metabelian, supersoluble, monomial
Aliases: C62.154C23, C32⋊102+ 1+4, (C3×D4)⋊20D6, (C2×C12)⋊10D6, (C3×Q8)⋊21D6, C3⋊5(D4○D12), (C6×C12)⋊16C22, C6.65(S3×C23), (C3×C6).64C24, C12⋊S3⋊30C22, C12.59D6⋊13C2, C12.26D6⋊10C2, C12.116(C22×S3), (C3×C12).158C23, (D4×C32)⋊27C22, C32⋊7D4⋊15C22, C3⋊Dic3.52C23, (Q8×C32)⋊24C22, C32⋊4Q8⋊28C22, D4⋊8(C2×C3⋊S3), Q8⋊8(C2×C3⋊S3), (D4×C3⋊S3)⋊10C2, (C3×C4○D4)⋊8S3, C4○D4⋊5(C3⋊S3), (C4×C3⋊S3)⋊9C22, (C32×C4○D4)⋊9C2, (C2×C12⋊S3)⋊22C2, C4.33(C22×C3⋊S3), C2.13(C23×C3⋊S3), (C2×C3⋊S3).56C23, (C2×C6).18(C22×S3), C22.3(C22×C3⋊S3), (C22×C3⋊S3)⋊12C22, (C2×C4)⋊4(C2×C3⋊S3), SmallGroup(288,1014)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.154C23
G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=b3, ab=ba, cac=a-1, dad=ab3, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >
Subgroups: 2052 in 498 conjugacy classes, 153 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C32, Dic3, C12, D6, C2×C6, C2×D4, C4○D4, C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, 2+ 1+4, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, D4×C32, Q8×C32, C22×C3⋊S3, D4○D12, C2×C12⋊S3, C12.59D6, D4×C3⋊S3, C12.26D6, C32×C4○D4, C62.154C23
Quotients: C1, C2, C22, S3, C23, D6, C24, C3⋊S3, C22×S3, 2+ 1+4, C2×C3⋊S3, S3×C23, C22×C3⋊S3, D4○D12, C23×C3⋊S3, C62.154C23
►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 35 31 6 11 9)(2 36 32 4 12 7)(3 34 33 5 10 8)(13 23 19 18 29 26)(14 24 20 16 30 27)(15 22 21 17 28 25)(37 72 65 40 69 62)(38 67 66 41 70 63)(39 68 61 42 71 64)(43 49 55 46 52 58)(44 50 56 47 53 59)(45 51 57 48 54 60)
(1 32)(2 31)(3 33)(4 9)(5 8)(6 7)(11 12)(13 16)(14 18)(15 17)(19 24)(20 23)(21 22)(25 28)(26 30)(27 29)(35 36)(37 42)(38 41)(39 40)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(55 57)(58 60)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)
(1 51)(2 49)(3 53)(4 52)(5 50)(6 54)(7 43)(8 47)(9 45)(10 56)(11 60)(12 58)(13 39)(14 37)(15 41)(16 40)(17 38)(18 42)(19 61)(20 65)(21 63)(22 70)(23 68)(24 72)(25 66)(26 64)(27 62)(28 67)(29 71)(30 69)(31 48)(32 46)(33 44)(34 59)(35 57)(36 55)
(1 19 6 26)(2 20 4 27)(3 21 5 25)(7 24 32 30)(8 22 33 28)(9 23 31 29)(10 15 34 17)(11 13 35 18)(12 14 36 16)(37 55 40 58)(38 56 41 59)(39 57 42 60)(43 72 46 69)(44 67 47 70)(45 68 48 71)(49 65 52 62)(50 66 53 63)(51 61 54 64)
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), (1,35,31,6,11,9)(2,36,32,4,12,7)(3,34,33,5,10,8)(13,23,19,18,29,26)(14,24,20,16,30,27)(15,22,21,17,28,25)(37,72,65,40,69,62)(38,67,66,41,70,63)(39,68,61,42,71,64)(43,49,55,46,52,58)(44,50,56,47,53,59)(45,51,57,48,54,60), (1,32)(2,31)(3,33)(4,9)(5,8)(6,7)(11,12)(13,16)(14,18)(15,17)(19,24)(20,23)(21,22)(25,28)(26,30)(27,29)(35,36)(37,42)(38,41)(39,40)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(55,57)(58,60)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67), (1,51)(2,49)(3,53)(4,52)(5,50)(6,54)(7,43)(8,47)(9,45)(10,56)(11,60)(12,58)(13,39)(14,37)(15,41)(16,40)(17,38)(18,42)(19,61)(20,65)(21,63)(22,70)(23,68)(24,72)(25,66)(26,64)(27,62)(28,67)(29,71)(30,69)(31,48)(32,46)(33,44)(34,59)(35,57)(36,55), (1,19,6,26)(2,20,4,27)(3,21,5,25)(7,24,32,30)(8,22,33,28)(9,23,31,29)(10,15,34,17)(11,13,35,18)(12,14,36,16)(37,55,40,58)(38,56,41,59)(39,57,42,60)(43,72,46,69)(44,67,47,70)(45,68,48,71)(49,65,52,62)(50,66,53,63)(51,61,54,64)>;
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), (1,35,31,6,11,9)(2,36,32,4,12,7)(3,34,33,5,10,8)(13,23,19,18,29,26)(14,24,20,16,30,27)(15,22,21,17,28,25)(37,72,65,40,69,62)(38,67,66,41,70,63)(39,68,61,42,71,64)(43,49,55,46,52,58)(44,50,56,47,53,59)(45,51,57,48,54,60), (1,32)(2,31)(3,33)(4,9)(5,8)(6,7)(11,12)(13,16)(14,18)(15,17)(19,24)(20,23)(21,22)(25,28)(26,30)(27,29)(35,36)(37,42)(38,41)(39,40)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(55,57)(58,60)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67), (1,51)(2,49)(3,53)(4,52)(5,50)(6,54)(7,43)(8,47)(9,45)(10,56)(11,60)(12,58)(13,39)(14,37)(15,41)(16,40)(17,38)(18,42)(19,61)(20,65)(21,63)(22,70)(23,68)(24,72)(25,66)(26,64)(27,62)(28,67)(29,71)(30,69)(31,48)(32,46)(33,44)(34,59)(35,57)(36,55), (1,19,6,26)(2,20,4,27)(3,21,5,25)(7,24,32,30)(8,22,33,28)(9,23,31,29)(10,15,34,17)(11,13,35,18)(12,14,36,16)(37,55,40,58)(38,56,41,59)(39,57,42,60)(43,72,46,69)(44,67,47,70)(45,68,48,71)(49,65,52,62)(50,66,53,63)(51,61,54,64) );
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)], [(1,35,31,6,11,9),(2,36,32,4,12,7),(3,34,33,5,10,8),(13,23,19,18,29,26),(14,24,20,16,30,27),(15,22,21,17,28,25),(37,72,65,40,69,62),(38,67,66,41,70,63),(39,68,61,42,71,64),(43,49,55,46,52,58),(44,50,56,47,53,59),(45,51,57,48,54,60)], [(1,32),(2,31),(3,33),(4,9),(5,8),(6,7),(11,12),(13,16),(14,18),(15,17),(19,24),(20,23),(21,22),(25,28),(26,30),(27,29),(35,36),(37,42),(38,41),(39,40),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(55,57),(58,60),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67)], [(1,51),(2,49),(3,53),(4,52),(5,50),(6,54),(7,43),(8,47),(9,45),(10,56),(11,60),(12,58),(13,39),(14,37),(15,41),(16,40),(17,38),(18,42),(19,61),(20,65),(21,63),(22,70),(23,68),(24,72),(25,66),(26,64),(27,62),(28,67),(29,71),(30,69),(31,48),(32,46),(33,44),(34,59),(35,57),(36,55)], [(1,19,6,26),(2,20,4,27),(3,21,5,25),(7,24,32,30),(8,22,33,28),(9,23,31,29),(10,15,34,17),(11,13,35,18),(12,14,36,16),(37,55,40,58),(38,56,41,59),(39,57,42,60),(43,72,46,69),(44,67,47,70),(45,68,48,71),(49,65,52,62),(50,66,53,63),(51,61,54,64)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | 2+ 1+4 | D4○D12 |
| kernel | C62.154C23 | C2×C12⋊S3 | C12.59D6 | D4×C3⋊S3 | C12.26D6 | C32×C4○D4 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C32 | C3 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 12 | 12 | 4 | 1 | 8 |
Matrix representation of C62.154C23 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 7 | 10 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10] >;
C62.154C23 in GAP, Magma, Sage, TeX
C_6^2._{154}C_2^3 % in TeX
G:=Group("C6^2.154C2^3"); // GroupNames label
G:=SmallGroup(288,1014);
// by ID
G=gap.SmallGroup(288,1014);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=1,e^2=b^3,a*b=b*a,c*a*c=a^-1,d*a*d=a*b^3,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,d*e=e*d>;
// generators/relations