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## G = Q8⋊He3order 216 = 23·33

### The semidirect product of Q8 and He3 acting via He3/C32=C3

Aliases: Q8⋊He3, C322SL2(𝔽3), (C3×C6).2A4, C6.11(C3×A4), C2.(C32⋊A4), (Q8×C32)⋊1C3, (C3×SL2(𝔽3))⋊C3, (C3×Q8).5C32, C3.5(C3×SL2(𝔽3)), SmallGroup(216,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — Q8⋊He3
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — Q8⋊He3
 Lower central Q8 — C3×Q8 — Q8⋊He3
 Upper central C1 — C6 — C3×C6

Generators and relations for Q8⋊He3
G = < a,b,c,d,e | a4=c3=d3=e3=1, b2=a2, bab-1=a-1, ac=ca, ad=da, eae-1=b, bc=cb, bd=db, ebe-1=ab, cd=dc, ece-1=cd-1, de=ed >

Smallest permutation representation of Q8⋊He3
On 72 points
Generators in S72
(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)
(1 7 3 5)(2 6 4 8)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)(33 39 35 37)(34 38 36 40)(41 47 43 45)(42 46 44 48)(49 55 51 53)(50 54 52 56)(57 63 59 61)(58 62 60 64)(65 71 67 69)(66 70 68 72)
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(1 25 57)(2 30 63)(3 27 59)(4 32 61)(5 28 64)(6 31 58)(7 26 62)(8 29 60)(9 33 65)(10 38 71)(11 35 67)(12 40 69)(13 36 72)(14 39 66)(15 34 70)(16 37 68)(17 41 49)(18 46 55)(19 43 51)(20 48 53)(21 44 56)(22 47 50)(23 42 54)(24 45 52)

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

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

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

Q8⋊He3 is a maximal subgroup of   C32⋊CSU2(𝔽3)  C322GL2(𝔽3)  C322CSU2(𝔽3)  C323GL2(𝔽3)  C6.(S3×A4)  Q8⋊He3⋊C2  C4○D4⋊He3

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3J 4 6A 6B 6C 6D 6E ··· 6J 12A ··· 12H order 1 2 3 3 3 3 3 ··· 3 4 6 6 6 6 6 ··· 6 12 ··· 12 size 1 1 1 1 3 3 12 ··· 12 6 1 1 3 3 12 ··· 12 6 ··· 6

31 irreducible representations

 dim 1 1 1 2 2 2 3 3 3 3 6 type + - + image C1 C3 C3 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 He3 C3×A4 C32⋊A4 Q8⋊He3 kernel Q8⋊He3 C3×SL2(𝔽3) Q8×C32 C32 C32 C3 C3×C6 Q8 C6 C2 C1 # reps 1 6 2 1 2 6 1 2 2 6 2

Matrix representation of Q8⋊He3 in GL5(𝔽13)

 0 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 3 4 0 0 0 4 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 8 0 0 0 0 12 1 0 0 0 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 0 1 0 0 0 4 10 0 0 0 0 0 9 0 7 0 0 12 0 4 0 0 9 3 4

G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,4,0,0,0,4,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,8,12,12,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[0,4,0,0,0,1,10,0,0,0,0,0,9,12,9,0,0,0,0,3,0,0,7,4,4] >;

Q8⋊He3 in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm He}_3
% in TeX

G:=Group("Q8:He3");
// GroupNames label

G:=SmallGroup(216,42);
// by ID

G=gap.SmallGroup(216,42);
# by ID

G:=PCGroup([6,-3,-3,-3,-2,2,-2,145,1299,117,2434,202,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^3=e^3=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=b,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations

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