non-abelian, supersoluble, monomial
Aliases: C62⋊4Dic3, C62.32D6, (C2×C62).4S3, He3⋊8(C22⋊C4), (C2×He3).38D4, (C22×He3)⋊5C4, C2.3(He3⋊7D4), (C23×He3).3C2, C3.2(C62⋊5C4), C22⋊2(He3⋊3C4), C6.31(C32⋊7D4), C32⋊4(C6.D4), C23.2(He3⋊C2), (C22×He3).25C22, (C2×He3⋊3C4)⋊3C2, C6.27(C2×C3⋊Dic3), C2.5(C2×He3⋊3C4), (C3×C6).39(C3⋊D4), (C22×C6).9(C3⋊S3), (C2×He3).34(C2×C4), (C2×C6).5(C3⋊Dic3), (C3×C6).20(C2×Dic3), C22.7(C2×He3⋊C2), (C2×C6).55(C2×C3⋊S3), SmallGroup(432,199)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊4Dic3
G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=ab4, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 605 in 187 conjugacy classes, 59 normal (13 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, He3, C3×Dic3, C62, C62, C6.D4, C3×C22⋊C4, C2×He3, C2×He3, C2×He3, C6×Dic3, C2×C62, He3⋊3C4, C22×He3, C22×He3, C22×He3, C3×C6.D4, C2×He3⋊3C4, C23×He3, C62⋊4Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C2×Dic3, C3⋊D4, C3⋊Dic3, C2×C3⋊S3, C6.D4, He3⋊C2, C2×C3⋊Dic3, C32⋊7D4, He3⋊3C4, C2×He3⋊C2, C62⋊5C4, C2×He3⋊3C4, He3⋊7D4, C62⋊4Dic3
►On 72 points
(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 22 12 16 13 21)(2 23 10 17 14 19)(3 24 11 18 15 20)(4 27 9 29 36 33)(5 25 7 30 34 31)(6 26 8 28 35 32)(37 69 48 40 72 45)(38 70 43 41 67 46)(39 71 44 42 68 47)(49 66 58 52 63 55)(50 61 59 53 64 56)(51 62 60 54 65 57)
(1 28 3 25 10 27)(2 33 13 26 15 31)(4 21 8 20 5 23)(6 18 34 19 36 16)(7 17 9 22 35 24)(11 30 14 29 12 32)(37 58 44 60 41 61)(38 64 40 55 47 57)(39 51 70 56 72 49)(42 54 67 59 69 52)(43 50 45 66 71 62)(46 53 48 63 68 65)
(1 39 25 56)(2 46 26 63)(3 72 27 51)(4 57 20 40)(5 64 21 47)(6 52 19 67)(7 50 22 71)(8 55 23 38)(9 62 24 45)(10 70 28 49)(11 37 29 60)(12 44 30 61)(13 68 31 53)(14 41 32 58)(15 48 33 65)(16 42 34 59)(17 43 35 66)(18 69 36 54)
G:=sub<Sym(72)| (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), (1,22,12,16,13,21)(2,23,10,17,14,19)(3,24,11,18,15,20)(4,27,9,29,36,33)(5,25,7,30,34,31)(6,26,8,28,35,32)(37,69,48,40,72,45)(38,70,43,41,67,46)(39,71,44,42,68,47)(49,66,58,52,63,55)(50,61,59,53,64,56)(51,62,60,54,65,57), (1,28,3,25,10,27)(2,33,13,26,15,31)(4,21,8,20,5,23)(6,18,34,19,36,16)(7,17,9,22,35,24)(11,30,14,29,12,32)(37,58,44,60,41,61)(38,64,40,55,47,57)(39,51,70,56,72,49)(42,54,67,59,69,52)(43,50,45,66,71,62)(46,53,48,63,68,65), (1,39,25,56)(2,46,26,63)(3,72,27,51)(4,57,20,40)(5,64,21,47)(6,52,19,67)(7,50,22,71)(8,55,23,38)(9,62,24,45)(10,70,28,49)(11,37,29,60)(12,44,30,61)(13,68,31,53)(14,41,32,58)(15,48,33,65)(16,42,34,59)(17,43,35,66)(18,69,36,54)>;
G:=Group( (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), (1,22,12,16,13,21)(2,23,10,17,14,19)(3,24,11,18,15,20)(4,27,9,29,36,33)(5,25,7,30,34,31)(6,26,8,28,35,32)(37,69,48,40,72,45)(38,70,43,41,67,46)(39,71,44,42,68,47)(49,66,58,52,63,55)(50,61,59,53,64,56)(51,62,60,54,65,57), (1,28,3,25,10,27)(2,33,13,26,15,31)(4,21,8,20,5,23)(6,18,34,19,36,16)(7,17,9,22,35,24)(11,30,14,29,12,32)(37,58,44,60,41,61)(38,64,40,55,47,57)(39,51,70,56,72,49)(42,54,67,59,69,52)(43,50,45,66,71,62)(46,53,48,63,68,65), (1,39,25,56)(2,46,26,63)(3,72,27,51)(4,57,20,40)(5,64,21,47)(6,52,19,67)(7,50,22,71)(8,55,23,38)(9,62,24,45)(10,70,28,49)(11,37,29,60)(12,44,30,61)(13,68,31,53)(14,41,32,58)(15,48,33,65)(16,42,34,59)(17,43,35,66)(18,69,36,54) );
G=PermutationGroup([[(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)], [(1,22,12,16,13,21),(2,23,10,17,14,19),(3,24,11,18,15,20),(4,27,9,29,36,33),(5,25,7,30,34,31),(6,26,8,28,35,32),(37,69,48,40,72,45),(38,70,43,41,67,46),(39,71,44,42,68,47),(49,66,58,52,63,55),(50,61,59,53,64,56),(51,62,60,54,65,57)], [(1,28,3,25,10,27),(2,33,13,26,15,31),(4,21,8,20,5,23),(6,18,34,19,36,16),(7,17,9,22,35,24),(11,30,14,29,12,32),(37,58,44,60,41,61),(38,64,40,55,47,57),(39,51,70,56,72,49),(42,54,67,59,69,52),(43,50,45,66,71,62),(46,53,48,63,68,65)], [(1,39,25,56),(2,46,26,63),(3,72,27,51),(4,57,20,40),(5,64,21,47),(6,52,19,67),(7,50,22,71),(8,55,23,38),(9,62,24,45),(10,70,28,49),(11,37,29,60),(12,44,30,61),(13,68,31,53),(14,41,32,58),(15,48,33,65),(16,42,34,59),(17,43,35,66),(18,69,36,54)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6AL | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 18 | ··· | 18 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | C3⋊D4 | He3⋊C2 | He3⋊3C4 | C2×He3⋊C2 | He3⋊7D4 |
| kernel | C62⋊4Dic3 | C2×He3⋊3C4 | C23×He3 | C22×He3 | C2×C62 | C2×He3 | C62 | C62 | C3×C6 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 2 | 8 | 4 | 16 | 4 | 8 | 4 | 4 |
Matrix representation of C62⋊4Dic3 ►in GL5(𝔽13)
| 9 | 0 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 4 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 | 0 |
| 2 | 8 | 0 | 0 | 0 |
| 1 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 5 | 0 | 0 |
G:=sub<GL(5,GF(13))| [9,10,0,0,0,0,10,0,0,0,0,0,0,0,4,0,0,12,0,0,0,0,0,10,0],[12,0,0,0,0,0,12,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[12,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,12,0],[2,1,0,0,0,8,11,0,0,0,0,0,0,0,5,0,0,0,5,0,0,0,5,0,0] >;
C62⋊4Dic3 in GAP, Magma, Sage, TeX
C_6^2\rtimes_4{\rm Dic}_3 % in TeX
G:=Group("C6^2:4Dic3"); // GroupNames label
G:=SmallGroup(432,199);
// by ID
G=gap.SmallGroup(432,199);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a*b^4,d*a*d^-1=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations