direct product, metabelian, nilpotent (class 2), monomial
Aliases: Q8×He3, C6.16C62, C4.(C2×He3), C12.7(C3×C6), (C3×C12).4C6, C32⋊4(C3×Q8), (C4×He3).5C2, (Q8×C32)⋊2C3, C3.2(Q8×C32), C2.3(C22×He3), (C3×Q8).7C32, (C2×He3).18C22, (C3×C6).15(C2×C6), SmallGroup(216,80)
Series: Derived ►Chief ►Lower central ►Upper central
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, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 114 in 66 conjugacy classes, 42 normal (9 characteristic)
C1, C2, C3, C3, C4, C6, C6, Q8, C32, C12, C12, C3×C6, C3×Q8, C3×Q8, He3, C3×C12, C2×He3, Q8×C32, C4×He3, Q8×He3
Quotients: C1, C2, C3, C22, C6, Q8, C32, C2×C6, C3×C6, C3×Q8, He3, C62, C2×He3, Q8×C32, C22×He3, 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 46 3 48)(2 45 4 47)(5 43 7 41)(6 42 8 44)(9 63 11 61)(10 62 12 64)(13 51 15 49)(14 50 16 52)(17 37 19 39)(18 40 20 38)(21 57 23 59)(22 60 24 58)(25 65 27 67)(26 68 28 66)(29 55 31 53)(30 54 32 56)(33 69 35 71)(34 72 36 70)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 38)(6 32 39)(7 29 40)(8 30 37)(9 15 60)(10 16 57)(11 13 58)(12 14 59)(17 42 56)(18 43 53)(19 44 54)(20 41 55)(21 64 50)(22 61 51)(23 62 52)(24 63 49)(45 67 70)(46 68 71)(47 65 72)(48 66 69)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 65 22)(18 66 23)(19 67 24)(20 68 21)(25 58 39)(26 59 40)(27 60 37)(28 57 38)(41 71 64)(42 72 61)(43 69 62)(44 70 63)(45 49 54)(46 50 55)(47 51 56)(48 52 53)
(1 26 12)(2 27 9)(3 28 10)(4 25 11)(5 16 57)(6 13 58)(7 14 59)(8 15 60)(17 72 56)(18 69 53)(19 70 54)(20 71 55)(21 41 50)(22 42 51)(23 43 52)(24 44 49)(29 40 35)(30 37 36)(31 38 33)(32 39 34)(45 67 63)(46 68 64)(47 65 61)(48 66 62)
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), (1,46,3,48)(2,45,4,47)(5,43,7,41)(6,42,8,44)(9,63,11,61)(10,62,12,64)(13,51,15,49)(14,50,16,52)(17,37,19,39)(18,40,20,38)(21,57,23,59)(22,60,24,58)(25,65,27,67)(26,68,28,66)(29,55,31,53)(30,54,32,56)(33,69,35,71)(34,72,36,70), (1,26,35)(2,27,36)(3,28,33)(4,25,34)(5,31,38)(6,32,39)(7,29,40)(8,30,37)(9,15,60)(10,16,57)(11,13,58)(12,14,59)(17,42,56)(18,43,53)(19,44,54)(20,41,55)(21,64,50)(22,61,51)(23,62,52)(24,63,49)(45,67,70)(46,68,71)(47,65,72)(48,66,69), (1,14,29)(2,15,30)(3,16,31)(4,13,32)(5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,65,22)(18,66,23)(19,67,24)(20,68,21)(25,58,39)(26,59,40)(27,60,37)(28,57,38)(41,71,64)(42,72,61)(43,69,62)(44,70,63)(45,49,54)(46,50,55)(47,51,56)(48,52,53), (1,26,12)(2,27,9)(3,28,10)(4,25,11)(5,16,57)(6,13,58)(7,14,59)(8,15,60)(17,72,56)(18,69,53)(19,70,54)(20,71,55)(21,41,50)(22,42,51)(23,43,52)(24,44,49)(29,40,35)(30,37,36)(31,38,33)(32,39,34)(45,67,63)(46,68,64)(47,65,61)(48,66,62)>;
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), (1,46,3,48)(2,45,4,47)(5,43,7,41)(6,42,8,44)(9,63,11,61)(10,62,12,64)(13,51,15,49)(14,50,16,52)(17,37,19,39)(18,40,20,38)(21,57,23,59)(22,60,24,58)(25,65,27,67)(26,68,28,66)(29,55,31,53)(30,54,32,56)(33,69,35,71)(34,72,36,70), (1,26,35)(2,27,36)(3,28,33)(4,25,34)(5,31,38)(6,32,39)(7,29,40)(8,30,37)(9,15,60)(10,16,57)(11,13,58)(12,14,59)(17,42,56)(18,43,53)(19,44,54)(20,41,55)(21,64,50)(22,61,51)(23,62,52)(24,63,49)(45,67,70)(46,68,71)(47,65,72)(48,66,69), (1,14,29)(2,15,30)(3,16,31)(4,13,32)(5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,65,22)(18,66,23)(19,67,24)(20,68,21)(25,58,39)(26,59,40)(27,60,37)(28,57,38)(41,71,64)(42,72,61)(43,69,62)(44,70,63)(45,49,54)(46,50,55)(47,51,56)(48,52,53), (1,26,12)(2,27,9)(3,28,10)(4,25,11)(5,16,57)(6,13,58)(7,14,59)(8,15,60)(17,72,56)(18,69,53)(19,70,54)(20,71,55)(21,41,50)(22,42,51)(23,43,52)(24,44,49)(29,40,35)(30,37,36)(31,38,33)(32,39,34)(45,67,63)(46,68,64)(47,65,61)(48,66,62) );
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)], [(1,46,3,48),(2,45,4,47),(5,43,7,41),(6,42,8,44),(9,63,11,61),(10,62,12,64),(13,51,15,49),(14,50,16,52),(17,37,19,39),(18,40,20,38),(21,57,23,59),(22,60,24,58),(25,65,27,67),(26,68,28,66),(29,55,31,53),(30,54,32,56),(33,69,35,71),(34,72,36,70)], [(1,26,35),(2,27,36),(3,28,33),(4,25,34),(5,31,38),(6,32,39),(7,29,40),(8,30,37),(9,15,60),(10,16,57),(11,13,58),(12,14,59),(17,42,56),(18,43,53),(19,44,54),(20,41,55),(21,64,50),(22,61,51),(23,62,52),(24,63,49),(45,67,70),(46,68,71),(47,65,72),(48,66,69)], [(1,14,29),(2,15,30),(3,16,31),(4,13,32),(5,33,10),(6,34,11),(7,35,12),(8,36,9),(17,65,22),(18,66,23),(19,67,24),(20,68,21),(25,58,39),(26,59,40),(27,60,37),(28,57,38),(41,71,64),(42,72,61),(43,69,62),(44,70,63),(45,49,54),(46,50,55),(47,51,56),(48,52,53)], [(1,26,12),(2,27,9),(3,28,10),(4,25,11),(5,16,57),(6,13,58),(7,14,59),(8,15,60),(17,72,56),(18,69,53),(19,70,54),(20,71,55),(21,41,50),(22,42,51),(23,43,52),(24,44,49),(29,40,35),(30,37,36),(31,38,33),(32,39,34),(45,67,63),(46,68,64),(47,65,61),(48,66,62)]])
Q8×He3 is a maximal subgroup of
He3⋊6Q16 He3⋊10SD16 He3⋊11SD16 He3⋊7Q16 (Q8×He3)⋊C2 He3⋊5D4⋊C2
55 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6J | 12A | ··· | 12F | 12G | ··· | 12AD |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 1 | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 6 | ··· | 6 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | - | ||||||
| image | C1 | C2 | C3 | C6 | Q8 | C3×Q8 | He3 | C2×He3 | Q8×He3 |
| kernel | Q8×He3 | C4×He3 | Q8×C32 | C3×C12 | He3 | C32 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 8 | 24 | 1 | 8 | 2 | 6 | 2 |
Matrix representation of Q8×He3 ►in GL5(𝔽13)
| 12 | 8 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 7 | 2 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 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 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [12,3,0,0,0,8,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,1,0,0,0,2,6,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[9,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,3,0] >;
Q8×He3 in GAP, Magma, Sage, TeX
Q_8\times {\rm He}_3 % in TeX
G:=Group("Q8xHe3"); // GroupNames label
G:=SmallGroup(216,80);
// by ID
G=gap.SmallGroup(216,80);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,457,223,519]);
// 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,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations