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G = C62.5D6  order 432 = 24·33

5th non-split extension by C62 of D6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C62.5D6, (C3xC6).7D12, C32:2(D6:C4), He3:4(C22:C4), (C2xHe3).16D4, C2.3(He3:3D4), C6.35(C3:D12), C6.16(C6.D6), C22.7(C32:D6), C3.2(C6.D12), (C22xHe3).5C22, (C2xC6).51S32, (C3xC6).9(C4xS3), (C2xC3:Dic3):2S3, (C2xC32:C12):2C2, (C2xHe3:C2):1C4, C2.5(He3:(C2xC4)), (C3xC6).11(C3:D4), (C2xHe3).16(C2xC4), (C22xHe3:C2).1C2, SmallGroup(432,98)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2xHe3 — C62.5D6
Chief series
C1 — C3 — C32 — He3 — C2xHe3 — C22xHe3 — C2xC32:C12 — C62.5D6
Lower central
He3 — C2xHe3 — C62.5D6
Upper central
C1 — C22

Generators and relations for C62.5D6
 G = < a,b,c,d | a6=b6=d2=1, c6=a3b3, ab=ba, cac-1=dad=a-1b4, cbc-1=b-1, bd=db, dcd=b3c5 >

Subgroups: 839 in 151 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, C23, C32, C32, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C3xS3, C3xC6, C3xC6, C2xDic3, C2xC12, C22xS3, C22xC6, He3, C3xDic3, C3:Dic3, S3xC6, C62, C62, D6:C4, C6.D4, He3:C2, C2xHe3, C2xHe3, C6xDic3, C2xC3:Dic3, S3xC2xC6, C32:C12, C2xHe3:C2, C2xHe3:C2, C22xHe3, D6:Dic3, C2xC32:C12, C22xHe3:C2, C62.5D6
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C4xS3, D12, C3:D4, S32, D6:C4, C6.D6, C3:D12, C32:D6, C6.D12, He3:(C2xC4), He3:3D4, C62.5D6

Smallest permutation representation of C62.5D6
►On 72 points
Generators in S72
(1 35 44 59 48 31)(2 52 61 60 6 17)(3 57 70 49 11 14)(4 34 39 50 47 26)(5 20 9 51 64 55)(7 29 38 53 42 25)(8 58 67 54 12 23)(10 28 45 56 41 32)(13 37 21 69 36 65)(15 63 30 71 19 43)(16 68 27 72 24 40)(18 46 22 62 33 66)

(1 14 40 53 64 33)(2 34 65 54 41 15)(3 16 42 55 66 35)(4 36 67 56 43 17)(5 18 44 57 68 25)(6 26 69 58 45 19)(7 20 46 59 70 27)(8 28 71 60 47 21)(9 22 48 49 72 29)(10 30 61 50 37 23)(11 24 38 51 62 31)(12 32 63 52 39 13)

(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 55)(2 8)(3 53)(4 6)(5 51)(7 49)(9 59)(10 12)(11 57)(13 23)(14 66)(15 21)(16 64)(17 19)(18 62)(20 72)(22 70)(24 68)(25 38)(26 36)(27 48)(28 34)(29 46)(30 32)(31 44)(33 42)(35 40)(37 39)(41 47)(43 45)(50 52)(54 60)(56 58)(61 63)(65 71)(67 69)
 
G:=sub<Sym(72)| (1,35,44,59,48,31)(2,52,61,60,6,17)(3,57,70,49,11,14)(4,34,39,50,47,26)(5,20,9,51,64,55)(7,29,38,53,42,25)(8,58,67,54,12,23)(10,28,45,56,41,32)(13,37,21,69,36,65)(15,63,30,71,19,43)(16,68,27,72,24,40)(18,46,22,62,33,66), (1,14,40,53,64,33)(2,34,65,54,41,15)(3,16,42,55,66,35)(4,36,67,56,43,17)(5,18,44,57,68,25)(6,26,69,58,45,19)(7,20,46,59,70,27)(8,28,71,60,47,21)(9,22,48,49,72,29)(10,30,61,50,37,23)(11,24,38,51,62,31)(12,32,63,52,39,13), (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), (1,55)(2,8)(3,53)(4,6)(5,51)(7,49)(9,59)(10,12)(11,57)(13,23)(14,66)(15,21)(16,64)(17,19)(18,62)(20,72)(22,70)(24,68)(25,38)(26,36)(27,48)(28,34)(29,46)(30,32)(31,44)(33,42)(35,40)(37,39)(41,47)(43,45)(50,52)(54,60)(56,58)(61,63)(65,71)(67,69)>;
 
G:=Group( (1,35,44,59,48,31)(2,52,61,60,6,17)(3,57,70,49,11,14)(4,34,39,50,47,26)(5,20,9,51,64,55)(7,29,38,53,42,25)(8,58,67,54,12,23)(10,28,45,56,41,32)(13,37,21,69,36,65)(15,63,30,71,19,43)(16,68,27,72,24,40)(18,46,22,62,33,66), (1,14,40,53,64,33)(2,34,65,54,41,15)(3,16,42,55,66,35)(4,36,67,56,43,17)(5,18,44,57,68,25)(6,26,69,58,45,19)(7,20,46,59,70,27)(8,28,71,60,47,21)(9,22,48,49,72,29)(10,30,61,50,37,23)(11,24,38,51,62,31)(12,32,63,52,39,13), (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), (1,55)(2,8)(3,53)(4,6)(5,51)(7,49)(9,59)(10,12)(11,57)(13,23)(14,66)(15,21)(16,64)(17,19)(18,62)(20,72)(22,70)(24,68)(25,38)(26,36)(27,48)(28,34)(29,46)(30,32)(31,44)(33,42)(35,40)(37,39)(41,47)(43,45)(50,52)(54,60)(56,58)(61,63)(65,71)(67,69) );
 
G=PermutationGroup([[(1,35,44,59,48,31),(2,52,61,60,6,17),(3,57,70,49,11,14),(4,34,39,50,47,26),(5,20,9,51,64,55),(7,29,38,53,42,25),(8,58,67,54,12,23),(10,28,45,56,41,32),(13,37,21,69,36,65),(15,63,30,71,19,43),(16,68,27,72,24,40),(18,46,22,62,33,66)], [(1,14,40,53,64,33),(2,34,65,54,41,15),(3,16,42,55,66,35),(4,36,67,56,43,17),(5,18,44,57,68,25),(6,26,69,58,45,19),(7,20,46,59,70,27),(8,28,71,60,47,21),(9,22,48,49,72,29),(10,30,61,50,37,23),(11,24,38,51,62,31),(12,32,63,52,39,13)], [(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)], [(1,55),(2,8),(3,53),(4,6),(5,51),(7,49),(9,59),(10,12),(11,57),(13,23),(14,66),(15,21),(16,64),(17,19),(18,62),(20,72),(22,70),(24,68),(25,38),(26,36),(27,48),(28,34),(29,46),(30,32),(31,44),(33,42),(35,40),(37,39),(41,47),(43,45),(50,52),(54,60),(56,58),(61,63),(65,71),(67,69)]])
 

38 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D6A6B6C6D···6I6J6K6L6M6N6O6P12A···12H
order122222333344446666···6666666612···12
size1111181826612181818182226···61212121818181818···18

38 irreducible representations

dim1111222222444666
type+++++++++++-
imageC1C2C2C4S3D4D6C4xS3D12C3:D4S32C6.D6C3:D12C32:D6He3:(C2xC4)He3:3D4
kernelC62.5D6C2xC32:C12C22xHe3:C2C2xHe3:C2C2xC3:Dic3C2xHe3C62C3xC6C3xC6C3xC6C2xC6C6C6C22C2C2
# reps1214222444112224

Matrix representation of C62.5D6 ►in GL10(F13)

11200000000
1000000000
0001000000
00121000000
0000090000
000012122000
0000141000
0000000445
0000000300
00000001099
,
12000000000
01200000000
00120000000
00012000000
0000400000
0000040000
0000004000
00000001000
00000000100
00000000010
,
00012000000
00112000000
01200000000
11200000000
000000012122
0000000100
0000000001
00001111000
00001200000
00000012000
,
12100000000
0100000000
00112000000
00012000000
000012122000
0000010000
0000001000
00000001111
00000000120
00000000012

G:=sub<GL(10,GF(13))| [1,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,9,12,4,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,4,3,10,0,0,0,0,0,0,0,4,0,9,0,0,0,0,0,0,0,5,0,9],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,10],[0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,12,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,2,0,1,0,0,0],[12,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,11,0,12] >;
 

C62.5D6 in GAP, Magma, Sage, TeX 

C_6^2._5D_6
 
% in TeX
 
G:=Group("C6^2.5D6");
 
// GroupNames label
 
G:=SmallGroup(432,98);
 
// by ID
 
G=gap.SmallGroup(432,98);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,92,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^6=a^3*b^3,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^3*c^5>;
 
// generators/relations
 

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