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, (C3×C6).64C24, C6.65(S3×C23), C12⋊S3⋊30C22, C12.59D6⋊13C2, C12.26D6⋊10C2, (C3×C12).158C23, C12.116(C22×S3), (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, C2.13(C23×C3⋊S3), C4.33(C22×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
Subgroups: 2052 in 498 conjugacy classes, 153 normal (12 characteristic)
C1, C2, C2 [×9], C3 [×4], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×24], C6 [×4], C6 [×12], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C32, Dic3 [×8], C12 [×16], D6 [×48], C2×C6 [×12], C2×D4 [×9], C4○D4, C4○D4 [×5], C3⋊S3 [×6], C3×C6, C3×C6 [×3], Dic6 [×4], C4×S3 [×24], D12 [×36], C3⋊D4 [×24], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], C22×S3 [×24], 2+ (1+4), C3⋊Dic3 [×2], C3×C12, C3×C12 [×3], C2×C3⋊S3 [×6], C2×C3⋊S3 [×6], C62 [×3], C2×D12 [×12], C4○D12 [×12], S3×D4 [×24], Q8⋊3S3 [×8], C3×C4○D4 [×4], C32⋊4Q8, C4×C3⋊S3 [×6], C12⋊S3 [×9], C32⋊7D4 [×6], C6×C12 [×3], D4×C32 [×3], Q8×C32, C22×C3⋊S3 [×6], D4○D12 [×4], C2×C12⋊S3 [×3], C12.59D6 [×3], D4×C3⋊S3 [×6], C12.26D6 [×2], C32×C4○D4, C62.154C23
Quotients:
C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C24, C3⋊S3, C22×S3 [×28], 2+ (1+4), C2×C3⋊S3 [×7], S3×C23 [×4], C22×C3⋊S3 [×7], D4○D12 [×4], C23×C3⋊S3, C62.154C23
Generators and relations
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 >
►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 21 4 16 12 22)(2 19 5 17 10 23)(3 20 6 18 11 24)(7 29 14 34 33 26)(8 30 15 35 31 27)(9 28 13 36 32 25)(37 62 59 40 65 56)(38 63 60 41 66 57)(39 64 55 42 61 58)(43 49 68 46 52 71)(44 50 69 47 53 72)(45 51 70 48 54 67)
(1 11)(2 10)(3 12)(4 6)(7 26)(8 25)(9 27)(13 35)(14 34)(15 36)(16 20)(17 19)(18 21)(22 24)(28 31)(29 33)(30 32)(37 41)(38 40)(43 51)(44 50)(45 49)(46 54)(47 53)(48 52)(55 61)(56 66)(57 65)(58 64)(59 63)(60 62)(67 68)(69 72)(70 71)
(1 66)(2 64)(3 62)(4 38)(5 42)(6 40)(7 50)(8 54)(9 52)(10 58)(11 56)(12 60)(13 43)(14 47)(15 45)(16 63)(17 61)(18 65)(19 55)(20 59)(21 57)(22 41)(23 39)(24 37)(25 46)(26 44)(27 48)(28 71)(29 69)(30 67)(31 70)(32 68)(33 72)(34 53)(35 51)(36 49)
(1 13 16 25)(2 14 17 26)(3 15 18 27)(4 32 22 28)(5 33 23 29)(6 31 24 30)(7 19 34 10)(8 20 35 11)(9 21 36 12)(37 67 40 70)(38 68 41 71)(39 69 42 72)(43 63 46 66)(44 64 47 61)(45 65 48 62)(49 60 52 57)(50 55 53 58)(51 56 54 59)
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), (1,21,4,16,12,22)(2,19,5,17,10,23)(3,20,6,18,11,24)(7,29,14,34,33,26)(8,30,15,35,31,27)(9,28,13,36,32,25)(37,62,59,40,65,56)(38,63,60,41,66,57)(39,64,55,42,61,58)(43,49,68,46,52,71)(44,50,69,47,53,72)(45,51,70,48,54,67), (1,11)(2,10)(3,12)(4,6)(7,26)(8,25)(9,27)(13,35)(14,34)(15,36)(16,20)(17,19)(18,21)(22,24)(28,31)(29,33)(30,32)(37,41)(38,40)(43,51)(44,50)(45,49)(46,54)(47,53)(48,52)(55,61)(56,66)(57,65)(58,64)(59,63)(60,62)(67,68)(69,72)(70,71), (1,66)(2,64)(3,62)(4,38)(5,42)(6,40)(7,50)(8,54)(9,52)(10,58)(11,56)(12,60)(13,43)(14,47)(15,45)(16,63)(17,61)(18,65)(19,55)(20,59)(21,57)(22,41)(23,39)(24,37)(25,46)(26,44)(27,48)(28,71)(29,69)(30,67)(31,70)(32,68)(33,72)(34,53)(35,51)(36,49), (1,13,16,25)(2,14,17,26)(3,15,18,27)(4,32,22,28)(5,33,23,29)(6,31,24,30)(7,19,34,10)(8,20,35,11)(9,21,36,12)(37,67,40,70)(38,68,41,71)(39,69,42,72)(43,63,46,66)(44,64,47,61)(45,65,48,62)(49,60,52,57)(50,55,53,58)(51,56,54,59)>;
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), (1,21,4,16,12,22)(2,19,5,17,10,23)(3,20,6,18,11,24)(7,29,14,34,33,26)(8,30,15,35,31,27)(9,28,13,36,32,25)(37,62,59,40,65,56)(38,63,60,41,66,57)(39,64,55,42,61,58)(43,49,68,46,52,71)(44,50,69,47,53,72)(45,51,70,48,54,67), (1,11)(2,10)(3,12)(4,6)(7,26)(8,25)(9,27)(13,35)(14,34)(15,36)(16,20)(17,19)(18,21)(22,24)(28,31)(29,33)(30,32)(37,41)(38,40)(43,51)(44,50)(45,49)(46,54)(47,53)(48,52)(55,61)(56,66)(57,65)(58,64)(59,63)(60,62)(67,68)(69,72)(70,71), (1,66)(2,64)(3,62)(4,38)(5,42)(6,40)(7,50)(8,54)(9,52)(10,58)(11,56)(12,60)(13,43)(14,47)(15,45)(16,63)(17,61)(18,65)(19,55)(20,59)(21,57)(22,41)(23,39)(24,37)(25,46)(26,44)(27,48)(28,71)(29,69)(30,67)(31,70)(32,68)(33,72)(34,53)(35,51)(36,49), (1,13,16,25)(2,14,17,26)(3,15,18,27)(4,32,22,28)(5,33,23,29)(6,31,24,30)(7,19,34,10)(8,20,35,11)(9,21,36,12)(37,67,40,70)(38,68,41,71)(39,69,42,72)(43,63,46,66)(44,64,47,61)(45,65,48,62)(49,60,52,57)(50,55,53,58)(51,56,54,59) );
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)], [(1,21,4,16,12,22),(2,19,5,17,10,23),(3,20,6,18,11,24),(7,29,14,34,33,26),(8,30,15,35,31,27),(9,28,13,36,32,25),(37,62,59,40,65,56),(38,63,60,41,66,57),(39,64,55,42,61,58),(43,49,68,46,52,71),(44,50,69,47,53,72),(45,51,70,48,54,67)], [(1,11),(2,10),(3,12),(4,6),(7,26),(8,25),(9,27),(13,35),(14,34),(15,36),(16,20),(17,19),(18,21),(22,24),(28,31),(29,33),(30,32),(37,41),(38,40),(43,51),(44,50),(45,49),(46,54),(47,53),(48,52),(55,61),(56,66),(57,65),(58,64),(59,63),(60,62),(67,68),(69,72),(70,71)], [(1,66),(2,64),(3,62),(4,38),(5,42),(6,40),(7,50),(8,54),(9,52),(10,58),(11,56),(12,60),(13,43),(14,47),(15,45),(16,63),(17,61),(18,65),(19,55),(20,59),(21,57),(22,41),(23,39),(24,37),(25,46),(26,44),(27,48),(28,71),(29,69),(30,67),(31,70),(32,68),(33,72),(34,53),(35,51),(36,49)], [(1,13,16,25),(2,14,17,26),(3,15,18,27),(4,32,22,28),(5,33,23,29),(6,31,24,30),(7,19,34,10),(8,20,35,11),(9,21,36,12),(37,67,40,70),(38,68,41,71),(39,69,42,72),(43,63,46,66),(44,64,47,61),(45,65,48,62),(49,60,52,57),(50,55,53,58),(51,56,54,59)])
Matrix representation ►G ⊆ 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] >;
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 |
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