metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊4D9, C23.31D18, C9⋊3C22≀C2, (C2×C18)⋊8D4, (C23×C18)⋊3C2, C18.63(C2×D4), (C23×C6).9S3, C3.(C24⋊4S3), C22⋊4(C9⋊D4), (C2×C18).61C23, (C2×Dic9)⋊3C22, (C22×C6).144D6, (C22×D9)⋊2C22, C18.D4⋊13C2, C22.66(C22×D9), (C22×C18).42C22, (C2×C9⋊D4)⋊8C2, C2.26(C2×C9⋊D4), C6.111(C2×C3⋊D4), (C2×C6).84(C3⋊D4), (C2×C6).218(C22×S3), SmallGroup(288,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊4D9
G = < a,b,c,d,e,f | a2=b2=c2=d2=e9=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 708 in 195 conjugacy classes, 56 normal (11 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, D9, C18, C18, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C22≀C2, Dic9, D18, C2×C18, C2×C18, C2×C18, C6.D4, C2×C3⋊D4, C23×C6, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C22×C18, C24⋊4S3, C18.D4, C2×C9⋊D4, C23×C18, C24⋊4D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C22≀C2, D18, C2×C3⋊D4, C9⋊D4, C22×D9, C24⋊4S3, C2×C9⋊D4, C24⋊4D9
►On 72 points
Generators in S72
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)
(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 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(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 39)(2 38)(3 37)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 54)(18 53)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 63)(27 62)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 72)(36 71)
G:=sub<Sym(72)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63), (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)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,39)(2,38)(3,37)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,54)(18,53)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,63)(27,62)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,72)(36,71)>;
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63), (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)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,39)(2,38)(3,37)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,54)(18,53)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,63)(27,62)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,72)(36,71) );
G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,55),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63)], [(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),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,55),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(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)], [(1,39),(2,38),(3,37),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,54),(18,53),(19,61),(20,60),(21,59),(22,58),(23,57),(24,56),(25,55),(26,63),(27,62),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,72),(36,71)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3 | 4A | 4B | 4C | 6A | ··· | 6O | 9A | 9B | 9C | 18A | ··· | 18AS |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 36 | 2 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D9 | C3⋊D4 | D18 | C9⋊D4 |
| kernel | C24⋊4D9 | C18.D4 | C2×C9⋊D4 | C23×C18 | C23×C6 | C2×C18 | C22×C6 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 1 | 6 | 3 | 3 | 12 | 9 | 36 |
Matrix representation of C24⋊4D9 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 33 | 36 |
| 1 | 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 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 26 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 7 | 34 |
| 0 | 10 | 0 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 0 | 24 | 12 |
| 0 | 0 | 23 | 13 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,33,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[26,0,0,0,0,10,0,0,0,0,12,7,0,0,0,34],[0,26,0,0,10,0,0,0,0,0,24,23,0,0,12,13] >;
C24⋊4D9 in GAP, Magma, Sage, TeX
C_2^4\rtimes_4D_9
% in TeX
G:=Group("C2^4:4D9"); // GroupNames label
G:=SmallGroup(288,163);
// by ID
G=gap.SmallGroup(288,163);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^9=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations