?
G = C32⋊82+ (1+4) order 288 = 25·32
2nd semidirect product of C32 and 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial
Aliases: C32⋊82+ (1+4), C62.279C23, (C6×D4)⋊7S3, (C2×C12)⋊8D6, (C3×D4)⋊18D6, (C22×C6)⋊11D6, C3⋊5(D4⋊6D6), (C6×C12)⋊14C22, (C3×C6).59C24, C6.60(S3×C23), C12.D6⋊9C2, (C2×C62)⋊12C22, C12⋊S3⋊26C22, C12.59D6⋊10C2, (C3×C12).130C23, C12.111(C22×S3), (D4×C32)⋊25C22, C32⋊7D4⋊13C22, C3⋊Dic3.48C23, C32⋊4Q8⋊24C22, D4⋊6(C2×C3⋊S3), (D4×C3⋊S3)⋊9C2, (D4×C3×C6)⋊14C2, C23⋊3(C2×C3⋊S3), (C2×D4)⋊7(C3⋊S3), (C4×C3⋊S3)⋊8C22, C2.8(C23×C3⋊S3), C4.21(C22×C3⋊S3), (C2×C3⋊S3).52C23, (C2×C32⋊7D4)⋊20C2, (C2×C6).16(C22×S3), C22.6(C22×C3⋊S3), (C22×C3⋊S3)⋊11C22, (C2×C3⋊Dic3)⋊14C22, (C2×C4)⋊3(C2×C3⋊S3), SmallGroup(288,1009)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1860 in 498 conjugacy classes, 153 normal (11 characteristic)
C1, C2, C2 [×9], C3 [×4], C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], S3 [×16], C6 [×4], C6 [×20], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C32, Dic3 [×16], C12 [×8], D6 [×32], C2×C6 [×20], C2×C6 [×8], C2×D4, C2×D4 [×8], C4○D4 [×6], C3⋊S3 [×4], C3×C6, C3×C6 [×5], Dic6 [×8], C4×S3 [×16], D12 [×8], C2×Dic3 [×16], C3⋊D4 [×48], C2×C12 [×4], C3×D4 [×16], C22×S3 [×16], C22×C6 [×8], 2+ (1+4), C3⋊Dic3 [×4], C3×C12 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C62 [×4], C62 [×2], C4○D12 [×8], S3×D4 [×16], D4⋊2S3 [×16], C2×C3⋊D4 [×16], C6×D4 [×4], C32⋊4Q8 [×2], C4×C3⋊S3 [×4], C12⋊S3 [×2], C2×C3⋊Dic3 [×4], C32⋊7D4 [×12], C6×C12, D4×C32 [×4], C22×C3⋊S3 [×4], C2×C62 [×2], D4⋊6D6 [×4], C12.59D6 [×2], D4×C3⋊S3 [×4], C12.D6 [×4], C2×C32⋊7D4 [×4], D4×C3×C6, C32⋊82+ (1+4)
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⋊6D6 [×4], C23×C3⋊S3, C32⋊82+ (1+4)
Generators and relations
G = < a,b,c,d,e,f | a3=b3=c4=d2=f2=1, e2=c2, ab=ba, ac=ca, ad=da, eae-1=a-1, af=fa, bc=cb, bd=db, ebe-1=b-1, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >
►On 72 points
(1 32 40)(2 29 37)(3 30 38)(4 31 39)(5 35 47)(6 36 48)(7 33 45)(8 34 46)(9 54 19)(10 55 20)(11 56 17)(12 53 18)(13 50 43)(14 51 44)(15 52 41)(16 49 42)(21 61 71)(22 62 72)(23 63 69)(24 64 70)(25 65 58)(26 66 59)(27 67 60)(28 68 57)
(1 23 19)(2 24 20)(3 21 17)(4 22 18)(5 66 49)(6 67 50)(7 68 51)(8 65 52)(9 32 63)(10 29 64)(11 30 61)(12 31 62)(13 48 27)(14 45 28)(15 46 25)(16 47 26)(33 57 44)(34 58 41)(35 59 42)(36 60 43)(37 70 55)(38 71 56)(39 72 53)(40 69 54)
(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 26)(2 25)(3 28)(4 27)(5 9)(6 12)(7 11)(8 10)(13 22)(14 21)(15 24)(16 23)(17 45)(18 48)(19 47)(20 46)(29 65)(30 68)(31 67)(32 66)(33 56)(34 55)(35 54)(36 53)(37 58)(38 57)(39 60)(40 59)(41 70)(42 69)(43 72)(44 71)(49 63)(50 62)(51 61)(52 64)
(1 2 3 4)(5 41 7 43)(6 42 8 44)(9 70 11 72)(10 71 12 69)(13 47 15 45)(14 48 16 46)(17 22 19 24)(18 23 20 21)(25 28 27 26)(29 38 31 40)(30 39 32 37)(33 50 35 52)(34 51 36 49)(53 63 55 61)(54 64 56 62)(57 67 59 65)(58 68 60 66)
(1 27)(2 28)(3 25)(4 26)(5 12)(6 9)(7 10)(8 11)(13 23)(14 24)(15 21)(16 22)(17 46)(18 47)(19 48)(20 45)(29 68)(30 65)(31 66)(32 67)(33 55)(34 56)(35 53)(36 54)(37 57)(38 58)(39 59)(40 60)(41 71)(42 72)(43 69)(44 70)(49 62)(50 63)(51 64)(52 61)
G:=sub<Sym(72)| (1,32,40)(2,29,37)(3,30,38)(4,31,39)(5,35,47)(6,36,48)(7,33,45)(8,34,46)(9,54,19)(10,55,20)(11,56,17)(12,53,18)(13,50,43)(14,51,44)(15,52,41)(16,49,42)(21,61,71)(22,62,72)(23,63,69)(24,64,70)(25,65,58)(26,66,59)(27,67,60)(28,68,57), (1,23,19)(2,24,20)(3,21,17)(4,22,18)(5,66,49)(6,67,50)(7,68,51)(8,65,52)(9,32,63)(10,29,64)(11,30,61)(12,31,62)(13,48,27)(14,45,28)(15,46,25)(16,47,26)(33,57,44)(34,58,41)(35,59,42)(36,60,43)(37,70,55)(38,71,56)(39,72,53)(40,69,54), (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), (1,26)(2,25)(3,28)(4,27)(5,9)(6,12)(7,11)(8,10)(13,22)(14,21)(15,24)(16,23)(17,45)(18,48)(19,47)(20,46)(29,65)(30,68)(31,67)(32,66)(33,56)(34,55)(35,54)(36,53)(37,58)(38,57)(39,60)(40,59)(41,70)(42,69)(43,72)(44,71)(49,63)(50,62)(51,61)(52,64), (1,2,3,4)(5,41,7,43)(6,42,8,44)(9,70,11,72)(10,71,12,69)(13,47,15,45)(14,48,16,46)(17,22,19,24)(18,23,20,21)(25,28,27,26)(29,38,31,40)(30,39,32,37)(33,50,35,52)(34,51,36,49)(53,63,55,61)(54,64,56,62)(57,67,59,65)(58,68,60,66), (1,27)(2,28)(3,25)(4,26)(5,12)(6,9)(7,10)(8,11)(13,23)(14,24)(15,21)(16,22)(17,46)(18,47)(19,48)(20,45)(29,68)(30,65)(31,66)(32,67)(33,55)(34,56)(35,53)(36,54)(37,57)(38,58)(39,59)(40,60)(41,71)(42,72)(43,69)(44,70)(49,62)(50,63)(51,64)(52,61)>;
G:=Group( (1,32,40)(2,29,37)(3,30,38)(4,31,39)(5,35,47)(6,36,48)(7,33,45)(8,34,46)(9,54,19)(10,55,20)(11,56,17)(12,53,18)(13,50,43)(14,51,44)(15,52,41)(16,49,42)(21,61,71)(22,62,72)(23,63,69)(24,64,70)(25,65,58)(26,66,59)(27,67,60)(28,68,57), (1,23,19)(2,24,20)(3,21,17)(4,22,18)(5,66,49)(6,67,50)(7,68,51)(8,65,52)(9,32,63)(10,29,64)(11,30,61)(12,31,62)(13,48,27)(14,45,28)(15,46,25)(16,47,26)(33,57,44)(34,58,41)(35,59,42)(36,60,43)(37,70,55)(38,71,56)(39,72,53)(40,69,54), (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), (1,26)(2,25)(3,28)(4,27)(5,9)(6,12)(7,11)(8,10)(13,22)(14,21)(15,24)(16,23)(17,45)(18,48)(19,47)(20,46)(29,65)(30,68)(31,67)(32,66)(33,56)(34,55)(35,54)(36,53)(37,58)(38,57)(39,60)(40,59)(41,70)(42,69)(43,72)(44,71)(49,63)(50,62)(51,61)(52,64), (1,2,3,4)(5,41,7,43)(6,42,8,44)(9,70,11,72)(10,71,12,69)(13,47,15,45)(14,48,16,46)(17,22,19,24)(18,23,20,21)(25,28,27,26)(29,38,31,40)(30,39,32,37)(33,50,35,52)(34,51,36,49)(53,63,55,61)(54,64,56,62)(57,67,59,65)(58,68,60,66), (1,27)(2,28)(3,25)(4,26)(5,12)(6,9)(7,10)(8,11)(13,23)(14,24)(15,21)(16,22)(17,46)(18,47)(19,48)(20,45)(29,68)(30,65)(31,66)(32,67)(33,55)(34,56)(35,53)(36,54)(37,57)(38,58)(39,59)(40,60)(41,71)(42,72)(43,69)(44,70)(49,62)(50,63)(51,64)(52,61) );
G=PermutationGroup([(1,32,40),(2,29,37),(3,30,38),(4,31,39),(5,35,47),(6,36,48),(7,33,45),(8,34,46),(9,54,19),(10,55,20),(11,56,17),(12,53,18),(13,50,43),(14,51,44),(15,52,41),(16,49,42),(21,61,71),(22,62,72),(23,63,69),(24,64,70),(25,65,58),(26,66,59),(27,67,60),(28,68,57)], [(1,23,19),(2,24,20),(3,21,17),(4,22,18),(5,66,49),(6,67,50),(7,68,51),(8,65,52),(9,32,63),(10,29,64),(11,30,61),(12,31,62),(13,48,27),(14,45,28),(15,46,25),(16,47,26),(33,57,44),(34,58,41),(35,59,42),(36,60,43),(37,70,55),(38,71,56),(39,72,53),(40,69,54)], [(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)], [(1,26),(2,25),(3,28),(4,27),(5,9),(6,12),(7,11),(8,10),(13,22),(14,21),(15,24),(16,23),(17,45),(18,48),(19,47),(20,46),(29,65),(30,68),(31,67),(32,66),(33,56),(34,55),(35,54),(36,53),(37,58),(38,57),(39,60),(40,59),(41,70),(42,69),(43,72),(44,71),(49,63),(50,62),(51,61),(52,64)], [(1,2,3,4),(5,41,7,43),(6,42,8,44),(9,70,11,72),(10,71,12,69),(13,47,15,45),(14,48,16,46),(17,22,19,24),(18,23,20,21),(25,28,27,26),(29,38,31,40),(30,39,32,37),(33,50,35,52),(34,51,36,49),(53,63,55,61),(54,64,56,62),(57,67,59,65),(58,68,60,66)], [(1,27),(2,28),(3,25),(4,26),(5,12),(6,9),(7,10),(8,11),(13,23),(14,24),(15,21),(16,22),(17,46),(18,47),(19,48),(20,45),(29,68),(30,65),(31,66),(32,67),(33,55),(34,56),(35,53),(36,54),(37,57),(38,58),(39,59),(40,60),(41,71),(42,72),(43,69),(44,70),(49,62),(50,63),(51,64),(52,61)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 2 | 11 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 9 | 11 |
G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,2,9,0,0,0,0,4,11,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,2,2,0,0,0,0,4,11,0,0,2,2,0,0,0,0,4,11,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11] >;
57 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 6M | ··· | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 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⋊6D6 |
| kernel | C32⋊82+ (1+4) | C12.59D6 | D4×C3⋊S3 | C12.D6 | C2×C32⋊7D4 | D4×C3×C6 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C32 | C3 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 4 | 4 | 16 | 8 | 1 | 8 |
In GAP, Magma, Sage, TeX
C_3^2\rtimes_82_+^{(1+4)} % in TeX
G:=Group("C3^2:8ES+(2,2)"); // GroupNames label
G:=SmallGroup(288,1009);
// by ID
G=gap.SmallGroup(288,1009);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=d^2=f^2=1,e^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c^2*e>;
// generators/relations