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G = C4xC9oHe3order 324 = 22·34

Direct product of C4 and C9oHe3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C4xC9oHe3, C36.C32, He3.6C12, C12.3C33, 3- 1+2:4C12, (C3xC36):5C3, (C3xC9):12C12, C9.2(C3xC12), C18.3(C3xC6), (C3xC18).20C6, (C4xHe3).2C3, C6.4(C32xC6), (C2xHe3).15C6, C3.3(C32xC12), C32.5(C3xC12), (C3xC12).5C32, (C4x3- 1+2):3C3, (C2x3- 1+2).7C6, C2.(C2xC9oHe3), (C3xC6).12(C3xC6), (C2xC9oHe3).3C2, SmallGroup(324,108)

Series: Derived Chief Lower central Upper central

C1C3 — C4xC9oHe3
C1C3C6C18C3xC18C2xC9oHe3 — C4xC9oHe3
C1C3 — C4xC9oHe3
C1C36 — C4xC9oHe3

Generators and relations for C4xC9oHe3
 G = < a,b,c,d,e | a4=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >

Subgroups: 123 in 99 conjugacy classes, 87 normal (15 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C9, C32, C12, C12, C18, C18, C3xC6, C3xC9, He3, 3- 1+2, C36, C36, C3xC12, C3xC18, C2xHe3, C2x3- 1+2, C9oHe3, C3xC36, C4xHe3, C4x3- 1+2, C2xC9oHe3, C4xC9oHe3
Quotients: C1, C2, C3, C4, C6, C32, C12, C3xC6, C33, C3xC12, C32xC6, C9oHe3, C32xC12, C2xC9oHe3, C4xC9oHe3

Smallest permutation representation of C4xC9oHe3
On 108 points
Generators in S108
(1 77 23 70)(2 78 24 71)(3 79 25 72)(4 80 26 64)(5 81 27 65)(6 73 19 66)(7 74 20 67)(8 75 21 68)(9 76 22 69)(10 55 102 48)(11 56 103 49)(12 57 104 50)(13 58 105 51)(14 59 106 52)(15 60 107 53)(16 61 108 54)(17 62 100 46)(18 63 101 47)(28 98 44 82)(29 99 45 83)(30 91 37 84)(31 92 38 85)(32 93 39 86)(33 94 40 87)(34 95 41 88)(35 96 42 89)(36 97 43 90)
(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(10 13 16)(11 14 17)(12 15 18)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 103 106)(101 104 107)(102 105 108)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)
(1 52 28)(2 53 29)(3 54 30)(4 46 31)(5 47 32)(6 48 33)(7 49 34)(8 50 35)(9 51 36)(10 94 73)(11 95 74)(12 96 75)(13 97 76)(14 98 77)(15 99 78)(16 91 79)(17 92 80)(18 93 81)(19 55 40)(20 56 41)(21 57 42)(22 58 43)(23 59 44)(24 60 45)(25 61 37)(26 62 38)(27 63 39)(64 100 85)(65 101 86)(66 102 87)(67 103 88)(68 104 89)(69 105 90)(70 106 82)(71 107 83)(72 108 84)

G:=sub<Sym(108)| (1,77,23,70)(2,78,24,71)(3,79,25,72)(4,80,26,64)(5,81,27,65)(6,73,19,66)(7,74,20,67)(8,75,21,68)(9,76,22,69)(10,55,102,48)(11,56,103,49)(12,57,104,50)(13,58,105,51)(14,59,106,52)(15,60,107,53)(16,61,108,54)(17,62,100,46)(18,63,101,47)(28,98,44,82)(29,99,45,83)(30,91,37,84)(31,92,38,85)(32,93,39,86)(33,94,40,87)(34,95,41,88)(35,96,42,89)(36,97,43,90), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (10,13,16)(11,14,17)(12,15,18)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,103,106)(101,104,107)(102,105,108), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105), (1,52,28)(2,53,29)(3,54,30)(4,46,31)(5,47,32)(6,48,33)(7,49,34)(8,50,35)(9,51,36)(10,94,73)(11,95,74)(12,96,75)(13,97,76)(14,98,77)(15,99,78)(16,91,79)(17,92,80)(18,93,81)(19,55,40)(20,56,41)(21,57,42)(22,58,43)(23,59,44)(24,60,45)(25,61,37)(26,62,38)(27,63,39)(64,100,85)(65,101,86)(66,102,87)(67,103,88)(68,104,89)(69,105,90)(70,106,82)(71,107,83)(72,108,84)>;

G:=Group( (1,77,23,70)(2,78,24,71)(3,79,25,72)(4,80,26,64)(5,81,27,65)(6,73,19,66)(7,74,20,67)(8,75,21,68)(9,76,22,69)(10,55,102,48)(11,56,103,49)(12,57,104,50)(13,58,105,51)(14,59,106,52)(15,60,107,53)(16,61,108,54)(17,62,100,46)(18,63,101,47)(28,98,44,82)(29,99,45,83)(30,91,37,84)(31,92,38,85)(32,93,39,86)(33,94,40,87)(34,95,41,88)(35,96,42,89)(36,97,43,90), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (10,13,16)(11,14,17)(12,15,18)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,103,106)(101,104,107)(102,105,108), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105), (1,52,28)(2,53,29)(3,54,30)(4,46,31)(5,47,32)(6,48,33)(7,49,34)(8,50,35)(9,51,36)(10,94,73)(11,95,74)(12,96,75)(13,97,76)(14,98,77)(15,99,78)(16,91,79)(17,92,80)(18,93,81)(19,55,40)(20,56,41)(21,57,42)(22,58,43)(23,59,44)(24,60,45)(25,61,37)(26,62,38)(27,63,39)(64,100,85)(65,101,86)(66,102,87)(67,103,88)(68,104,89)(69,105,90)(70,106,82)(71,107,83)(72,108,84) );

G=PermutationGroup([[(1,77,23,70),(2,78,24,71),(3,79,25,72),(4,80,26,64),(5,81,27,65),(6,73,19,66),(7,74,20,67),(8,75,21,68),(9,76,22,69),(10,55,102,48),(11,56,103,49),(12,57,104,50),(13,58,105,51),(14,59,106,52),(15,60,107,53),(16,61,108,54),(17,62,100,46),(18,63,101,47),(28,98,44,82),(29,99,45,83),(30,91,37,84),(31,92,38,85),(32,93,39,86),(33,94,40,87),(34,95,41,88),(35,96,42,89),(36,97,43,90)], [(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(10,13,16),(11,14,17),(12,15,18),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,103,106),(101,104,107),(102,105,108)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,106,103),(101,107,104),(102,108,105)], [(1,52,28),(2,53,29),(3,54,30),(4,46,31),(5,47,32),(6,48,33),(7,49,34),(8,50,35),(9,51,36),(10,94,73),(11,95,74),(12,96,75),(13,97,76),(14,98,77),(15,99,78),(16,91,79),(17,92,80),(18,93,81),(19,55,40),(20,56,41),(21,57,42),(22,58,43),(23,59,44),(24,60,45),(25,61,37),(26,62,38),(27,63,39),(64,100,85),(65,101,86),(66,102,87),(67,103,88),(68,104,89),(69,105,90),(70,106,82),(71,107,83),(72,108,84)]])

132 conjugacy classes

class 1  2 3A3B3C···3J4A4B6A6B6C···6J9A···9F9G···9V12A12B12C12D12E···12T18A···18F18G···18V36A···36L36M···36AR
order12333···344666···69···99···91212121212···1218···1818···1836···3636···36
size11113···311113···31···13···311113···31···13···31···13···3

132 irreducible representations

dim111111111111333
type++
imageC1C2C3C3C3C4C6C6C6C12C12C12C9oHe3C2xC9oHe3C4xC9oHe3
kernelC4xC9oHe3C2xC9oHe3C3xC36C4xHe3C4x3- 1+2C9oHe3C3xC18C2xHe3C2x3- 1+2C3xC9He33- 1+2C4C2C1
# reps11821628216164326612

Matrix representation of C4xC9oHe3 in GL4(F37) generated by

6000
0100
0010
0001
,
1000
01200
00120
00012
,
10000
0187
00100
00026
,
1000
01000
00100
00010
,
10000
04149
00026
092933
G:=sub<GL(4,GF(37))| [6,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[10,0,0,0,0,1,0,0,0,8,10,0,0,7,0,26],[1,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,4,0,9,0,14,0,29,0,9,26,33] >;

C4xC9oHe3 in GAP, Magma, Sage, TeX

C_4\times C_9\circ {\rm He}_3
% in TeX

G:=Group("C4xC9oHe3");
// GroupNames label

G:=SmallGroup(324,108);
// by ID

G=gap.SmallGroup(324,108);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,-3,324,1034,165]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^9=c^3=e^3=1,d^1=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,d*e=e*d>;
// generators/relations

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