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G = C62.79D6  order 432 = 24·33

27th non-split extension by C62 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C62.79D6, (C6xDic3):4S3, (C3xC6).42D12, C32:9(D6:C4), (C32xC6).43D4, C33:12(C22:C4), C6.17(C12:S3), C2.2(C33:8D4), C2.2(C33:7D4), C6.6(C32:7D4), (C3xC62).9C22, C6.27(C3:D12), C3:1(C6.11D12), C3:1(C6.D12), C6.10(C6.D6), (C2xC6).33S32, C6.4(C4xC3:S3), (Dic3xC3xC6):4C2, (C3xC6).50(C4xS3), (C6xC3:Dic3):3C2, (C2xC3:Dic3):8S3, C22.7(S3xC3:S3), (C2xC33:C2):3C4, C2.4(C33:8(C2xC4)), (C2xDic3):2(C3:S3), (C3xC6).62(C3:D4), (C32xC6).40(C2xC4), (C22xC33:C2).1C2, (C2xC6).15(C2xC3:S3), SmallGroup(432,451)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — C62.79D6
Chief series
C1 — C3 — C32 — C33 — C32xC6 — C3xC62 — Dic3xC3xC6 — C62.79D6
Lower central
C33 — C32xC6 — C62.79D6
Upper central
C1 — C22

Generators and relations for C62.79D6
 G = < a,b,c,d | a6=b6=d2=1, c6=a3, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=b-1, dcd=b3c5 >

Subgroups: 2456 in 332 conjugacy classes, 72 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, C23, C32, C32, C32, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C3:S3, C3xC6, C3xC6, C3xC6, C2xDic3, C2xDic3, C2xC12, C22xS3, C33, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C62, C62, C62, D6:C4, C33:C2, C32xC6, C6xDic3, C6xDic3, C2xC3:Dic3, C6xC12, C22xC3:S3, C32xDic3, C3xC3:Dic3, C2xC33:C2, C2xC33:C2, C3xC62, C6.D12, C6.11D12, Dic3xC3xC6, C6xC3:Dic3, C22xC33:C2, C62.79D6
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C3:S3, C4xS3, D12, C3:D4, S32, C2xC3:S3, D6:C4, C6.D6, C3:D12, C4xC3:S3, C12:S3, C32:7D4, S3xC3:S3, C6.D12, C6.11D12, C33:8(C2xC4), C33:7D4, C33:8D4, C62.79D6

Smallest permutation representation of C62.79D6
►On 72 points
Generators in S72
(1 33 42 7 27 48)(2 34 43 8 28 37)(3 35 44 9 29 38)(4 36 45 10 30 39)(5 25 46 11 31 40)(6 26 47 12 32 41)(13 59 63 19 53 69)(14 60 64 20 54 70)(15 49 65 21 55 71)(16 50 66 22 56 72)(17 51 67 23 57 61)(18 52 68 24 58 62)

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

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

66 conjugacy classes

class 1 2A2B2C2D2E3A···3E3F3G3H3I4A4B4C4D6A···6O6P···6AA12A···12P12Q12R12S12T
order1222223···3333344446···66···612···1212121212
size111154542···244446618182···24···46···618181818

66 irreducible representations

dim111112222222444
type++++++++++++
imageC1C2C2C2C4S3S3D4D6C4xS3D12C3:D4S32C6.D6C3:D12
kernelC62.79D6Dic3xC3xC6C6xC3:Dic3C22xC33:C2C2xC33:C2C6xDic3C2xC3:Dic3C32xC6C62C3xC6C3xC6C3xC6C2xC6C6C6
# reps111144125101010448

Matrix representation of C62.79D6 ►in GL8(F13)

10000000
01000000
00010000
0012120000
000011200
00001000
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
01000000
10000000
00100000
00010000
00000800
00005800
00000001
00000010
,
120000000
01000000
00100000
0012120000
000012100
00000100
00000001
00000010

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62.79D6 in GAP, Magma, Sage, TeX 

C_6^2._{79}D_6
 
% in TeX
 
G:=Group("C6^2.79D6");
 
// GroupNames label
 
G:=SmallGroup(432,451);
 
// by ID
 
G=gap.SmallGroup(432,451);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,92,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^6=a^3,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^5>;
 
// generators/relations
 

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