direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×M4(2), C4.C36, C72⋊7C2, C8⋊3C18, C24.7C6, C36.4C4, C22.C36, C12.5C12, C36.22C22, (C2×C36).8C2, (C2×C18).1C4, C4.5(C2×C18), (C2×C6).3C12, (C2×C12).9C6, (C2×C4).2C18, C2.3(C2×C36), C3.(C3×M4(2)), C18.12(C2×C4), C6.12(C2×C12), C12.28(C2×C6), C36○(C3×M4(2)), (C3×M4(2)).C3, SmallGroup(144,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×M4(2)
G = < a,b,c | a9=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of C9×M4(2)
►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 66 53 11 40 28 59 21)(2 67 54 12 41 29 60 22)(3 68 46 13 42 30 61 23)(4 69 47 14 43 31 62 24)(5 70 48 15 44 32 63 25)(6 71 49 16 45 33 55 26)(7 72 50 17 37 34 56 27)(8 64 51 18 38 35 57 19)(9 65 52 10 39 36 58 20)
(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 64)(36 65)
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), (1,66,53,11,40,28,59,21)(2,67,54,12,41,29,60,22)(3,68,46,13,42,30,61,23)(4,69,47,14,43,31,62,24)(5,70,48,15,44,32,63,25)(6,71,49,16,45,33,55,26)(7,72,50,17,37,34,56,27)(8,64,51,18,38,35,57,19)(9,65,52,10,39,36,58,20), (10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,64)(36,65)>;
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), (1,66,53,11,40,28,59,21)(2,67,54,12,41,29,60,22)(3,68,46,13,42,30,61,23)(4,69,47,14,43,31,62,24)(5,70,48,15,44,32,63,25)(6,71,49,16,45,33,55,26)(7,72,50,17,37,34,56,27)(8,64,51,18,38,35,57,19)(9,65,52,10,39,36,58,20), (10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,64)(36,65) );
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)], [(1,66,53,11,40,28,59,21),(2,67,54,12,41,29,60,22),(3,68,46,13,42,30,61,23),(4,69,47,14,43,31,62,24),(5,70,48,15,44,32,63,25),(6,71,49,16,45,33,55,26),(7,72,50,17,37,34,56,27),(8,64,51,18,38,35,57,19),(9,65,52,10,39,36,58,20)], [(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,19),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,64),(36,65)]])
C9×M4(2) is a maximal subgroup of
C36.53D4 C4.D36 C36.48D4 Dic18⋊C4 D36.C4 C8⋊D18 C8.D18
90 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | ··· | 18F | 18G | ··· | 18L | 24A | ··· | 24H | 36A | ··· | 36L | 36M | ··· | 36R | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C36 | C36 | M4(2) | C3×M4(2) | C9×M4(2) |
| kernel | C9×M4(2) | C72 | C2×C36 | C3×M4(2) | C36 | C2×C18 | C24 | C2×C12 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 6 | 4 | 4 | 12 | 6 | 12 | 12 | 2 | 4 | 12 |
Matrix representation of C9×M4(2) ►in GL2(𝔽37) generated by
| 34 | 0 |
| 0 | 34 |
| 0 | 14 |
| 26 | 0 |
| 36 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(37))| [34,0,0,34],[0,26,14,0],[36,0,0,1] >;
C9×M4(2) in GAP, Magma, Sage, TeX
C_9\times M_4(2)
% in TeX
G:=Group("C9xM4(2)"); // GroupNames label
G:=SmallGroup(144,24);
// by ID
G=gap.SmallGroup(144,24);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-2,72,889,122,165]);
// Polycyclic
G:=Group<a,b,c|a^9=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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