direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×2+ (1+4), D4⋊4C62, Q8⋊5C62, C23⋊2C62, C62.157C23, (C6×D4)⋊15C6, (C2×C4)⋊2C62, C4.9(C2×C62), (C6×C12)⋊27C22, C6.24(C23×C6), (C3×C6).71C24, (C2×C62)⋊2C22, C12.63(C22×C6), C22.2(C2×C62), C2.4(C22×C62), (C3×C12).192C23, (D4×C32)⋊31C22, (Q8×C32)⋊28C22, (D4×C3×C6)⋊24C2, (C2×C12)⋊9(C2×C6), C4○D4⋊5(C3×C6), (C2×D4)⋊6(C3×C6), (C3×C4○D4)⋊12C6, (C3×D4)⋊13(C2×C6), (C22×C6)⋊3(C2×C6), (C3×Q8)⋊14(C2×C6), (C32×C4○D4)⋊13C2, (C2×C6).12(C22×C6), SmallGroup(288,1022)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 660 in 498 conjugacy classes, 408 normal (6 characteristic)
C1, C2, C2 [×9], C3 [×4], C4 [×6], C22 [×9], C22 [×6], C6 [×4], C6 [×36], C2×C4 [×9], D4 [×18], Q8 [×2], C23 [×6], C32, C12 [×24], C2×C6 [×36], C2×C6 [×24], C2×D4 [×9], C4○D4 [×6], C3×C6, C3×C6 [×9], C2×C12 [×36], C3×D4 [×72], C3×Q8 [×8], C22×C6 [×24], 2+ (1+4), C3×C12 [×6], C62 [×9], C62 [×6], C6×D4 [×36], C3×C4○D4 [×24], C6×C12 [×9], D4×C32 [×18], Q8×C32 [×2], C2×C62 [×6], C3×2+ (1+4) [×4], D4×C3×C6 [×9], C32×C4○D4 [×6], C32×2+ (1+4)
Quotients:
C1, C2 [×15], C3 [×4], C22 [×35], C6 [×60], C23 [×15], C32, C2×C6 [×140], C24, C3×C6 [×15], C22×C6 [×60], 2+ (1+4), C62 [×35], C23×C6 [×4], C2×C62 [×15], C3×2+ (1+4) [×4], C22×C62, C32×2+ (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, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >
►On 72 points
Generators in S72
(1 56 18)(2 53 19)(3 54 20)(4 55 17)(5 33 9)(6 34 10)(7 35 11)(8 36 12)(13 40 66)(14 37 67)(15 38 68)(16 39 65)(21 69 25)(22 70 26)(23 71 27)(24 72 28)(29 64 60)(30 61 57)(31 62 58)(32 63 59)(41 49 45)(42 50 46)(43 51 47)(44 52 48)
(1 65 21)(2 66 22)(3 67 23)(4 68 24)(5 43 29)(6 44 30)(7 41 31)(8 42 32)(9 47 60)(10 48 57)(11 45 58)(12 46 59)(13 70 53)(14 71 54)(15 72 55)(16 69 56)(17 38 28)(18 39 25)(19 40 26)(20 37 27)(33 51 64)(34 52 61)(35 49 62)(36 50 63)
(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 42)(2 41)(3 44)(4 43)(5 24)(6 23)(7 22)(8 21)(9 28)(10 27)(11 26)(12 25)(13 62)(14 61)(15 64)(16 63)(17 47)(18 46)(19 45)(20 48)(29 68)(30 67)(31 66)(32 65)(33 72)(34 71)(35 70)(36 69)(37 57)(38 60)(39 59)(40 58)(49 53)(50 56)(51 55)(52 54)
(1 4 3 2)(5 6 7 8)(9 10 11 12)(13 16 15 14)(17 20 19 18)(21 24 23 22)(25 28 27 26)(29 30 31 32)(33 34 35 36)(37 40 39 38)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 56 55 54)(57 58 59 60)(61 62 63 64)(65 68 67 66)(69 72 71 70)
(1 41)(2 42)(3 43)(4 44)(5 23)(6 24)(7 21)(8 22)(9 27)(10 28)(11 25)(12 26)(13 63)(14 64)(15 61)(16 62)(17 48)(18 45)(19 46)(20 47)(29 67)(30 68)(31 65)(32 66)(33 71)(34 72)(35 69)(36 70)(37 60)(38 57)(39 58)(40 59)(49 56)(50 53)(51 54)(52 55)
G:=sub<Sym(72)| (1,56,18)(2,53,19)(3,54,20)(4,55,17)(5,33,9)(6,34,10)(7,35,11)(8,36,12)(13,40,66)(14,37,67)(15,38,68)(16,39,65)(21,69,25)(22,70,26)(23,71,27)(24,72,28)(29,64,60)(30,61,57)(31,62,58)(32,63,59)(41,49,45)(42,50,46)(43,51,47)(44,52,48), (1,65,21)(2,66,22)(3,67,23)(4,68,24)(5,43,29)(6,44,30)(7,41,31)(8,42,32)(9,47,60)(10,48,57)(11,45,58)(12,46,59)(13,70,53)(14,71,54)(15,72,55)(16,69,56)(17,38,28)(18,39,25)(19,40,26)(20,37,27)(33,51,64)(34,52,61)(35,49,62)(36,50,63), (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), (1,42)(2,41)(3,44)(4,43)(5,24)(6,23)(7,22)(8,21)(9,28)(10,27)(11,26)(12,25)(13,62)(14,61)(15,64)(16,63)(17,47)(18,46)(19,45)(20,48)(29,68)(30,67)(31,66)(32,65)(33,72)(34,71)(35,70)(36,69)(37,57)(38,60)(39,59)(40,58)(49,53)(50,56)(51,55)(52,54), (1,4,3,2)(5,6,7,8)(9,10,11,12)(13,16,15,14)(17,20,19,18)(21,24,23,22)(25,28,27,26)(29,30,31,32)(33,34,35,36)(37,40,39,38)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,56,55,54)(57,58,59,60)(61,62,63,64)(65,68,67,66)(69,72,71,70), (1,41)(2,42)(3,43)(4,44)(5,23)(6,24)(7,21)(8,22)(9,27)(10,28)(11,25)(12,26)(13,63)(14,64)(15,61)(16,62)(17,48)(18,45)(19,46)(20,47)(29,67)(30,68)(31,65)(32,66)(33,71)(34,72)(35,69)(36,70)(37,60)(38,57)(39,58)(40,59)(49,56)(50,53)(51,54)(52,55)>;
G:=Group( (1,56,18)(2,53,19)(3,54,20)(4,55,17)(5,33,9)(6,34,10)(7,35,11)(8,36,12)(13,40,66)(14,37,67)(15,38,68)(16,39,65)(21,69,25)(22,70,26)(23,71,27)(24,72,28)(29,64,60)(30,61,57)(31,62,58)(32,63,59)(41,49,45)(42,50,46)(43,51,47)(44,52,48), (1,65,21)(2,66,22)(3,67,23)(4,68,24)(5,43,29)(6,44,30)(7,41,31)(8,42,32)(9,47,60)(10,48,57)(11,45,58)(12,46,59)(13,70,53)(14,71,54)(15,72,55)(16,69,56)(17,38,28)(18,39,25)(19,40,26)(20,37,27)(33,51,64)(34,52,61)(35,49,62)(36,50,63), (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), (1,42)(2,41)(3,44)(4,43)(5,24)(6,23)(7,22)(8,21)(9,28)(10,27)(11,26)(12,25)(13,62)(14,61)(15,64)(16,63)(17,47)(18,46)(19,45)(20,48)(29,68)(30,67)(31,66)(32,65)(33,72)(34,71)(35,70)(36,69)(37,57)(38,60)(39,59)(40,58)(49,53)(50,56)(51,55)(52,54), (1,4,3,2)(5,6,7,8)(9,10,11,12)(13,16,15,14)(17,20,19,18)(21,24,23,22)(25,28,27,26)(29,30,31,32)(33,34,35,36)(37,40,39,38)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,56,55,54)(57,58,59,60)(61,62,63,64)(65,68,67,66)(69,72,71,70), (1,41)(2,42)(3,43)(4,44)(5,23)(6,24)(7,21)(8,22)(9,27)(10,28)(11,25)(12,26)(13,63)(14,64)(15,61)(16,62)(17,48)(18,45)(19,46)(20,47)(29,67)(30,68)(31,65)(32,66)(33,71)(34,72)(35,69)(36,70)(37,60)(38,57)(39,58)(40,59)(49,56)(50,53)(51,54)(52,55) );
G=PermutationGroup([(1,56,18),(2,53,19),(3,54,20),(4,55,17),(5,33,9),(6,34,10),(7,35,11),(8,36,12),(13,40,66),(14,37,67),(15,38,68),(16,39,65),(21,69,25),(22,70,26),(23,71,27),(24,72,28),(29,64,60),(30,61,57),(31,62,58),(32,63,59),(41,49,45),(42,50,46),(43,51,47),(44,52,48)], [(1,65,21),(2,66,22),(3,67,23),(4,68,24),(5,43,29),(6,44,30),(7,41,31),(8,42,32),(9,47,60),(10,48,57),(11,45,58),(12,46,59),(13,70,53),(14,71,54),(15,72,55),(16,69,56),(17,38,28),(18,39,25),(19,40,26),(20,37,27),(33,51,64),(34,52,61),(35,49,62),(36,50,63)], [(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)], [(1,42),(2,41),(3,44),(4,43),(5,24),(6,23),(7,22),(8,21),(9,28),(10,27),(11,26),(12,25),(13,62),(14,61),(15,64),(16,63),(17,47),(18,46),(19,45),(20,48),(29,68),(30,67),(31,66),(32,65),(33,72),(34,71),(35,70),(36,69),(37,57),(38,60),(39,59),(40,58),(49,53),(50,56),(51,55),(52,54)], [(1,4,3,2),(5,6,7,8),(9,10,11,12),(13,16,15,14),(17,20,19,18),(21,24,23,22),(25,28,27,26),(29,30,31,32),(33,34,35,36),(37,40,39,38),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,56,55,54),(57,58,59,60),(61,62,63,64),(65,68,67,66),(69,72,71,70)], [(1,41),(2,42),(3,43),(4,44),(5,23),(6,24),(7,21),(8,22),(9,27),(10,28),(11,25),(12,26),(13,63),(14,64),(15,61),(16,62),(17,48),(18,45),(19,46),(20,47),(29,67),(30,68),(31,65),(32,66),(33,71),(34,72),(35,69),(36,70),(37,60),(38,57),(39,58),(40,59),(49,56),(50,53),(51,54),(52,55)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 6 | 1 | 2 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 11 | 6 | 12 | 4 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 4 | 7 | 12 | 11 |
| 0 | 8 | 5 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 7 | 12 | 11 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 2 | 8 | 8 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 4 | 7 | 12 | 11 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 7 | 11 | 5 | 6 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[9,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,9,0,0,11,0,6,0,12,6,0,1,1,0,12,0,2,0,0,4],[12,0,0,0,0,0,0,1,4,8,0,1,0,7,5,0,0,0,12,0,0,0,0,11,1],[1,0,0,0,0,0,4,0,0,2,0,7,0,12,8,0,12,1,0,8,0,11,0,0,9],[1,0,0,0,0,0,0,4,1,7,0,0,7,0,11,0,1,12,0,5,0,0,11,0,6] >;
153 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2J | 3A | ··· | 3H | 4A | ··· | 4F | 6A | ··· | 6H | 6I | ··· | 6CB | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
153 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | 2+ (1+4) | C3×2+ (1+4) |
| kernel | C32×2+ (1+4) | D4×C3×C6 | C32×C4○D4 | C3×2+ (1+4) | C6×D4 | C3×C4○D4 | C32 | C3 |
| # reps | 1 | 9 | 6 | 8 | 72 | 48 | 1 | 8 |
In GAP, Magma, Sage, TeX
C_3^2\times 2_+^{(1+4)} % in TeX
G:=Group("C3^2xES+(2,2)"); // GroupNames label
G:=SmallGroup(288,1022);
// by ID
G=gap.SmallGroup(288,1022);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,2045,1563,4259]);
// 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,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,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
×
⋊
ℤ
𝔽
○
≀