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## G = He3.5C12order 324 = 22·34

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3.5C12
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C9○He3 — He3.5C12
 Lower central He3 — He3.5C12
 Upper central C1 — C18

Generators and relations for He3.5C12
G = < a,b,c,d | a3=b3=c3=1, d12=b, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 147 in 67 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C9, C32, Dic3, C12, C18, C18, C3×C6, C3×C9, He3, 3- 1+2, C36, C3×Dic3, C3×C18, C2×He3, C2×3- 1+2, C9○He3, C9×Dic3, He33C4, C2×C9○He3, He3.5C12
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, C3×Dic3, C3⋊Dic3, C3×C3⋊S3, C3×C3⋊Dic3, He3.4C6, He3.5C12

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 type + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 He3.4C6 He3.5C12 kernel He3.5C12 C2×C9○He3 He3⋊3C4 C9○He3 C2×He3 He3 C3×C18 C3×C9 C3×C6 C32 C2 C1 # reps 1 1 2 2 2 4 4 4 8 8 12 12

Matrix representation of He3.5C12 in GL5(𝔽37)

 36 36 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 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 0 0 10 0 0 1 0 0 0 0 0 26 0
,
 29 0 0 0 0 8 8 0 0 0 0 0 0 3 0 0 0 3 0 0 0 0 0 0 3

G:=sub<GL(5,GF(37))| [36,1,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[36,1,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,26,0,0,10,0,0],[29,8,0,0,0,0,8,0,0,0,0,0,0,3,0,0,0,3,0,0,0,0,0,0,3] >;

He3.5C12 in GAP, Magma, Sage, TeX

{\rm He}_3._5C_{12}
% in TeX

G:=Group("He3.5C12");
// GroupNames label

G:=SmallGroup(324,102);
// by ID

G=gap.SmallGroup(324,102);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,655,579,2164,382]);
// 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*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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