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G = C62.21D6  order 432 = 24·33

4th non-split extension by C62 of D6 acting via D6/C2=S3

metabelian, supersoluble, monomial

Aliases: C62.21D6, (C6×C12)⋊1S3, (C6×C12)⋊1C6, C6.16(S3×C12), C6.11D12⋊C3, C6.13(C3×D12), (C3×C6).20D12, C62.7(C2×C6), C32⋊5(D6⋊C4), He3⋊5(C22⋊C4), (C2×He3).21D4, C2.2(He3⋊4D4), C2.2(He3⋊6D4), (C22×He3).19C22, (C2×C3⋊S3)⋊C12, (C2×C4×He3)⋊1C2, C3.2(C3×D6⋊C4), (C2×C32⋊C6)⋊2C4, (C2×C6).41(S3×C6), (C3×C6).4(C2×C12), (C3×C6).16(C4×S3), (C2×C12).6(C3×S3), (C2×C3⋊Dic3)⋊1C6, (C3×C6).10(C3×D4), C6.15(C3×C3⋊D4), C2.5(C4×C32⋊C6), (C2×C32⋊C12)⋊3C2, (C2×C4)⋊1(C32⋊C6), (C22×C3⋊S3).1C6, C32⋊2(C3×C22⋊C4), (C3×C6).20(C3⋊D4), (C2×He3).20(C2×C4), C22.6(C2×C32⋊C6), (C22×C32⋊C6).2C2, SmallGroup(432,141)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62.21D6
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C22×He3 — C22×C32⋊C6 — C62.21D6
Lower central
C32 — C3×C6 — C62.21D6
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.21D6
 G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 677 in 135 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, D6⋊C4, C3×C22⋊C4, C32⋊C6, C2×He3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, S3×C2×C6, C22×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C2×C32⋊C6, C22×He3, C3×D6⋊C4, C6.11D12, C2×C32⋊C12, C2×C4×He3, C22×C32⋊C6, C62.21D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, S3×C6, D6⋊C4, C3×C22⋊C4, C32⋊C6, S3×C12, C3×D12, C3×C3⋊D4, C2×C32⋊C6, C3×D6⋊C4, C4×C32⋊C6, He3⋊4D4, He3⋊6D4, C62.21D6

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B4C4D6A6B6C6D···6I6J···6R6S6T6U6V12A12B12C12D12E···12T12U12V12W12X
order12222233333344446666···66···666661212121212···1212121212
size111118182336662218182223···36···61818181822226···618181818

62 irreducible representations

dim111111111122222222222266666
type+++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D4D6C3×S3C4×S3D12C3⋊D4C3×D4S3×C6S3×C12C3×D12C3×C3⋊D4C32⋊C6C2×C32⋊C6C4×C32⋊C6He3⋊4D4He3⋊6D4
kernelC62.21D6C2×C32⋊C12C2×C4×He3C22×C32⋊C6C6.11D12C2×C32⋊C6C2×C3⋊Dic3C6×C12C22×C3⋊S3C2×C3⋊S3C6×C12C2×He3C62C2×C12C3×C6C3×C6C3×C6C3×C6C2×C6C6C6C6C2×C4C22C2C2C2
# reps111124222812122224244411222

Matrix representation of C62.21D6 ►in GL8(𝔽13)

40000000
04000000
00100000
00010000
000012100
000012000
000000012
000000112
,
120000000
012000000
00010000
001210000
00000100
000012100
00000001
000000121
,
111000000
112000000
000000114
00000092
001140000
00920000
000011400
00009200
,
122000000
01000000
000000411
00000029
000041100
00002900
004110000
00290000

G:=sub<GL(8,GF(13))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1],[1,1,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0],[12,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0] >;
 

C62.21D6 in GAP, Magma, Sage, TeX 

C_6^2._{21}D_6
 
% in TeX
 
G:=Group("C6^2.21D6");
 
// GroupNames label
 
G:=SmallGroup(432,141);
 
// by ID
 
G=gap.SmallGroup(432,141);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,4037,2035,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
 
// generators/relations
 

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