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## G = Q8⋊He3⋊C2order 432 = 24·33

### 4th semidirect product of Q8⋊He3 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8⋊He3⋊C2
G = < a,b,c,d,e,f | a4=c3=d3=e3=f2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, eae-1=b, af=fa, bc=cb, bd=db, ebe-1=ab, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, fdf=d-1, ef=fe >

Subgroups: 608 in 79 conjugacy classes, 16 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C32, C32, Dic3, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, SL2(𝔽3), Dic6, C4×S3, C3×Q8, C3×Q8, He3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×SL2(𝔽3), S3×Q8, C32⋊C6, C2×He3, C3×SL2(𝔽3), C3×SL2(𝔽3), C324Q8, C4×C3⋊S3, Q8×C32, C2×C32⋊C6, S3×SL2(𝔽3), Q8×C3⋊S3, Q8⋊He3, Q8⋊He3⋊C2
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, SL2(𝔽3), C2×A4, C2×SL2(𝔽3), C32⋊C6, S3×A4, S3×SL2(𝔽3), C62⋊C6, Q8⋊He3⋊C2

Character table of Q8⋊He3⋊C2

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D size 1 1 9 9 2 6 12 12 24 24 6 54 2 6 12 12 24 24 36 36 36 36 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 ζ32 ζ3 ζ3 ζ32 1 -1 1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 1 1 1 1 linear of order 6 ρ4 1 1 -1 -1 1 1 ζ3 ζ32 ζ32 ζ3 1 -1 1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 1 1 1 1 linear of order 6 ρ5 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 1 linear of order 3 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 1 linear of order 3 ρ7 2 2 0 0 2 -1 2 2 -1 -1 2 0 2 -1 2 2 -1 -1 0 0 0 0 -1 -1 2 -1 orthogonal lifted from S3 ρ8 2 -2 2 -2 2 2 -1 -1 -1 -1 0 0 -2 -2 1 1 1 1 -1 1 -1 1 0 0 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 -2 -2 2 2 2 -1 -1 -1 -1 0 0 -2 -2 1 1 1 1 1 -1 1 -1 0 0 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ10 2 -2 -2 2 2 2 ζ65 ζ6 ζ6 ζ65 0 0 -2 -2 ζ32 ζ3 ζ3 ζ32 ζ3 ζ6 ζ32 ζ65 0 0 0 0 complex lifted from SL2(𝔽3) ρ11 2 -2 2 -2 2 2 ζ65 ζ6 ζ6 ζ65 0 0 -2 -2 ζ32 ζ3 ζ3 ζ32 ζ65 ζ32 ζ6 ζ3 0 0 0 0 complex lifted from SL2(𝔽3) ρ12 2 2 0 0 2 -1 -1+√-3 -1-√-3 ζ6 ζ65 2 0 2 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 0 0 -1 -1 2 -1 complex lifted from C3×S3 ρ13 2 -2 2 -2 2 2 ζ6 ζ65 ζ65 ζ6 0 0 -2 -2 ζ3 ζ32 ζ32 ζ3 ζ6 ζ3 ζ65 ζ32 0 0 0 0 complex lifted from SL2(𝔽3) ρ14 2 -2 -2 2 2 2 ζ6 ζ65 ζ65 ζ6 0 0 -2 -2 ζ3 ζ32 ζ32 ζ3 ζ32 ζ65 ζ3 ζ6 0 0 0 0 complex lifted from SL2(𝔽3) ρ15 2 2 0 0 2 -1 -1-√-3 -1+√-3 ζ65 ζ6 2 0 2 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 0 0 -1 -1 2 -1 complex lifted from C3×S3 ρ16 3 3 -3 -3 3 3 0 0 0 0 -1 1 3 3 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from C2×A4 ρ17 3 3 3 3 3 3 0 0 0 0 -1 -1 3 3 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from A4 ρ18 4 -4 0 0 4 -2 -2 -2 1 1 0 0 -4 2 2 2 -1 -1 0 0 0 0 0 0 0 0 symplectic lifted from S3×SL2(𝔽3), Schur index 2 ρ19 4 -4 0 0 4 -2 1-√-3 1+√-3 ζ32 ζ3 0 0 -4 2 -1-√-3 -1+√-3 ζ65 ζ6 0 0 0 0 0 0 0 0 complex lifted from S3×SL2(𝔽3) ρ20 4 -4 0 0 4 -2 1+√-3 1-√-3 ζ3 ζ32 0 0 -4 2 -1+√-3 -1-√-3 ζ6 ζ65 0 0 0 0 0 0 0 0 complex lifted from S3×SL2(𝔽3) ρ21 6 6 0 0 -3 0 0 0 0 0 -2 0 -3 0 0 0 0 0 0 0 0 0 -2 4 1 -2 orthogonal lifted from C62⋊C6 ρ22 6 6 0 0 6 -3 0 0 0 0 -2 0 6 -3 0 0 0 0 0 0 0 0 1 1 -2 1 orthogonal lifted from S3×A4 ρ23 6 6 0 0 -3 0 0 0 0 0 -2 0 -3 0 0 0 0 0 0 0 0 0 4 -2 1 -2 orthogonal lifted from C62⋊C6 ρ24 6 6 0 0 -3 0 0 0 0 0 -2 0 -3 0 0 0 0 0 0 0 0 0 -2 -2 1 4 orthogonal lifted from C62⋊C6 ρ25 6 6 0 0 -3 0 0 0 0 0 6 0 -3 0 0 0 0 0 0 0 0 0 0 0 -3 0 orthogonal lifted from C32⋊C6 ρ26 12 -12 0 0 -6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8⋊He3⋊C2
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 21 3 23)(2 24 4 22)(5 12 7 10)(6 11 8 9)(13 63 15 61)(14 62 16 64)(17 36 19 34)(18 35 20 33)(25 47 27 45)(26 46 28 48)(29 37 31 39)(30 40 32 38)(41 51 43 49)(42 50 44 52)(53 65 55 67)(54 68 56 66)(57 70 59 72)(58 69 60 71)
(13 71 65)(14 72 66)(15 69 67)(16 70 68)(25 43 32)(26 44 29)(27 41 30)(28 42 31)(37 46 52)(38 47 49)(39 48 50)(40 45 51)(53 61 60)(54 62 57)(55 63 58)(56 64 59)
(1 19 11)(2 20 12)(3 17 9)(4 18 10)(5 22 35)(6 23 36)(7 24 33)(8 21 34)(13 65 71)(14 66 72)(15 67 69)(16 68 70)(25 43 32)(26 44 29)(27 41 30)(28 42 31)(37 46 52)(38 47 49)(39 48 50)(40 45 51)(53 60 61)(54 57 62)(55 58 63)(56 59 64)
(1 59 47)(2 71 27)(3 57 45)(4 69 25)(5 68 37)(6 55 31)(7 66 39)(8 53 29)(9 54 40)(10 67 32)(11 56 38)(12 65 30)(13 41 20)(14 50 33)(15 43 18)(16 52 35)(17 62 51)(19 64 49)(21 60 26)(22 70 46)(23 58 28)(24 72 48)(34 61 44)(36 63 42)
(5 35)(6 36)(7 33)(8 34)(9 17)(10 18)(11 19)(12 20)(13 65)(14 66)(15 67)(16 68)(29 44)(30 41)(31 42)(32 43)(37 52)(38 49)(39 50)(40 51)(53 61)(54 62)(55 63)(56 64)

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

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

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

Matrix representation of Q8⋊He3⋊C2 in GL10(𝔽13)

 8 0 10 0 0 0 0 0 0 0 0 8 0 10 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 11 0 0 0 0 0 0 0 0 1 0 11 0 0 0 0 0 0 1 0 12 0 0 0 0 0 0 0 0 1 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 12 12 0 0 0 0 0 0 0 2 0 0 12 12 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 11 2 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 12 2 12 12 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 12 2 0 0 12 12
,
 12 0 1 0 0 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 6 0 11 0 0 0 0 0 0 0 0 6 0 11 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 5 0 0 0 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 12 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 12 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 12 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 12 12

G:=sub<GL(10,GF(13))| [8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,10,0,5,0,0,0,0,0,0,0,0,10,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,11,0,12,0,0,0,0,0,0,0,0,11,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,2,2,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,5,12,12,12,12,0,0,0,0,2,1,0,2,0,2,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[12,0,6,0,0,0,0,0,0,0,0,12,0,6,0,0,0,0,0,0,1,0,11,0,0,0,0,0,0,0,0,1,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,5,12,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,2,1,0,2,0,2,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12] >;

Q8⋊He3⋊C2 in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm He}_3\rtimes C_2
% in TeX

G:=Group("Q8:He3:C2");
// GroupNames label

G:=SmallGroup(432,270);
// by ID

G=gap.SmallGroup(432,270);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-3,-2,198,772,94,1081,528,1684,1691,6053]);
// Polycyclic

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

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