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## G = C22×He3.C3order 324 = 22·34

### Direct product of C22 and He3.C3

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

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 160 in 70 conjugacy classes, 40 normal (12 characteristic)
C1, C2 [×3], C3, C3 [×2], C22, C6 [×3], C6 [×6], C9 [×3], C32, C32, C2×C6, C2×C6 [×2], C18 [×9], C3×C6 [×3], C3×C6 [×3], C3×C9, He3, 3- 1+2 [×2], C2×C18 [×3], C62, C62, C3×C18 [×3], C2×He3 [×3], C2×3- 1+2 [×6], He3.C3, C6×C18, C22×He3, C22×3- 1+2 [×2], C2×He3.C3 [×3], C22×He3.C3
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C32, C2×C6 [×4], C3×C6 [×3], He3, C62, C2×He3 [×3], He3.C3, C22×He3, C2×He3.C3 [×3], C22×He3.C3

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

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 6A ··· 6F 6G ··· 6L 6M ··· 6R 9A ··· 9F 9G 9H 9I 9J 18A ··· 18R 18S ··· 18AD order 1 2 2 2 3 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 9 9 9 18 ··· 18 18 ··· 18 size 1 1 1 1 1 1 3 3 9 9 1 ··· 1 3 ··· 3 9 ··· 9 3 ··· 3 9 9 9 9 3 ··· 3 9 ··· 9

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 He3 C2×He3 He3.C3 C2×He3.C3 kernel C22×He3.C3 C2×He3.C3 C6×C18 C22×He3 C22×3- 1+2 C3×C18 C2×He3 C2×3- 1+2 C2×C6 C6 C22 C2 # reps 1 3 2 2 4 6 6 12 2 6 6 18

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

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

C22×He3.C3 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xHe3.C3");
// GroupNames label

G:=SmallGroup(324,87);
// by ID

G=gap.SmallGroup(324,87);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,237,1096]);
// Polycyclic

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

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