metabelian, supersoluble, monomial
Aliases: D36.1C6, 3- 1+2:3SD16, C9:C8:3C6, C9:C24:3C2, Q8:2D9:C3, (Q8xC9):1C6, C36.4(C2xC6), Q8:3(C9:C6), C9:3(C3xSD16), C12.12(S3xC6), D36:C3.1C2, (C3xC12).15D6, C18.10(C3xD4), (Q8xC32).3S3, C2.7(Dic9:C6), C32.(Q8:2S3), (Q8x3- 1+2):1C2, (C2x3- 1+2).10D4, (C4x3- 1+2).4C22, C4.4(C2xC9:C6), C6.29(C3xC3:D4), (C3xQ8).26(C3xS3), C3.3(C3xQ8:2S3), (C3xC6).30(C3:D4), SmallGroup(432,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D36.C6
G = < a,b,c | a36=b2=1, c6=a18, bab=a-1, cac-1=a7, cbc-1=a15b >
Subgroups: 294 in 66 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, C12, C12, D6, C2xC6, SD16, D9, C18, C18, C3xS3, C3xC6, C3:C8, C24, D12, C3xD4, C3xQ8, C3xQ8, 3- 1+2, C36, C36, D18, C3xC12, C3xC12, S3xC6, Q8:2S3, C3xSD16, C9:C6, C2x3- 1+2, C9:C8, D36, Q8xC9, Q8xC9, C3xC3:C8, C3xD12, Q8xC32, C4x3- 1+2, C4x3- 1+2, C2xC9:C6, Q8:2D9, C3xQ8:2S3, C9:C24, D36:C3, Q8x3- 1+2, D36.C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, SD16, C3xS3, C3:D4, C3xD4, S3xC6, Q8:2S3, C3xSD16, C9:C6, C3xC3:D4, C2xC9:C6, C3xQ8:2S3, Dic9:C6, D36.C6
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(37 61)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(62 72)(63 71)(64 70)(65 69)(66 68)
(1 63 19 45)(2 58 8 64 14 70 20 40 26 46 32 52)(3 53 33 47 27 41 21 71 15 65 9 59)(4 48 22 66)(5 43 11 49 17 55 23 61 29 67 35 37)(6 38 36 68 30 62 24 56 18 50 12 44)(7 69 25 51)(10 54 28 72)(13 39 31 57)(16 60 34 42)
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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(62,72)(63,71)(64,70)(65,69)(66,68), (1,63,19,45)(2,58,8,64,14,70,20,40,26,46,32,52)(3,53,33,47,27,41,21,71,15,65,9,59)(4,48,22,66)(5,43,11,49,17,55,23,61,29,67,35,37)(6,38,36,68,30,62,24,56,18,50,12,44)(7,69,25,51)(10,54,28,72)(13,39,31,57)(16,60,34,42)>;
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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(62,72)(63,71)(64,70)(65,69)(66,68), (1,63,19,45)(2,58,8,64,14,70,20,40,26,46,32,52)(3,53,33,47,27,41,21,71,15,65,9,59)(4,48,22,66)(5,43,11,49,17,55,23,61,29,67,35,37)(6,38,36,68,30,62,24,56,18,50,12,44)(7,69,25,51)(10,54,28,72)(13,39,31,57)(16,60,34,42) );
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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(37,61),(38,60),(39,59),(40,58),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(62,72),(63,71),(64,70),(65,69),(66,68)], [(1,63,19,45),(2,58,8,64,14,70,20,40,26,46,32,52),(3,53,33,47,27,41,21,71,15,65,9,59),(4,48,22,66),(5,43,11,49,17,55,23,61,29,67,35,37),(6,38,36,68,30,62,24,56,18,50,12,44),(7,69,25,51),(10,54,28,72),(13,39,31,57),(16,60,34,42)]])
41 conjugacy classes
class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 36A | ··· | 36I |
order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 |
size | 1 | 1 | 36 | 2 | 3 | 3 | 2 | 4 | 2 | 3 | 3 | 36 | 36 | 18 | 18 | 6 | 6 | 6 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 12 | ··· | 12 |
41 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 |
type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D36.C6 | S3 | D4 | D6 | SD16 | C3xS3 | C3xD4 | C3:D4 | S3xC6 | C3xSD16 | C3xC3:D4 | Q8:2S3 | C3xQ8:2S3 | C9:C6 | C2xC9:C6 | Dic9:C6 |
kernel | D36.C6 | C9:C24 | D36:C3 | Q8x3- 1+2 | Q8:2D9 | C9:C8 | D36 | Q8xC9 | C1 | Q8xC32 | C2x3- 1+2 | C3xC12 | 3- 1+2 | C3xQ8 | C18 | C3xC6 | C12 | C9 | C6 | C32 | C3 | Q8 | C4 | C2 |
# reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of D36.C6 ►in GL10(F73)
72 | 72 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
72 | 72 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
24 | 23 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
24 | 24 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
72 | 72 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
72 | 72 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
72 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 72 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
41 | 0 | 32 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
41 | 53 | 20 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
3 | 8 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
10 | 3 | 32 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 43 | 60 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 13 | 30 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 43 | 43 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 30 | 13 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 | 43 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 30 | 30 |
G:=sub<GL(10,GF(73))| [72,72,24,24,0,0,0,0,0,0,72,72,23,24,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,1,2,1,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0],[72,72,72,0,0,0,0,0,0,0,72,72,0,72,0,0,0,0,0,0,1,2,1,1,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0],[41,41,3,10,0,0,0,0,0,0,0,53,8,3,0,0,0,0,0,0,32,20,20,32,0,0,0,0,0,0,12,32,0,32,0,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,0,0,43,30,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,43,30] >;
D36.C6 in GAP, Magma, Sage, TeX
D_{36}.C_6
% in TeX
G:=Group("D36.C6");
// GroupNames label
G:=SmallGroup(432,163);
// by ID
G=gap.SmallGroup(432,163);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,10085,2035,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=b^2=1,c^6=a^18,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^15*b>;
// generators/relations