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## G = He3⋊4Q8order 216 = 23·33

### 2nd semidirect product of He3 and Q8 acting via Q8/C4=C2

Aliases: He34Q8, C324Dic6, (C3×C12).4S3, (C3×C6).16D6, C12.7(C3⋊S3), C4.(He3⋊C2), (C4×He3).2C2, He33C4.3C2, C3.2(C324Q8), (C2×He3).11C22, C6.27(C2×C3⋊S3), C2.3(C2×He3⋊C2), SmallGroup(216,66)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊4Q8
 Chief series C1 — C3 — C32 — He3 — C2×He3 — He3⋊3C4 — He3⋊4Q8
 Lower central He3 — C2×He3 — He3⋊4Q8
 Upper central C1 — C6 — C12

Generators and relations for He34Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 186 in 66 conjugacy classes, 24 normal (10 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, Q8, C32, Dic3, C12, C12, C3×C6, Dic6, C3×Q8, He3, C3×Dic3, C3×C12, C2×He3, C3×Dic6, He33C4, C4×He3, He34Q8
Quotients: C1, C2, C22, S3, Q8, D6, C3⋊S3, Dic6, C2×C3⋊S3, He3⋊C2, C324Q8, C2×He3⋊C2, He34Q8

Smallest permutation representation of He34Q8
On 72 points
Generators in S72
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 43 69)(10 44 70)(11 41 71)(12 42 72)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 53 45)(18 54 46)(19 55 47)(20 56 48)(25 68 61)(26 65 62)(27 66 63)(28 67 64)
(1 40 33)(2 37 34)(3 38 35)(4 39 36)(5 31 13)(6 32 14)(7 29 15)(8 30 16)(9 27 54)(10 28 55)(11 25 56)(12 26 53)(17 72 62)(18 69 63)(19 70 64)(20 71 61)(21 58 49)(22 59 50)(23 60 51)(24 57 52)(41 68 48)(42 65 45)(43 66 46)(44 67 47)
(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 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)

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

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

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

He34Q8 is a maximal subgroup of
He35SD16  He33Q16  He37SD16  He35Q16  He39SD16  He37Q16  C3⋊S3⋊Dic6  C12⋊S3⋊S3  C62.47D6  C62.16D6  Q8×He3⋊C2
He34Q8 is a maximal quotient of
C62.29D6  C62.30D6

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 4C 6A 6B 6C 6D 6E 6F 12A 12B 12C ··· 12J 12K 12L 12M 12N order 1 2 3 3 3 3 3 3 4 4 4 6 6 6 6 6 6 12 12 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 6 6 2 18 18 1 1 6 6 6 6 2 2 6 ··· 6 18 18 18 18

31 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 type + + + + - + - image C1 C2 C2 S3 Q8 D6 Dic6 He3⋊C2 C2×He3⋊C2 He3⋊4Q8 kernel He3⋊4Q8 He3⋊3C4 C4×He3 C3×C12 He3 C3×C6 C32 C4 C2 C1 # reps 1 2 1 4 1 4 8 4 4 2

Matrix representation of He34Q8 in GL5(𝔽13)

 0 12 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 12 12 2 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 12 1 0 0 0 12 0 0 0 0 0 0 0 9 0 0 0 10 10 6 0 0 12 3 3
,
 3 7 0 0 0 6 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 4 0 0 0 2 2 0 0 0 0 0 12 12 2 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(13))| [0,1,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,2,1],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[12,12,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,9,10,3,0,0,0,6,3],[3,6,0,0,0,7,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,2,0,0,0,4,2,0,0,0,0,0,12,0,0,0,0,12,1,0,0,0,2,0,1] >;

He34Q8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4Q_8
% in TeX

G:=Group("He3:4Q8");
// GroupNames label

G:=SmallGroup(216,66);
// by ID

G=gap.SmallGroup(216,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,387,1444,382]);
// Polycyclic

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

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