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G = C62.8D6  order 432 = 24·33

8th non-split extension by C62 of D6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C62.8D6, He3⋊5(C4○D4), He3⋊7D4⋊1C2, He3⋊6D4⋊1C2, He3⋊3D4⋊2C2, He3⋊2Q8⋊4C2, C32⋊7D4⋊1S3, C3⋊Dic3.9D6, C32⋊4(C4○D12), C32⋊3(D4⋊2S3), C3.3(D6.3D6), C32⋊C12.8C22, (C2×He3).12C23, He3⋊3C4.6C22, C22.1(C32⋊D6), (C22×He3).8C22, (C2×C6).8S32, C6.86(C2×S32), (C2×C3⋊S3).3D6, C6.S32⋊3C2, He3⋊(C2×C4)⋊2C2, (C2×C3⋊Dic3)⋊4S3, (C2×C32⋊C12)⋊6C2, C2.13(C2×C32⋊D6), (C3×C6).12(C22×S3), (C2×C32⋊C6).2C22, (C2×He3⋊C2).5C22, SmallGroup(432,318)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — C62.8D6
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C6.S32 — C62.8D6
Lower central
He3 — C2×He3 — C62.8D6
Upper central
C1 — C2 — C22

Generators and relations for C62.8D6
 G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=a-1b4, dad-1=a-1b, cbc-1=b-1, bd=db, dcd-1=c5 >

Subgroups: 895 in 156 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C4○D12, D4⋊2S3, C32⋊C6, He3⋊C2, C2×He3, C2×He3, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C32⋊C12, He3⋊3C4, C2×C32⋊C6, C2×He3⋊C2, C22×He3, D6.3D6, D6.4D6, He3⋊2Q8, C6.S32, He3⋊(C2×C4), He3⋊3D4, C2×C32⋊C12, He3⋊6D4, He3⋊7D4, C62.8D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, C2×S32, C32⋊D6, D6.3D6, C2×C32⋊D6, C62.8D6

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

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

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

32 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L12A12B12C12D12E12F
order12222333344444666666666666121212121212
size11218182661299181818246666121212123636181818183636

32 irreducible representations

dim11111111122222222444466
type++++++++-++++++-+++
imageC1C2C2C2C2C2C2C2C62.8D6S3S3D6D6D6C4○D4C4○D12S32D4⋊2S3C2×S32D6.3D6C32⋊D6C2×C32⋊D6
kernelC62.8D6He3⋊2Q8C6.S32He3⋊(C2×C4)He3⋊3D4C2×C32⋊C12He3⋊6D4He3⋊7D4C1C2×C3⋊Dic3C32⋊7D4C3⋊Dic3C2×C3⋊S3C62He3C32C2×C6C32C6C3C22C2
# reps1111111111131224111222

Matrix representation of C62.8D6 ►in GL10(𝔽13)

112103000000
10100000000
00121000000
00120000000
00001200000
00000120000
00000001200
000012121100
000012120011
00000000120
,
12000000000
01200000000
00120000000
00012000000
00000120000
00001120000
000010121200
00000121000
000010001212
00000120010
,
0500000000
5000000000
0005000000
0050000000
000011001112
00000000121
00000000120
00000100120
00000010120
0000111212120
,
01010000000
10100000000
05012000000
50120000000
00000000112
000012120021
00000012010
00000001210
0000000010
00001200010

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

C62.8D6 in GAP, Magma, Sage, TeX 

C_6^2._8D_6
 
% in TeX
 
G:=Group("C6^2.8D6");
 
// GroupNames label
 
G:=SmallGroup(432,318);
 
// by ID
 
G=gap.SmallGroup(432,318);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^4,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^5>;
 
// generators/relations
 

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