metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊5D4, C23⋊2D18, C9⋊2C22≀C2, (C2×D4)⋊3D9, (C2×C18)⋊2D4, (C2×C4)⋊2D18, (D4×C18)⋊8C2, C2.25(D4×D9), D18⋊C4⋊14C2, C3.(C23⋊2D6), (C2×C36)⋊7C22, (C6×D4).20S3, C6.100(S3×D4), C18.50(C2×D4), (C23×D9)⋊2C2, (C2×C12).216D6, C22⋊3(C9⋊D4), (C22×C6).50D6, (C2×C18).52C23, (C22×C18)⋊3C22, (C2×Dic9)⋊2C22, C18.D4⋊10C2, C22.59(C22×D9), (C22×D9).25C22, (C2×C9⋊D4)⋊4C2, C2.13(C2×C9⋊D4), C6.97(C2×C3⋊D4), (C2×C6).5(C3⋊D4), (C2×C6).209(C22×S3), SmallGroup(288,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊2D18
G = < a,b,c,d,e | a2=b2=c2=d18=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1028 in 195 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C2×D4, C24, D9, C18, C18, C18, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22≀C2, Dic9, C36, D18, D18, C2×C18, C2×C18, C2×C18, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, C2×Dic9, C9⋊D4, C2×C36, D4×C9, C22×D9, C22×D9, C22×C18, C23⋊2D6, D18⋊C4, C18.D4, C2×C9⋊D4, D4×C18, C23×D9, C23⋊2D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C22≀C2, D18, S3×D4, C2×C3⋊D4, C9⋊D4, C22×D9, C23⋊2D6, D4×D9, C2×C9⋊D4, C23⋊2D18
►On 72 points
Generators in S72
(1 51)(2 43)(3 53)(4 45)(5 37)(6 47)(7 39)(8 49)(9 41)(10 38)(11 48)(12 40)(13 50)(14 42)(15 52)(16 44)(17 54)(18 46)(19 57)(20 67)(21 59)(22 69)(23 61)(24 71)(25 63)(26 55)(27 65)(28 66)(29 58)(30 68)(31 60)(32 70)(33 62)(34 72)(35 64)(36 56)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(37 71)(38 72)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 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 3)(4 9)(5 8)(6 7)(10 11)(12 18)(13 17)(14 16)(19 23)(20 22)(24 27)(25 26)(28 32)(29 31)(33 36)(34 35)(37 56)(38 55)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)
G:=sub<Sym(72)| (1,51)(2,43)(3,53)(4,45)(5,37)(6,47)(7,39)(8,49)(9,41)(10,38)(11,48)(12,40)(13,50)(14,42)(15,52)(16,44)(17,54)(18,46)(19,57)(20,67)(21,59)(22,69)(23,61)(24,71)(25,63)(26,55)(27,65)(28,66)(29,58)(30,68)(31,60)(32,70)(33,62)(34,72)(35,64)(36,56), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,71)(38,72)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,3)(4,9)(5,8)(6,7)(10,11)(12,18)(13,17)(14,16)(19,23)(20,22)(24,27)(25,26)(28,32)(29,31)(33,36)(34,35)(37,56)(38,55)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)>;
G:=Group( (1,51)(2,43)(3,53)(4,45)(5,37)(6,47)(7,39)(8,49)(9,41)(10,38)(11,48)(12,40)(13,50)(14,42)(15,52)(16,44)(17,54)(18,46)(19,57)(20,67)(21,59)(22,69)(23,61)(24,71)(25,63)(26,55)(27,65)(28,66)(29,58)(30,68)(31,60)(32,70)(33,62)(34,72)(35,64)(36,56), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,71)(38,72)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (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), (1,3)(4,9)(5,8)(6,7)(10,11)(12,18)(13,17)(14,16)(19,23)(20,22)(24,27)(25,26)(28,32)(29,31)(33,36)(34,35)(37,56)(38,55)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57) );
G=PermutationGroup([[(1,51),(2,43),(3,53),(4,45),(5,37),(6,47),(7,39),(8,49),(9,41),(10,38),(11,48),(12,40),(13,50),(14,42),(15,52),(16,44),(17,54),(18,46),(19,57),(20,67),(21,59),(22,69),(23,61),(24,71),(25,63),(26,55),(27,65),(28,66),(29,58),(30,68),(31,60),(32,70),(33,62),(34,72),(35,64),(36,56)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,19),(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(37,71),(38,72),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72)], [(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)], [(1,3),(4,9),(5,8),(6,7),(10,11),(12,18),(13,17),(14,16),(19,23),(20,22),(24,27),(25,26),(28,32),(29,31),(33,36),(34,35),(37,56),(38,55),(39,72),(40,71),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 2 | 4 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | C3⋊D4 | D18 | D18 | C9⋊D4 | S3×D4 | D4×D9 |
| kernel | C23⋊2D18 | D18⋊C4 | C18.D4 | C2×C9⋊D4 | D4×C18 | C23×D9 | C6×D4 | D18 | C2×C18 | C2×C12 | C22×C6 | C2×D4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 2 | 6 |
Matrix representation of C23⋊2D18 ►in GL4(𝔽37) generated by
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 30 | 14 |
| 0 | 0 | 23 | 7 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 6 | 11 |
| 0 | 0 | 26 | 17 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 17 |
| 0 | 0 | 6 | 11 |
G:=sub<GL(4,GF(37))| [0,1,0,0,1,0,0,0,0,0,30,23,0,0,14,7],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,36,0,0,0,0,6,26,0,0,11,17],[1,0,0,0,0,1,0,0,0,0,26,6,0,0,17,11] >;
C23⋊2D18 in GAP, Magma, Sage, TeX
C_2^3\rtimes_2D_{18} % in TeX
G:=Group("C2^3:2D18"); // GroupNames label
G:=SmallGroup(288,147);
// by ID
G=gap.SmallGroup(288,147);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^18=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations