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## G = He3.2C12order 324 = 22·34

### 2nd non-split extension by He3 of C12 acting via C12/C2=C6

Aliases: He3.2C12, (C3×C18).4S3, (C3×C9)⋊3Dic3, He33C42C3, He3⋊C31C4, (C2×He3).3C6, C3.8(C32⋊C12), C6.13(C32⋊C6), C2.(He3.2C6), C32.3(C3×Dic3), (C3×C6).3(C3×S3), (C2×He3⋊C3).1C2, SmallGroup(324,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3.2C12
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C3 — He3.2C12
 Lower central He3 — He3.2C12
 Upper central C1 — C6

Generators and relations for He3.2C12
G = < a,b,c,d | a3=b3=c3=1, d12=b, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=ac-1 >

Smallest permutation representation of He3.2C12
On 108 points
Generators in S108
(1 73 39)(2 52 98)(3 75 41)(4 54 100)(5 77 43)(6 56 102)(7 79 45)(8 58 104)(9 81 47)(10 60 106)(11 83 49)(12 62 108)(13 85 51)(14 64 74)(15 87 53)(16 66 76)(17 89 55)(18 68 78)(19 91 57)(20 70 80)(21 93 59)(22 72 82)(23 95 61)(24 38 84)(25 97 63)(26 40 86)(27 99 65)(28 42 88)(29 101 67)(30 44 90)(31 103 69)(32 46 92)(33 105 71)(34 48 94)(35 107 37)(36 50 96)
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)(73 85 97)(74 86 98)(75 87 99)(76 88 100)(77 89 101)(78 90 102)(79 91 103)(80 92 104)(81 93 105)(82 94 106)(83 95 107)(84 96 108)
(1 81 29)(2 90 10)(3 49 45)(4 32 84)(5 13 93)(6 48 52)(7 87 35)(8 96 16)(9 55 51)(11 19 99)(12 54 58)(14 102 22)(15 61 57)(17 25 105)(18 60 64)(20 108 28)(21 67 63)(23 31 75)(24 66 70)(26 78 34)(27 37 69)(30 72 40)(33 43 39)(36 42 46)(38 88 104)(41 107 91)(44 94 74)(47 77 97)(50 100 80)(53 83 103)(56 106 86)(59 89 73)(62 76 92)(65 95 79)(68 82 98)(71 101 85)
(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)| (1,73,39)(2,52,98)(3,75,41)(4,54,100)(5,77,43)(6,56,102)(7,79,45)(8,58,104)(9,81,47)(10,60,106)(11,83,49)(12,62,108)(13,85,51)(14,64,74)(15,87,53)(16,66,76)(17,89,55)(18,68,78)(19,91,57)(20,70,80)(21,93,59)(22,72,82)(23,95,61)(24,38,84)(25,97,63)(26,40,86)(27,99,65)(28,42,88)(29,101,67)(30,44,90)(31,103,69)(32,46,92)(33,105,71)(34,48,94)(35,107,37)(36,50,96), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,85,97)(74,86,98)(75,87,99)(76,88,100)(77,89,101)(78,90,102)(79,91,103)(80,92,104)(81,93,105)(82,94,106)(83,95,107)(84,96,108), (1,81,29)(2,90,10)(3,49,45)(4,32,84)(5,13,93)(6,48,52)(7,87,35)(8,96,16)(9,55,51)(11,19,99)(12,54,58)(14,102,22)(15,61,57)(17,25,105)(18,60,64)(20,108,28)(21,67,63)(23,31,75)(24,66,70)(26,78,34)(27,37,69)(30,72,40)(33,43,39)(36,42,46)(38,88,104)(41,107,91)(44,94,74)(47,77,97)(50,100,80)(53,83,103)(56,106,86)(59,89,73)(62,76,92)(65,95,79)(68,82,98)(71,101,85), (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( (1,73,39)(2,52,98)(3,75,41)(4,54,100)(5,77,43)(6,56,102)(7,79,45)(8,58,104)(9,81,47)(10,60,106)(11,83,49)(12,62,108)(13,85,51)(14,64,74)(15,87,53)(16,66,76)(17,89,55)(18,68,78)(19,91,57)(20,70,80)(21,93,59)(22,72,82)(23,95,61)(24,38,84)(25,97,63)(26,40,86)(27,99,65)(28,42,88)(29,101,67)(30,44,90)(31,103,69)(32,46,92)(33,105,71)(34,48,94)(35,107,37)(36,50,96), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,85,97)(74,86,98)(75,87,99)(76,88,100)(77,89,101)(78,90,102)(79,91,103)(80,92,104)(81,93,105)(82,94,106)(83,95,107)(84,96,108), (1,81,29)(2,90,10)(3,49,45)(4,32,84)(5,13,93)(6,48,52)(7,87,35)(8,96,16)(9,55,51)(11,19,99)(12,54,58)(14,102,22)(15,61,57)(17,25,105)(18,60,64)(20,108,28)(21,67,63)(23,31,75)(24,66,70)(26,78,34)(27,37,69)(30,72,40)(33,43,39)(36,42,46)(38,88,104)(41,107,91)(44,94,74)(47,77,97)(50,100,80)(53,83,103)(56,106,86)(59,89,73)(62,76,92)(65,95,79)(68,82,98)(71,101,85), (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([[(1,73,39),(2,52,98),(3,75,41),(4,54,100),(5,77,43),(6,56,102),(7,79,45),(8,58,104),(9,81,47),(10,60,106),(11,83,49),(12,62,108),(13,85,51),(14,64,74),(15,87,53),(16,66,76),(17,89,55),(18,68,78),(19,91,57),(20,70,80),(21,93,59),(22,72,82),(23,95,61),(24,38,84),(25,97,63),(26,40,86),(27,99,65),(28,42,88),(29,101,67),(30,44,90),(31,103,69),(32,46,92),(33,105,71),(34,48,94),(35,107,37),(36,50,96)], [(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,17,29),(6,18,30),(7,19,31),(8,20,32),(9,21,33),(10,22,34),(11,23,35),(12,24,36),(37,49,61),(38,50,62),(39,51,63),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69),(46,58,70),(47,59,71),(48,60,72),(73,85,97),(74,86,98),(75,87,99),(76,88,100),(77,89,101),(78,90,102),(79,91,103),(80,92,104),(81,93,105),(82,94,106),(83,95,107),(84,96,108)], [(1,81,29),(2,90,10),(3,49,45),(4,32,84),(5,13,93),(6,48,52),(7,87,35),(8,96,16),(9,55,51),(11,19,99),(12,54,58),(14,102,22),(15,61,57),(17,25,105),(18,60,64),(20,108,28),(21,67,63),(23,31,75),(24,66,70),(26,78,34),(27,37,69),(30,72,40),(33,43,39),(36,42,46),(38,88,104),(41,107,91),(44,94,74),(47,77,97),(50,100,80),(53,83,103),(56,106,86),(59,89,73),(62,76,92),(65,95,79),(68,82,98),(71,101,85)], [(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 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 36A ··· 36L order 1 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 6 18 18 18 9 9 1 1 6 18 18 18 3 ··· 3 9 9 9 9 3 ··· 3 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 He3.2C6 He3.2C12 C32⋊C6 C32⋊C12 kernel He3.2C12 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 He3.2C12 in GL3(𝔽37) generated by

 10 0 16 0 0 27 0 10 27
,
 10 0 0 0 10 0 0 0 10
,
 34 33 7 34 0 3 4 33 3
,
 20 16 13 20 25 6 35 11 16
G:=sub<GL(3,GF(37))| [10,0,0,0,0,10,16,27,27],[10,0,0,0,10,0,0,0,10],[34,34,4,33,0,33,7,3,3],[20,20,35,16,25,11,13,6,16] >;

He3.2C12 in GAP, Magma, Sage, TeX

{\rm He}_3._2C_{12}
% in TeX

G:=Group("He3.2C12");
// GroupNames label

G:=SmallGroup(324,17);
// by ID

G=gap.SmallGroup(324,17);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,655,579,585,5404,382]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^12=b,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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