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G = C62:3C12order 432 = 24·33

2nd semidirect product of C62 and C12 acting via C12/C2=C6

metabelian, supersoluble, monomial

Aliases: C62:3C12, C62:3Dic3, C62.26D6, C62:5C4:C3, (C2xC62).2S3, (C2xC62).4C6, C62.8(C2xC6), He3:7(C22:C4), (C2xHe3).31D4, (C22xHe3):4C4, C6.19(C6xDic3), C2.3(He3:6D4), (C23xHe3).2C2, C22:3(C32:C12), C23.3(C32:C6), C32:3(C6.D4), (C22xHe3).20C22, (C3xC6).9(C2xC12), (C2xC6).47(S3xC6), (C2xC3:Dic3):2C6, (C3xC6).16(C3xD4), C6.31(C3xC3:D4), (C2xC32:C12):4C2, C2.5(C2xC32:C12), C32:3(C3xC22:C4), (C3xC6).31(C3:D4), (C2xHe3).30(C2xC4), (C22xC6).25(C3xS3), (C2xC6).19(C3xDic3), (C3xC6).14(C2xDic3), C3.2(C3xC6.D4), C22.7(C2xC32:C6), SmallGroup(432,166)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C62:3C12
C1C3C32C3xC6C62C22xHe3C2xC32:C12 — C62:3C12
C32C3xC6 — C62:3C12
C1C22C23

Generators and relations for C62:3C12
 G = < a,b,c | a6=b6=c12=1, ab=ba, cac-1=a-1b, cbc-1=b-1 >

Subgroups: 549 in 151 conjugacy classes, 46 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2xC4, C23, C32, C32, Dic3, C12, C2xC6, C2xC6, C2xC6, C22:C4, C3xC6, C3xC6, C3xC6, C2xDic3, C2xC12, C22xC6, C22xC6, He3, C3xDic3, C3:Dic3, C62, C62, C62, C6.D4, C3xC22:C4, C2xHe3, C2xHe3, C2xHe3, C6xDic3, C2xC3:Dic3, C2xC62, C2xC62, C32:C12, C22xHe3, C22xHe3, C22xHe3, C3xC6.D4, C62:5C4, C2xC32:C12, C23xHe3, C62:3C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, D4, Dic3, C12, D6, C2xC6, C22:C4, C3xS3, C2xDic3, C3:D4, C2xC12, C3xD4, C3xDic3, S3xC6, C6.D4, C3xC22:C4, C32:C6, C6xDic3, C3xC3:D4, C32:C12, C2xC32:C6, C3xC6.D4, C2xC32:C12, He3:6D4, C62:3C12

Smallest permutation representation of C62:3C12
On 72 points
Generators in S72
(1 11 33 7 5 27)(2 53 30 57 10 45)(3 72 70 9 66 64)(4 39 8 59 36 51)(6 15 69 49 61 23)(12 21 63 55 67 17)(13 32 43 71 47 28)(14 24 52 20 18 58)(16 40 38 22 46 44)(19 26 37 65 41 34)(25 35 68 31 29 62)(42 56 54 48 50 60)
(1 20 25 50 72 46)(2 47 61 51 26 21)(3 22 27 52 62 48)(4 37 63 53 28 23)(5 24 29 54 64 38)(6 39 65 55 30 13)(7 14 31 56 66 40)(8 41 67 57 32 15)(9 16 33 58 68 42)(10 43 69 59 34 17)(11 18 35 60 70 44)(12 45 71 49 36 19)
(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)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B4C4D6A···6G6H···6M6N···6AL12A···12H
order12222233333344446···66···66···612···12
size111122233666181818182···23···36···618···18

62 irreducible representations

dim1111111122222222226666
type+++++-++-+
imageC1C2C2C3C4C6C6C12S3D4Dic3D6C3xS3C3:D4C3xD4C3xDic3S3xC6C3xC3:D4C32:C6C32:C12C2xC32:C6He3:6D4
kernelC62:3C12C2xC32:C12C23xHe3C62:5C4C22xHe3C2xC3:Dic3C2xC62C62C2xC62C2xHe3C62C62C22xC6C3xC6C3xC6C2xC6C2xC6C6C23C22C22C2
# reps1212442812212444281214

Matrix representation of C62:3C12 in GL8(F13)

90000000
03000000
000012000
001200000
000120000
00000010
00000001
00000100
,
10000000
01000000
001000000
000100000
000010000
00000400
00000040
00000004
,
01000000
120000000
00000009
00000100
00000030
00004000
001200000
000100000

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0] >;

C62:3C12 in GAP, Magma, Sage, TeX

C_6^2\rtimes_3C_{12}
% in TeX

G:=Group("C6^2:3C12");
// GroupNames label

G:=SmallGroup(432,166);
// by ID

G=gap.SmallGroup(432,166);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,4037,2035,14118]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=b^-1>;
// generators/relations

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