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## G = C22×C3.He3order 324 = 22·34

### Direct product of C22 and C3.He3

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C3.He3
 Chief series C1 — C3 — C32 — C3×C9 — C3.He3 — C2×C3.He3 — C22×C3.He3
 Lower central C1 — C3 — C32 — C22×C3.He3
 Upper central C1 — C2×C6 — C62 — C22×C3.He3

Generators and relations for C22×C3.He3
G = < a,b,c,d,e,f | a2=b2=c3=e3=1, d3=c-1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=cde-1, fef-1=c-1e >

Subgroups: 115 in 65 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C3, C3, C22, C6, C6, C9, C32, C2×C6, C2×C6, C18, C3×C6, C3×C9, 3- 1+2, C2×C18, C62, C3×C18, C2×3- 1+2, C3.He3, C6×C18, C22×3- 1+2, C2×C3.He3, C22×C3.He3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, He3, C62, C2×He3, C3.He3, C22×He3, C2×C3.He3, C22×C3.He3

Smallest permutation representation of C22×C3.He3
On 108 points
Generators in S108
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)
(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 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 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)(64 70 67)(65 71 68)(66 72 69)(73 76 79)(74 77 80)(75 78 81)(91 97 94)(92 98 95)(93 99 96)(100 103 106)(101 104 107)(102 105 108)
(1 26 11 7 23 17 4 20 14)(2 21 12 8 27 18 5 24 15)(3 25 13 9 22 10 6 19 16)(28 46 43 34 52 40 31 49 37)(29 50 44 35 47 41 32 53 38)(30 54 45 36 51 42 33 48 39)(55 73 70 61 79 67 58 76 64)(56 77 71 62 74 68 59 80 65)(57 81 72 63 78 69 60 75 66)(82 100 97 88 106 94 85 103 91)(83 104 98 89 101 95 86 107 92)(84 108 99 90 105 96 87 102 93)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A ··· 6F 6G ··· 6L 9A ··· 9F 9G ··· 9L 18A ··· 18R 18S ··· 18AJ order 1 2 2 2 3 3 3 3 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 1 1 1 3 3 1 ··· 1 3 ··· 3 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

68 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 C3.He3 C2×C3.He3 kernel C22×C3.He3 C2×C3.He3 C6×C18 C22×3- 1+2 C3×C18 C2×3- 1+2 C2×C6 C6 C22 C2 # reps 1 3 2 6 6 18 2 6 6 18

Matrix representation of C22×C3.He3 in GL4(𝔽19) generated by

 1 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 7 0 0 0 0 9 0 0 0 0 9 0 0 13 15 4
,
 1 0 0 0 0 1 0 0 0 0 11 0 0 11 12 7
,
 7 0 0 0 0 0 1 0 0 12 8 10 0 1 0 11
G:=sub<GL(4,GF(19))| [1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,9,0,13,0,0,9,15,0,0,0,4],[1,0,0,0,0,1,0,11,0,0,11,12,0,0,0,7],[7,0,0,0,0,0,12,1,0,1,8,0,0,0,10,11] >;

C22×C3.He3 in GAP, Magma, Sage, TeX

C_2^2\times C_3.{\rm He}_3
% in TeX

G:=Group("C2^2xC3.He3");
// GroupNames label

G:=SmallGroup(324,89);
// by ID

G=gap.SmallGroup(324,89);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,303,453,1096]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=e^3=1,d^3=c^-1,f^3=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=c*d*e^-1,f*e*f^-1=c^-1*e>;
// generators/relations

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