direct product, metabelian, soluble, monomial
Aliases: Q8×C3.A4, C3.(Q8×A4), C22⋊(Q8×C9), C12.8(C2×A4), (C22×C4).C18, (C3×Q8).3A4, (C22×Q8)⋊2C9, (C22×C12).4C6, C6.15(C22×A4), C23.9(C2×C18), (Q8×C2×C6).2C3, C4.1(C2×C3.A4), (C2×C6).2(C3×Q8), (C4×C3.A4).3C2, C2.4(C22×C3.A4), (C22×C6).37(C2×C6), (C2×C3.A4).8C22, SmallGroup(288,346)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C3.A4
G = < a,b,c,d,e,f | a4=c3=d2=e2=1, b2=a2, f3=c, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
Subgroups: 186 in 80 conjugacy classes, 30 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, Q8, Q8, C23, C9, C12, C12, C2×C6, C2×C6, C22×C4, C2×Q8, C18, C2×C12, C3×Q8, C3×Q8, C22×C6, C22×Q8, C36, C3.A4, C22×C12, C6×Q8, Q8×C9, C2×C3.A4, Q8×C2×C6, C4×C3.A4, Q8×C3.A4
Quotients: C1, C2, C3, C22, C6, Q8, C9, A4, C2×C6, C18, C3×Q8, C2×A4, C3.A4, C2×C18, C22×A4, Q8×C9, C2×C3.A4, Q8×A4, C22×C3.A4, Q8×C3.A4
►On 72 points
Generators in S72
(1 43 25 31)(2 44 26 32)(3 45 27 33)(4 37 19 34)(5 38 20 35)(6 39 21 36)(7 40 22 28)(8 41 23 29)(9 42 24 30)(10 52 70 55)(11 53 71 56)(12 54 72 57)(13 46 64 58)(14 47 65 59)(15 48 66 60)(16 49 67 61)(17 50 68 62)(18 51 69 63)
(1 61 25 49)(2 62 26 50)(3 63 27 51)(4 55 19 52)(5 56 20 53)(6 57 21 54)(7 58 22 46)(8 59 23 47)(9 60 24 48)(10 37 70 34)(11 38 71 35)(12 39 72 36)(13 40 64 28)(14 41 65 29)(15 42 66 30)(16 43 67 31)(17 44 68 32)(18 45 69 33)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(2 26)(3 27)(5 20)(6 21)(8 23)(9 24)(11 71)(12 72)(14 65)(15 66)(17 68)(18 69)(29 41)(30 42)(32 44)(33 45)(35 38)(36 39)(47 59)(48 60)(50 62)(51 63)(53 56)(54 57)
(1 25)(3 27)(4 19)(6 21)(7 22)(9 24)(10 70)(12 72)(13 64)(15 66)(16 67)(18 69)(28 40)(30 42)(31 43)(33 45)(34 37)(36 39)(46 58)(48 60)(49 61)(51 63)(52 55)(54 57)
(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)
G:=sub<Sym(72)| (1,43,25,31)(2,44,26,32)(3,45,27,33)(4,37,19,34)(5,38,20,35)(6,39,21,36)(7,40,22,28)(8,41,23,29)(9,42,24,30)(10,52,70,55)(11,53,71,56)(12,54,72,57)(13,46,64,58)(14,47,65,59)(15,48,66,60)(16,49,67,61)(17,50,68,62)(18,51,69,63), (1,61,25,49)(2,62,26,50)(3,63,27,51)(4,55,19,52)(5,56,20,53)(6,57,21,54)(7,58,22,46)(8,59,23,47)(9,60,24,48)(10,37,70,34)(11,38,71,35)(12,39,72,36)(13,40,64,28)(14,41,65,29)(15,42,66,30)(16,43,67,31)(17,44,68,32)(18,45,69,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (2,26)(3,27)(5,20)(6,21)(8,23)(9,24)(11,71)(12,72)(14,65)(15,66)(17,68)(18,69)(29,41)(30,42)(32,44)(33,45)(35,38)(36,39)(47,59)(48,60)(50,62)(51,63)(53,56)(54,57), (1,25)(3,27)(4,19)(6,21)(7,22)(9,24)(10,70)(12,72)(13,64)(15,66)(16,67)(18,69)(28,40)(30,42)(31,43)(33,45)(34,37)(36,39)(46,58)(48,60)(49,61)(51,63)(52,55)(54,57), (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)>;
G:=Group( (1,43,25,31)(2,44,26,32)(3,45,27,33)(4,37,19,34)(5,38,20,35)(6,39,21,36)(7,40,22,28)(8,41,23,29)(9,42,24,30)(10,52,70,55)(11,53,71,56)(12,54,72,57)(13,46,64,58)(14,47,65,59)(15,48,66,60)(16,49,67,61)(17,50,68,62)(18,51,69,63), (1,61,25,49)(2,62,26,50)(3,63,27,51)(4,55,19,52)(5,56,20,53)(6,57,21,54)(7,58,22,46)(8,59,23,47)(9,60,24,48)(10,37,70,34)(11,38,71,35)(12,39,72,36)(13,40,64,28)(14,41,65,29)(15,42,66,30)(16,43,67,31)(17,44,68,32)(18,45,69,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (2,26)(3,27)(5,20)(6,21)(8,23)(9,24)(11,71)(12,72)(14,65)(15,66)(17,68)(18,69)(29,41)(30,42)(32,44)(33,45)(35,38)(36,39)(47,59)(48,60)(50,62)(51,63)(53,56)(54,57), (1,25)(3,27)(4,19)(6,21)(7,22)(9,24)(10,70)(12,72)(13,64)(15,66)(16,67)(18,69)(28,40)(30,42)(31,43)(33,45)(34,37)(36,39)(46,58)(48,60)(49,61)(51,63)(52,55)(54,57), (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) );
G=PermutationGroup([[(1,43,25,31),(2,44,26,32),(3,45,27,33),(4,37,19,34),(5,38,20,35),(6,39,21,36),(7,40,22,28),(8,41,23,29),(9,42,24,30),(10,52,70,55),(11,53,71,56),(12,54,72,57),(13,46,64,58),(14,47,65,59),(15,48,66,60),(16,49,67,61),(17,50,68,62),(18,51,69,63)], [(1,61,25,49),(2,62,26,50),(3,63,27,51),(4,55,19,52),(5,56,20,53),(6,57,21,54),(7,58,22,46),(8,59,23,47),(9,60,24,48),(10,37,70,34),(11,38,71,35),(12,39,72,36),(13,40,64,28),(14,41,65,29),(15,42,66,30),(16,43,67,31),(17,44,68,32),(18,45,69,33)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72)], [(2,26),(3,27),(5,20),(6,21),(8,23),(9,24),(11,71),(12,72),(14,65),(15,66),(17,68),(18,69),(29,41),(30,42),(32,44),(33,45),(35,38),(36,39),(47,59),(48,60),(50,62),(51,63),(53,56),(54,57)], [(1,25),(3,27),(4,19),(6,21),(7,22),(9,24),(10,70),(12,72),(13,64),(15,66),(16,67),(18,69),(28,40),(30,42),(31,43),(33,45),(34,37),(36,39),(46,58),(48,60),(49,61),(51,63),(52,55),(54,57)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 12A | ··· | 12F | 12G | ··· | 12L | 18A | ··· | 18F | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | ··· | 8 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | Q8 | C3×Q8 | Q8×C9 | A4 | C2×A4 | C3.A4 | C2×C3.A4 | Q8×A4 | Q8×C3.A4 |
| kernel | Q8×C3.A4 | C4×C3.A4 | Q8×C2×C6 | C22×C12 | C22×Q8 | C22×C4 | C3.A4 | C2×C6 | C22 | C3×Q8 | C12 | Q8 | C4 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 1 | 2 | 6 | 1 | 3 | 2 | 6 | 1 | 2 |
Matrix representation of Q8×C3.A4 ►in GL5(𝔽37)
| 34 | 8 | 0 | 0 | 0 |
| 8 | 3 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 0 | 36 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 26 | 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 | 36 | 36 | 36 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 36 | 36 |
| 0 | 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(37))| [34,8,0,0,0,8,3,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[0,1,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;
Q8×C3.A4 in GAP, Magma, Sage, TeX
Q_8\times C_3.A_4
% in TeX
G:=Group("Q8xC3.A4"); // GroupNames label
G:=SmallGroup(288,346);
// by ID
G=gap.SmallGroup(288,346);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,84,197,92,142,1531,2666]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^3=d^2=e^2=1,b^2=a^2,f^3=c,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations