direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×C23⋊C4, C23⋊C36, (C2×C4)⋊C36, (C2×C36)⋊2C4, (C6×D4).6C6, C22⋊C4⋊1C18, (C2×C12).1C12, (C22×C18)⋊1C4, (D4×C18).7C2, (C2×D4).1C18, (C2×C18).22D4, C22.2(D4×C9), C22.2(C2×C36), C23.1(C2×C18), (C22×C6).2C12, C18.21(C22⋊C4), (C22×C18).1C22, C3.(C3×C23⋊C4), (C3×C23⋊C4).C3, (C9×C22⋊C4)⋊2C2, (C2×C6).25(C3×D4), C2.3(C9×C22⋊C4), (C2×C6).23(C2×C12), (C2×C18).19(C2×C4), (C3×C22⋊C4).1C6, C6.21(C3×C22⋊C4), (C22×C6).6(C2×C6), SmallGroup(288,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C23⋊C4
G = < a,b,c,d,e | a9=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 150 in 78 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C9, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C18, C18, C2×C12, C2×C12, C3×D4, C22×C6, C23⋊C4, C36, C2×C18, C2×C18, C2×C18, C3×C22⋊C4, C6×D4, C2×C36, C2×C36, D4×C9, C22×C18, C3×C23⋊C4, C9×C22⋊C4, D4×C18, C9×C23⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, C18, C2×C12, C3×D4, C23⋊C4, C36, C2×C18, C3×C22⋊C4, C2×C36, D4×C9, C3×C23⋊C4, C9×C22⋊C4, C9×C23⋊C4
►On 72 points
Generators in S72
(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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)(19 45)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 59)(29 60)(30 61)(31 62)(32 63)(33 55)(34 56)(35 57)(36 58)(46 71)(47 72)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 46)(7 47)(8 48)(9 49)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 64)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 28)(26 29)(27 30)(37 63)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)
(1 42)(2 43)(3 44)(4 45)(5 37)(6 38)(7 39)(8 40)(9 41)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 64)(36 65)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)
(10 65 24 36)(11 66 25 28)(12 67 26 29)(13 68 27 30)(14 69 19 31)(15 70 20 32)(16 71 21 33)(17 72 22 34)(18 64 23 35)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)
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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10)(19,45)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,59)(29,60)(30,61)(31,62)(32,63)(33,55)(34,56)(35,57)(36,58)(46,71)(47,72)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70), (1,50)(2,51)(3,52)(4,53)(5,54)(6,46)(7,47)(8,48)(9,49)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,64)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,28)(26,29)(27,30)(37,63)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62), (1,42)(2,43)(3,44)(4,45)(5,37)(6,38)(7,39)(8,40)(9,41)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,64)(36,65)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63), (10,65,24,36)(11,66,25,28)(12,67,26,29)(13,68,27,30)(14,69,19,31)(15,70,20,32)(16,71,21,33)(17,72,22,34)(18,64,23,35)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)>;
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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10)(19,45)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,59)(29,60)(30,61)(31,62)(32,63)(33,55)(34,56)(35,57)(36,58)(46,71)(47,72)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70), (1,50)(2,51)(3,52)(4,53)(5,54)(6,46)(7,47)(8,48)(9,49)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,64)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,28)(26,29)(27,30)(37,63)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62), (1,42)(2,43)(3,44)(4,45)(5,37)(6,38)(7,39)(8,40)(9,41)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,64)(36,65)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63), (10,65,24,36)(11,66,25,28)(12,67,26,29)(13,68,27,30)(14,69,19,31)(15,70,20,32)(16,71,21,33)(17,72,22,34)(18,64,23,35)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63) );
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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,10),(19,45),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,59),(29,60),(30,61),(31,62),(32,63),(33,55),(34,56),(35,57),(36,58),(46,71),(47,72),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,46),(7,47),(8,48),(9,49),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,64),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,28),(26,29),(27,30),(37,63),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62)], [(1,42),(2,43),(3,44),(4,45),(5,37),(6,38),(7,39),(8,40),(9,41),(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,64),(36,65),(46,55),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63)], [(10,65,24,36),(11,66,25,28),(12,67,26,29),(13,68,27,30),(14,69,19,31),(15,70,20,32),(16,71,21,33),(17,72,22,34),(18,64,23,35),(46,55),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4E | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 9A | ··· | 9F | 12A | ··· | 12J | 18A | ··· | 18F | 18G | ··· | 18X | 18Y | ··· | 18AD | 36A | ··· | 36AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C36 | C36 | D4 | C3×D4 | D4×C9 | C23⋊C4 | C3×C23⋊C4 | C9×C23⋊C4 |
| kernel | C9×C23⋊C4 | C9×C22⋊C4 | D4×C18 | C3×C23⋊C4 | C2×C36 | C22×C18 | C3×C22⋊C4 | C6×D4 | C23⋊C4 | C2×C12 | C22×C6 | C22⋊C4 | C2×D4 | C2×C4 | C23 | C2×C18 | C2×C6 | C22 | C9 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 6 | 4 | 4 | 12 | 6 | 12 | 12 | 2 | 4 | 12 | 1 | 2 | 6 |
Matrix representation of C9×C23⋊C4 ►in GL4(𝔽37) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(37))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,0,1,0,0,36,0] >;
C9×C23⋊C4 in GAP, Magma, Sage, TeX
C_9\times C_2^3\rtimes C_4
% in TeX
G:=Group("C9xC2^3:C4"); // GroupNames label
G:=SmallGroup(288,49);
// by ID
G=gap.SmallGroup(288,49);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2951]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations