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## G = He3.C12order 324 = 22·34

### 1st non-split extension by He3 of C12 acting via C12/C2=C6

Aliases: He3.1C12, (C3×C18).3S3, (C3×C9)⋊2Dic3, He33C4.C3, He3.C33C4, (C2×He3).2C6, C6.12(C32⋊C6), C3.7(C32⋊C12), C2.(He3.C6), C32.2(C3×Dic3), (C3×C6).2(C3×S3), (C2×He3.C3).3C2, SmallGroup(324,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3.C12
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3.C3 — He3.C12
 Lower central He3 — He3.C12
 Upper central C1 — C6

Generators and relations for He3.C12
G = < a,b,c,d | a3=b3=c3=1, d12=b-1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=ac-1 >

Smallest permutation representation of He3.C12
On 108 points
Generators in S108
(1 52 86)(2 75 65)(3 54 88)(4 77 67)(5 56 90)(6 79 69)(7 58 92)(8 81 71)(9 60 94)(10 83 37)(11 62 96)(12 85 39)(13 64 98)(14 87 41)(15 66 100)(16 89 43)(17 68 102)(18 91 45)(19 70 104)(20 93 47)(21 72 106)(22 95 49)(23 38 108)(24 97 51)(25 40 74)(26 99 53)(27 42 76)(28 101 55)(29 44 78)(30 103 57)(31 46 80)(32 105 59)(33 48 82)(34 107 61)(35 50 84)(36 73 63)
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)
(2 65 87)(3 54 76)(5 90 68)(6 79 57)(8 71 93)(9 60 82)(11 96 38)(12 85 63)(14 41 99)(15 66 88)(17 102 44)(18 91 69)(20 47 105)(21 72 94)(23 108 50)(24 97 39)(26 53 75)(27 42 100)(29 78 56)(30 103 45)(32 59 81)(33 48 106)(35 84 62)(36 73 51)(37 49 61)(40 64 52)(43 55 67)(46 70 58)(74 86 98)(77 101 89)(80 92 104)(83 107 95)
(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)| (1,52,86)(2,75,65)(3,54,88)(4,77,67)(5,56,90)(6,79,69)(7,58,92)(8,81,71)(9,60,94)(10,83,37)(11,62,96)(12,85,39)(13,64,98)(14,87,41)(15,66,100)(16,89,43)(17,68,102)(18,91,45)(19,70,104)(20,93,47)(21,72,106)(22,95,49)(23,38,108)(24,97,51)(25,40,74)(26,99,53)(27,42,76)(28,101,55)(29,44,78)(30,103,57)(31,46,80)(32,105,59)(33,48,82)(34,107,61)(35,50,84)(36,73,63), (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96), (2,65,87)(3,54,76)(5,90,68)(6,79,57)(8,71,93)(9,60,82)(11,96,38)(12,85,63)(14,41,99)(15,66,88)(17,102,44)(18,91,69)(20,47,105)(21,72,94)(23,108,50)(24,97,39)(26,53,75)(27,42,100)(29,78,56)(30,103,45)(32,59,81)(33,48,106)(35,84,62)(36,73,51)(37,49,61)(40,64,52)(43,55,67)(46,70,58)(74,86,98)(77,101,89)(80,92,104)(83,107,95), (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( (1,52,86)(2,75,65)(3,54,88)(4,77,67)(5,56,90)(6,79,69)(7,58,92)(8,81,71)(9,60,94)(10,83,37)(11,62,96)(12,85,39)(13,64,98)(14,87,41)(15,66,100)(16,89,43)(17,68,102)(18,91,45)(19,70,104)(20,93,47)(21,72,106)(22,95,49)(23,38,108)(24,97,51)(25,40,74)(26,99,53)(27,42,76)(28,101,55)(29,44,78)(30,103,57)(31,46,80)(32,105,59)(33,48,82)(34,107,61)(35,50,84)(36,73,63), (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96), (2,65,87)(3,54,76)(5,90,68)(6,79,57)(8,71,93)(9,60,82)(11,96,38)(12,85,63)(14,41,99)(15,66,88)(17,102,44)(18,91,69)(20,47,105)(21,72,94)(23,108,50)(24,97,39)(26,53,75)(27,42,100)(29,78,56)(30,103,45)(32,59,81)(33,48,106)(35,84,62)(36,73,51)(37,49,61)(40,64,52)(43,55,67)(46,70,58)(74,86,98)(77,101,89)(80,92,104)(83,107,95), (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([[(1,52,86),(2,75,65),(3,54,88),(4,77,67),(5,56,90),(6,79,69),(7,58,92),(8,81,71),(9,60,94),(10,83,37),(11,62,96),(12,85,39),(13,64,98),(14,87,41),(15,66,100),(16,89,43),(17,68,102),(18,91,45),(19,70,104),(20,93,47),(21,72,106),(22,95,49),(23,38,108),(24,97,51),(25,40,74),(26,99,53),(27,42,76),(28,101,55),(29,44,78),(30,103,57),(31,46,80),(32,105,59),(33,48,82),(34,107,61),(35,50,84),(36,73,63)], [(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,61,49),(38,62,50),(39,63,51),(40,64,52),(41,65,53),(42,66,54),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60),(73,97,85),(74,98,86),(75,99,87),(76,100,88),(77,101,89),(78,102,90),(79,103,91),(80,104,92),(81,105,93),(82,106,94),(83,107,95),(84,108,96)], [(2,65,87),(3,54,76),(5,90,68),(6,79,57),(8,71,93),(9,60,82),(11,96,38),(12,85,63),(14,41,99),(15,66,88),(17,102,44),(18,91,69),(20,47,105),(21,72,94),(23,108,50),(24,97,39),(26,53,75),(27,42,100),(29,78,56),(30,103,45),(32,59,81),(33,48,106),(35,84,62),(36,73,51),(37,49,61),(40,64,52),(43,55,67),(46,70,58),(74,86,98),(77,101,89),(80,92,104),(83,107,95)], [(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)]])

44 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 9A ··· 9F 9G 9H 12A 12B 12C 12D 18A ··· 18F 18G 18H 36A ··· 36L order 1 2 3 3 3 3 4 4 6 6 6 6 9 ··· 9 9 9 12 12 12 12 18 ··· 18 18 18 36 ··· 36 size 1 1 1 1 6 18 9 9 1 1 6 18 3 ··· 3 18 18 9 9 9 9 3 ··· 3 18 18 9 ··· 9

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 6 6 type + + + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 He3.C6 He3.C12 C32⋊C6 C32⋊C12 kernel He3.C12 C2×He3.C3 He3⋊3C4 He3.C3 C2×He3 He3 C3×C18 C3×C9 C3×C6 C32 C2 C1 C6 C3 # reps 1 1 2 2 2 4 1 1 2 2 12 12 1 1

Matrix representation of He3.C12 in GL5(𝔽37)

 1 0 0 0 0 0 1 0 0 0 0 0 10 0 28 0 0 0 0 27 0 0 0 10 27
,
 1 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10
,
 36 36 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 27 10 0 0 0 1 0 26
,
 14 0 0 0 0 23 23 0 0 0 0 0 9 28 21 0 0 21 28 0 0 0 9 28 12

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,28,27,27],[1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[36,1,0,0,0,36,0,0,0,0,0,0,1,27,1,0,0,0,10,0,0,0,0,0,26],[14,23,0,0,0,0,23,0,0,0,0,0,9,21,9,0,0,28,28,28,0,0,21,0,12] >;

He3.C12 in GAP, Magma, Sage, TeX

{\rm He}_3.C_{12}
% in TeX

G:=Group("He3.C12");
// GroupNames label

G:=SmallGroup(324,15);
// by ID

G=gap.SmallGroup(324,15);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,655,579,585,8644,652]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^12=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

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