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G = D36.C6order 432 = 24·33

1st non-split extension by D36 of C6 acting faithfully

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

C1C36 — D36.C6
C1C3C9C18C36C4x3- 1+2D36:C3 — D36.C6
C9C18C36 — D36.C6
C1C2C4Q8

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

Smallest permutation representation of D36.C6
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 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 2A2B3A3B3C4A4B6A6B6C6D6E8A8B9A9B9C12A12B12C12D12E12F12G18A18B18C24A24B24C24D36A···36I
order122333446666688999121212121212121818182424242436···36
size113623324233363618186664446612126661818181812···12

41 irreducible representations

dim1111111112222222222244666
type+++++++++++
imageC1C2C2C2C3C6C6C6D36.C6S3D4D6SD16C3xS3C3xD4C3:D4S3xC6C3xSD16C3xC3:D4Q8:2S3C3xQ8:2S3C9:C6C2xC9:C6Dic9:C6
kernelD36.C6C9:C24D36:C3Q8x3- 1+2Q8:2D9C9:C8D36Q8xC9C1Q8xC32C2x3- 1+2C3xC123- 1+2C3xQ8C18C3xC6C12C9C6C32C3Q8C4C2
# reps111122221111222224412112

Matrix representation of D36.C6 in GL10(F73)

727221000000
727212000000
242311000000
242411000000
00000000172
0000000010
00007200000
00000720000
00000072000
00000007200
,
727212000000
727221000000
72011000000
07211000000
0000000010
00000000172
0000001000
00000017200
0000100000
00001720000
,
4103212000000
41532032000000
38200000000
1033232000000
000043600000
000013300000
000000434300
000000301300
000000006043
000000003030

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

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