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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, (C6×Dic3)⋊4S3, (C3×C6).42D12, C32⋊9(D6⋊C4), (C32×C6).43D4, C33⋊12(C22⋊C4), C6.17(C12⋊S3), C2.2(C33⋊8D4), C2.2(C33⋊7D4), C6.6(C32⋊7D4), (C3×C62).9C22, C6.27(C3⋊D12), C3⋊1(C6.11D12), C3⋊1(C6.D12), C6.10(C6.D6), (C2×C6).33S32, C6.4(C4×C3⋊S3), (Dic3×C3×C6)⋊4C2, (C3×C6).50(C4×S3), (C6×C3⋊Dic3)⋊3C2, (C2×C3⋊Dic3)⋊8S3, C22.7(S3×C3⋊S3), (C2×C33⋊C2)⋊3C4, C2.4(C33⋊8(C2×C4)), (C2×Dic3)⋊2(C3⋊S3), (C3×C6).62(C3⋊D4), (C32×C6).40(C2×C4), (C22×C33⋊C2).1C2, (C2×C6).15(C2×C3⋊S3), SmallGroup(432,451)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C62.79D6
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — Dic3×C3×C6 — C62.79D6
Lower central
C33 — C32×C6 — 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, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C33⋊C2, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C2×C33⋊C2, C3×C62, C6.D12, C6.11D12, Dic3×C3×C6, C6×C3⋊Dic3, C22×C33⋊C2, C62.79D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, S32, C2×C3⋊S3, D6⋊C4, C6.D6, C3⋊D12, C4×C3⋊S3, C12⋊S3, C32⋊7D4, S3×C3⋊S3, C6.D12, C6.11D12, C33⋊8(C2×C4), 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++++++++++++
imageC1C2C2C2C4S3S3D4D6C4×S3D12C3⋊D4S32C6.D6C3⋊D12
kernelC62.79D6Dic3×C3×C6C6×C3⋊Dic3C22×C33⋊C2C2×C33⋊C2C6×Dic3C2×C3⋊Dic3C32×C6C62C3×C6C3×C6C3×C6C2×C6C6C6
# 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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