Aliases: C6.8(S3×A4), Q8⋊He3⋊4C2, (Q8×C32)⋊2C6, Q8⋊2(C32⋊C6), C3⋊S3⋊2SL2(𝔽3), (C3×SL2(𝔽3))⋊2S3, C2.3(C62⋊C6), C3.3(S3×SL2(𝔽3)), C32⋊2(C2×SL2(𝔽3)), (Q8×C3⋊S3)⋊1C3, (C2×C3⋊S3).2A4, (C3×C6).2(C2×A4), (C3×Q8).16(C3×S3), SmallGroup(432,270)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8×C32 — Q8⋊He3⋊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), C32⋊4Q8, 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 |
►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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