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## G = He3⋊3C8order 216 = 23·33

### 1st semidirect product of He3 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3⋊3C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3C8
 Lower central C32 — He3⋊3C8
 Upper central C1 — C4

Generators and relations for He33C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, dbd-1=b-1, cd=dc >

Smallest permutation representation of He33C8
On 72 points
Generators in S72
(1 68 47)(2 48 69)(3 70 41)(4 42 71)(5 72 43)(6 44 65)(7 66 45)(8 46 67)(9 17 31)(10 32 18)(11 19 25)(12 26 20)(13 21 27)(14 28 22)(15 23 29)(16 30 24)(33 54 57)(34 58 55)(35 56 59)(36 60 49)(37 50 61)(38 62 51)(39 52 63)(40 64 53)
(1 52 31)(2 32 53)(3 54 25)(4 26 55)(5 56 27)(6 28 49)(7 50 29)(8 30 51)(9 68 63)(10 64 69)(11 70 57)(12 58 71)(13 72 59)(14 60 65)(15 66 61)(16 62 67)(17 47 39)(18 40 48)(19 41 33)(20 34 42)(21 43 35)(22 36 44)(23 45 37)(24 38 46)
(1 47 9)(2 48 10)(3 41 11)(4 42 12)(5 43 13)(6 44 14)(7 45 15)(8 46 16)(17 63 31)(18 64 32)(19 57 25)(20 58 26)(21 59 27)(22 60 28)(23 61 29)(24 62 30)(33 70 54)(34 71 55)(35 72 56)(36 65 49)(37 66 50)(38 67 51)(39 68 52)(40 69 53)
(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)

G:=sub<Sym(72)| (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,17,31)(10,32,18)(11,19,25)(12,26,20)(13,21,27)(14,28,22)(15,23,29)(16,30,24)(33,54,57)(34,58,55)(35,56,59)(36,60,49)(37,50,61)(38,62,51)(39,52,63)(40,64,53), (1,52,31)(2,32,53)(3,54,25)(4,26,55)(5,56,27)(6,28,49)(7,50,29)(8,30,51)(9,68,63)(10,64,69)(11,70,57)(12,58,71)(13,72,59)(14,60,65)(15,66,61)(16,62,67)(17,47,39)(18,40,48)(19,41,33)(20,34,42)(21,43,35)(22,36,44)(23,45,37)(24,38,46), (1,47,9)(2,48,10)(3,41,11)(4,42,12)(5,43,13)(6,44,14)(7,45,15)(8,46,16)(17,63,31)(18,64,32)(19,57,25)(20,58,26)(21,59,27)(22,60,28)(23,61,29)(24,62,30)(33,70,54)(34,71,55)(35,72,56)(36,65,49)(37,66,50)(38,67,51)(39,68,52)(40,69,53), (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)>;

G:=Group( (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,17,31)(10,32,18)(11,19,25)(12,26,20)(13,21,27)(14,28,22)(15,23,29)(16,30,24)(33,54,57)(34,58,55)(35,56,59)(36,60,49)(37,50,61)(38,62,51)(39,52,63)(40,64,53), (1,52,31)(2,32,53)(3,54,25)(4,26,55)(5,56,27)(6,28,49)(7,50,29)(8,30,51)(9,68,63)(10,64,69)(11,70,57)(12,58,71)(13,72,59)(14,60,65)(15,66,61)(16,62,67)(17,47,39)(18,40,48)(19,41,33)(20,34,42)(21,43,35)(22,36,44)(23,45,37)(24,38,46), (1,47,9)(2,48,10)(3,41,11)(4,42,12)(5,43,13)(6,44,14)(7,45,15)(8,46,16)(17,63,31)(18,64,32)(19,57,25)(20,58,26)(21,59,27)(22,60,28)(23,61,29)(24,62,30)(33,70,54)(34,71,55)(35,72,56)(36,65,49)(37,66,50)(38,67,51)(39,68,52)(40,69,53), (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) );

G=PermutationGroup([[(1,68,47),(2,48,69),(3,70,41),(4,42,71),(5,72,43),(6,44,65),(7,66,45),(8,46,67),(9,17,31),(10,32,18),(11,19,25),(12,26,20),(13,21,27),(14,28,22),(15,23,29),(16,30,24),(33,54,57),(34,58,55),(35,56,59),(36,60,49),(37,50,61),(38,62,51),(39,52,63),(40,64,53)], [(1,52,31),(2,32,53),(3,54,25),(4,26,55),(5,56,27),(6,28,49),(7,50,29),(8,30,51),(9,68,63),(10,64,69),(11,70,57),(12,58,71),(13,72,59),(14,60,65),(15,66,61),(16,62,67),(17,47,39),(18,40,48),(19,41,33),(20,34,42),(21,43,35),(22,36,44),(23,45,37),(24,38,46)], [(1,47,9),(2,48,10),(3,41,11),(4,42,12),(5,43,13),(6,44,14),(7,45,15),(8,46,16),(17,63,31),(18,64,32),(19,57,25),(20,58,26),(21,59,27),(22,60,28),(23,61,29),(24,62,30),(33,70,54),(34,71,55),(35,72,56),(36,65,49),(37,66,50),(38,67,51),(39,68,52),(40,69,53)], [(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)]])

He33C8 is a maximal subgroup of
C32⋊C6⋊C8  He3⋊M4(2)  C12.89S32  He33M4(2)  He33D8  He34SD16  He35SD16  He33Q16  C8×C32⋊C6  He35M4(2)  He37M4(2)  He38SD16  He36D8  He36Q16  He310SD16
He33C8 is a maximal quotient of
He33C16

40 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12L 24A ··· 24H order 1 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 3 3 6 6 6 1 1 2 3 3 6 6 6 9 9 9 9 2 2 3 3 3 3 6 ··· 6 9 ··· 9

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 type + + + - + - image C1 C2 C3 C4 C6 C8 C12 C24 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3×C3⋊C8 C32⋊C6 C32⋊C12 He3⋊3C8 kernel He3⋊3C8 C4×He3 C32⋊4C8 C2×He3 C3×C12 He3 C3×C6 C32 C3×C12 C3×C6 C12 C32 C6 C3 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 1 1 2 2 2 4 1 1 2

Matrix representation of He33C8 in GL6(𝔽73)

 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
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 0 0 0 0 0 1 0 0 0 0 72 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0
,
 9 20 11 64 9 20 11 64 53 62 11 64 11 64 9 20 9 20 53 62 11 64 11 64 9 20 9 20 11 64 11 64 11 64 53 62

G:=sub<GL(6,GF(73))| [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],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,72,0,0,0,0,1,72,0,0,0,0],[9,11,11,53,9,11,20,64,64,62,20,64,11,53,9,11,9,11,64,62,20,64,20,64,9,11,9,11,11,53,20,64,20,64,64,62] >;

He33C8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3C_8
% in TeX

G:=Group("He3:3C8");
// GroupNames label

G:=SmallGroup(216,14);
// by ID

G=gap.SmallGroup(216,14);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,1444,1450,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations

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