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## G = He3.4Dic3order 324 = 22·34

### The non-split extension by He3 of Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — He3.4Dic3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×C9○He3 — He3.4Dic3
 Lower central C3×C9 — He3.4Dic3
 Upper central C1 — C2

Generators and relations for He3.4Dic3
G = < a,b,c,d,e | a3=b3=c3=1, d6=b, e2=bd3, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, cd=dc, ce=ec, ede-1=b-1d5 >

Subgroups: 207 in 63 conjugacy classes, 28 normal (20 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C9, C9, C32, C32, Dic3, C12, C18, C18, C18, C3×C6, C3×C6, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, Dic9, C3×Dic3, C3⋊Dic3, C3×C18, C3×C18, C2×He3, C2×3- 1+2, C2×3- 1+2, C9○He3, C3×Dic9, C32⋊C12, C9⋊C12, C9⋊Dic3, C2×C9○He3, He3.4Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, C3×C3⋊S3, C3×C3⋊Dic3, He3.4S3, He3.4Dic3

Smallest permutation representation of He3.4Dic3
On 108 points
Generators in S108
(1 77 62)(2 78 63)(3 79 64)(4 80 65)(5 81 66)(6 82 67)(7 83 68)(8 84 69)(9 85 70)(10 86 71)(11 87 72)(12 88 55)(13 89 56)(14 90 57)(15 73 58)(16 74 59)(17 75 60)(18 76 61)(19 50 103)(20 51 104)(21 52 105)(22 53 106)(23 54 107)(24 37 108)(25 38 91)(26 39 92)(27 40 93)(28 41 94)(29 42 95)(30 43 96)(31 44 97)(32 45 98)(33 46 99)(34 47 100)(35 48 101)(36 49 102)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 79 85)(74 80 86)(75 81 87)(76 82 88)(77 83 89)(78 84 90)(91 97 103)(92 98 104)(93 99 105)(94 100 106)(95 101 107)(96 102 108)
(1 62 89)(2 63 90)(3 64 73)(4 65 74)(5 66 75)(6 67 76)(7 68 77)(8 69 78)(9 70 79)(10 71 80)(11 72 81)(12 55 82)(13 56 83)(14 57 84)(15 58 85)(16 59 86)(17 60 87)(18 61 88)(19 44 97)(20 45 98)(21 46 99)(22 47 100)(23 48 101)(24 49 102)(25 50 103)(26 51 104)(27 52 105)(28 53 106)(29 54 107)(30 37 108)(31 38 91)(32 39 92)(33 40 93)(34 41 94)(35 42 95)(36 43 96)
(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)
(1 39 10 48)(2 38 11 47)(3 37 12 46)(4 54 13 45)(5 53 14 44)(6 52 15 43)(7 51 16 42)(8 50 17 41)(9 49 18 40)(19 75 28 84)(20 74 29 83)(21 73 30 82)(22 90 31 81)(23 89 32 80)(24 88 33 79)(25 87 34 78)(26 86 35 77)(27 85 36 76)(55 99 64 108)(56 98 65 107)(57 97 66 106)(58 96 67 105)(59 95 68 104)(60 94 69 103)(61 93 70 102)(62 92 71 101)(63 91 72 100)

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

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

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

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 9A 9B 9C 9D ··· 9K 12A 12B 12C 12D 18A 18B 18C 18D ··· 18K order 1 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 9 9 9 9 ··· 9 12 12 12 12 18 18 18 18 ··· 18 size 1 1 2 3 3 6 6 6 27 27 2 3 3 6 6 6 2 2 2 6 ··· 6 27 27 27 27 2 2 2 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 type + + + + + - - - + - image C1 C2 C3 C4 C6 C12 S3 S3 S3 Dic3 Dic3 Dic3 C3×S3 C3×S3 C3×Dic3 C3×Dic3 He3.4S3 He3.4Dic3 kernel He3.4Dic3 C2×C9○He3 C9⋊Dic3 C9○He3 C3×C18 C3×C9 C3×C18 C2×He3 C2×3- 1+2 C3×C9 He3 3- 1+2 C18 C3×C6 C9 C32 C2 C1 # reps 1 1 2 2 2 4 1 1 2 1 1 2 6 2 6 2 3 3

Matrix representation of He3.4Dic3 in GL6(𝔽37)

 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 36 1 0 0 0 0 36 0 0 0 0 0 0 0 36 1 0 0 0 0 36 0 0 0 0 0 0 0 36 1 0 0 0 0 36 0
,
 0 0 0 0 36 1 0 0 0 0 36 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 1 36 0 0
,
 17 31 0 0 0 0 6 11 0 0 0 0 0 0 17 31 0 0 0 0 6 11 0 0 0 0 0 0 17 31 0 0 0 0 6 11
,
 10 18 18 9 10 18 28 27 27 19 28 27 18 9 10 18 10 18 27 19 28 27 28 27 10 18 10 18 18 9 28 27 28 27 27 19

G:=sub<GL(6,GF(37))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[36,36,0,0,0,0,1,0,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,36,36,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,36,36,36,36,0,0,0,0,1,0,0,0,0,0],[17,6,0,0,0,0,31,11,0,0,0,0,0,0,17,6,0,0,0,0,31,11,0,0,0,0,0,0,17,6,0,0,0,0,31,11],[10,28,18,27,10,28,18,27,9,19,18,27,18,27,10,28,10,28,9,19,18,27,18,27,10,28,10,28,18,27,18,27,18,27,9,19] >;

He3.4Dic3 in GAP, Magma, Sage, TeX

{\rm He}_3._4{\rm Dic}_3
% in TeX

G:=Group("He3.4Dic3");
// GroupNames label

G:=SmallGroup(324,101);
// by ID

G=gap.SmallGroup(324,101);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,3171,453,2164,2170,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=1,d^6=b,e^2=b*d^3,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*d^5>;
// generators/relations

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