12,16),(17,31),(18,32),(19,29),(20,30),(21,55),(22,56),(23,53),(24,54),(25,57),(26,58),(27,59),(28,60),(37,65),(38,66),(39,67),(40,68),(45,70),(46,71),(47,72),(48,69),(49,64),(50,61),(51,62),(52,63)], [(1,11,26),(2,32,62),(3,69,29),(4,25,55),(5,53,72),(6,28,54),(7,64,10),(8,31,61),(9,39,63),(12,27,38),(13,67,52),(14,43,49),(15,58,35),(16,59,66),(17,50,44),(18,51,36),(19,33,48),(20,68,45),(21,34,57),(22,46,65),(23,47,41),(24,42,60),(30,40,70),(37,56,71)], [(1,37,8),(2,38,5),(3,39,6),(4,40,7),(9,54,29),(10,55,30),(11,56,31),(12,53,32),(13,24,19),(14,21,20),(15,22,17),(16,23,18),(25,70,64),(26,71,61),(27,72,62),(28,69,63),(33,67,42),(34,68,43),(35,65,44),(36,66,41),(45,49,57),(46,50,58),(47,51,59),(48,52,60)], [(2,32,27),(4,25,30),(5,53,62),(7,64,55),(9,29,54),(10,40,70),(11,56,31),(12,72,38),(13,19,24),(14,68,45),(15,22,17),(16,47,66),(18,59,36),(20,34,57),(21,43,49),(23,51,41),(26,61,71),(28,69,63),(46,58,50),(48,52,60)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4H | 6A | ··· | 6F | 6G | ··· | 6N | 6O | ··· | 6T | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 1 | 1 | 12 | 12 | 9 | ··· | 9 | 1 | ··· | 1 | 9 | ··· | 9 | 12 | ··· | 12 | 9 | ··· | 9 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | He3⋊C4 | C2×He3⋊C4 | C32⋊C4 | C2×C32⋊C4 |
| kernel | C22×He3⋊C4 | C2×He3⋊C4 | C22×He3⋊C2 | C2×He3⋊C2 | C22×He3 | C22 | C2 | C2×C6 | C6 |
| # reps | 1 | 6 | 1 | 6 | 2 | 8 | 24 | 2 | 6 |
Matrix representation of C22×He3⋊C4 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 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 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 11 | 7 |
| 0 | 11 | 7 | 11 |
| 0 | 8 | 7 | 7 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[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],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[1,0,0,0,0,11,11,8,0,11,7,7,0,7,11,7] >;
C22×He3⋊C4 in GAP, Magma, Sage, TeX
C_2^2\times {\rm He}_3\rtimes C_4 % in TeX
G:=Group("C2^2xHe3:C4"); // GroupNames label
G:=SmallGroup(432,543);
// by ID
G=gap.SmallGroup(432,543);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,3924,165,5381,1286,537]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^3=f^4=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,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,e*c*e^-1=c*d^-1,f*c*f^-1=c*d*e,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations