direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C22⋊C4, C22⋊2C36, C18.12D4, C23.2C18, (C2×C36)⋊2C2, (C2×C18)⋊1C4, (C2×C4)⋊1C18, C2.1(D4×C9), C2.1(C2×C36), (C2×C12).1C6, (C2×C6).2C12, C6.12(C3×D4), C6.10(C2×C12), C18.10(C2×C4), (C22×C6).5C6, C22.2(C2×C18), (C22×C18).1C2, (C2×C18).13C22, C3.(C3×C22⋊C4), (C3×C22⋊C4).C3, (C2×C6).16(C2×C6), (C2×C18)○(C3×C22⋊C4), SmallGroup(144,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C22⋊C4
G = < a,b,c,d | a9=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C9×C22⋊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 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)(28 50)(29 51)(30 52)(31 53)(32 54)(33 46)(34 47)(35 48)(36 49)(55 68)(56 69)(57 70)(58 71)(59 72)(60 64)(61 65)(62 66)(63 67)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 62)(11 63)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(37 52)(38 53)(39 54)(40 46)(41 47)(42 48)(43 49)(44 50)(45 51)
(1 70 41 14)(2 71 42 15)(3 72 43 16)(4 64 44 17)(5 65 45 18)(6 66 37 10)(7 67 38 11)(8 68 39 12)(9 69 40 13)(19 49 59 36)(20 50 60 28)(21 51 61 29)(22 52 62 30)(23 53 63 31)(24 54 55 32)(25 46 56 33)(26 47 57 34)(27 48 58 35)
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), (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)(28,50)(29,51)(30,52)(31,53)(32,54)(33,46)(34,47)(35,48)(36,49)(55,68)(56,69)(57,70)(58,71)(59,72)(60,64)(61,65)(62,66)(63,67), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,62)(11,63)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(37,52)(38,53)(39,54)(40,46)(41,47)(42,48)(43,49)(44,50)(45,51), (1,70,41,14)(2,71,42,15)(3,72,43,16)(4,64,44,17)(5,65,45,18)(6,66,37,10)(7,67,38,11)(8,68,39,12)(9,69,40,13)(19,49,59,36)(20,50,60,28)(21,51,61,29)(22,52,62,30)(23,53,63,31)(24,54,55,32)(25,46,56,33)(26,47,57,34)(27,48,58,35)>;
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), (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)(28,50)(29,51)(30,52)(31,53)(32,54)(33,46)(34,47)(35,48)(36,49)(55,68)(56,69)(57,70)(58,71)(59,72)(60,64)(61,65)(62,66)(63,67), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,62)(11,63)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(37,52)(38,53)(39,54)(40,46)(41,47)(42,48)(43,49)(44,50)(45,51), (1,70,41,14)(2,71,42,15)(3,72,43,16)(4,64,44,17)(5,65,45,18)(6,66,37,10)(7,67,38,11)(8,68,39,12)(9,69,40,13)(19,49,59,36)(20,50,60,28)(21,51,61,29)(22,52,62,30)(23,53,63,31)(24,54,55,32)(25,46,56,33)(26,47,57,34)(27,48,58,35) );
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)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,37),(7,38),(8,39),(9,40),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,19),(17,20),(18,21),(28,50),(29,51),(30,52),(31,53),(32,54),(33,46),(34,47),(35,48),(36,49),(55,68),(56,69),(57,70),(58,71),(59,72),(60,64),(61,65),(62,66),(63,67)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,62),(11,63),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,72),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(37,52),(38,53),(39,54),(40,46),(41,47),(42,48),(43,49),(44,50),(45,51)], [(1,70,41,14),(2,71,42,15),(3,72,43,16),(4,64,44,17),(5,65,45,18),(6,66,37,10),(7,67,38,11),(8,68,39,12),(9,69,40,13),(19,49,59,36),(20,50,60,28),(21,51,61,29),(22,52,62,30),(23,53,63,31),(24,54,55,32),(25,46,56,33),(26,47,57,34),(27,48,58,35)]])
C9×C22⋊C4 is a maximal subgroup of
C22.D36 C23.16D18 C22⋊2Dic18 C23.8D18 Dic9⋊4D4 C22⋊3D36 C23.9D18 D18⋊D4 Dic9.D4 C22.4D36 D4×C36
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 9A | ··· | 9F | 12A | ··· | 12H | 18A | ··· | 18R | 18S | ··· | 18AD | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | D4 | C3×D4 | D4×C9 |
| kernel | C9×C22⋊C4 | C2×C36 | C22×C18 | C3×C22⋊C4 | C2×C18 | C2×C12 | C22×C6 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C18 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 24 | 2 | 4 | 12 |
Matrix representation of C9×C22⋊C4 ►in GL4(𝔽37) generated by
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 22 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 15 | 2 |
| 0 | 0 | 36 | 22 |
G:=sub<GL(4,GF(37))| [7,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,1,0,0,0,0,1,22,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,31,0,0,0,0,15,36,0,0,2,22] >;
C9×C22⋊C4 in GAP, Magma, Sage, TeX
C_9\times C_2^2\rtimes C_4
% in TeX
G:=Group("C9xC2^2:C4"); // GroupNames label
G:=SmallGroup(144,21);
// by ID
G=gap.SmallGroup(144,21);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,230]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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