direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C4×C3.A4, C24.C18, C23⋊2C36, (C23×C4)⋊C9, C22⋊(C2×C36), C6.7(C4×A4), (C2×C12).2A4, (C23×C12).C3, C12.16(C2×A4), (C22×C4)⋊2C18, (C23×C6).4C6, (C22×C12).7C6, C6.12(C22×A4), C23.7(C2×C18), (C22×C6).7C12, C3.(C2×C4×A4), (C2×C6).23(C2×A4), (C2×C6).8(C2×C12), C22.7(C2×C3.A4), C2.1(C22×C3.A4), (C22×C6).35(C2×C6), (C22×C3.A4).2C2, (C2×C3.A4).6C22, SmallGroup(288,343)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C2×C4×C3.A4 |
Generators and relations for C2×C4×C3.A4
G = < a,b,c,d,e,f | a2=b4=c3=d2=e2=1, f3=c, ab=ba, 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: 276 in 116 conjugacy classes, 40 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C9, C12, C12, C2×C6, C2×C6, C22×C4, C22×C4, C24, C18, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C23×C4, C36, C3.A4, C2×C18, C22×C12, C22×C12, C23×C6, C2×C36, C2×C3.A4, C2×C3.A4, C23×C12, C4×C3.A4, C22×C3.A4, C2×C4×C3.A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C9, C12, A4, C2×C6, C18, C2×C12, C2×A4, C36, C3.A4, C2×C18, C4×A4, C22×A4, C2×C36, C2×C3.A4, C2×C4×A4, C4×C3.A4, C22×C3.A4, C2×C4×C3.A4
►On 72 points
Generators in S72
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 64)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 46)
(1 69 54 27)(2 70 46 19)(3 71 47 20)(4 72 48 21)(5 64 49 22)(6 65 50 23)(7 66 51 24)(8 67 52 25)(9 68 53 26)(10 61 34 37)(11 62 35 38)(12 63 36 39)(13 55 28 40)(14 56 29 41)(15 57 30 42)(16 58 31 43)(17 59 32 44)(18 60 33 45)
(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)
(1 54)(2 60)(3 37)(4 48)(5 63)(6 40)(7 51)(8 57)(9 43)(10 71)(11 35)(12 22)(13 65)(14 29)(15 25)(16 68)(17 32)(18 19)(20 34)(21 72)(23 28)(24 66)(26 31)(27 69)(30 67)(33 70)(36 64)(38 62)(39 49)(41 56)(42 52)(44 59)(45 46)(47 61)(50 55)(53 58)
(1 44)(2 46)(3 61)(4 38)(5 49)(6 55)(7 41)(8 52)(9 58)(10 20)(11 72)(12 36)(13 23)(14 66)(15 30)(16 26)(17 69)(18 33)(19 70)(21 35)(22 64)(24 29)(25 67)(27 32)(28 65)(31 68)(34 71)(37 47)(39 63)(40 50)(42 57)(43 53)(45 60)(48 62)(51 56)(54 59)
(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)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,64)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,46), (1,69,54,27)(2,70,46,19)(3,71,47,20)(4,72,48,21)(5,64,49,22)(6,65,50,23)(7,66,51,24)(8,67,52,25)(9,68,53,26)(10,61,34,37)(11,62,35,38)(12,63,36,39)(13,55,28,40)(14,56,29,41)(15,57,30,42)(16,58,31,43)(17,59,32,44)(18,60,33,45), (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), (1,54)(2,60)(3,37)(4,48)(5,63)(6,40)(7,51)(8,57)(9,43)(10,71)(11,35)(12,22)(13,65)(14,29)(15,25)(16,68)(17,32)(18,19)(20,34)(21,72)(23,28)(24,66)(26,31)(27,69)(30,67)(33,70)(36,64)(38,62)(39,49)(41,56)(42,52)(44,59)(45,46)(47,61)(50,55)(53,58), (1,44)(2,46)(3,61)(4,38)(5,49)(6,55)(7,41)(8,52)(9,58)(10,20)(11,72)(12,36)(13,23)(14,66)(15,30)(16,26)(17,69)(18,33)(19,70)(21,35)(22,64)(24,29)(25,67)(27,32)(28,65)(31,68)(34,71)(37,47)(39,63)(40,50)(42,57)(43,53)(45,60)(48,62)(51,56)(54,59), (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( (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,64)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,46), (1,69,54,27)(2,70,46,19)(3,71,47,20)(4,72,48,21)(5,64,49,22)(6,65,50,23)(7,66,51,24)(8,67,52,25)(9,68,53,26)(10,61,34,37)(11,62,35,38)(12,63,36,39)(13,55,28,40)(14,56,29,41)(15,57,30,42)(16,58,31,43)(17,59,32,44)(18,60,33,45), (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), (1,54)(2,60)(3,37)(4,48)(5,63)(6,40)(7,51)(8,57)(9,43)(10,71)(11,35)(12,22)(13,65)(14,29)(15,25)(16,68)(17,32)(18,19)(20,34)(21,72)(23,28)(24,66)(26,31)(27,69)(30,67)(33,70)(36,64)(38,62)(39,49)(41,56)(42,52)(44,59)(45,46)(47,61)(50,55)(53,58), (1,44)(2,46)(3,61)(4,38)(5,49)(6,55)(7,41)(8,52)(9,58)(10,20)(11,72)(12,36)(13,23)(14,66)(15,30)(16,26)(17,69)(18,33)(19,70)(21,35)(22,64)(24,29)(25,67)(27,32)(28,65)(31,68)(34,71)(37,47)(39,63)(40,50)(42,57)(43,53)(45,60)(48,62)(51,56)(54,59), (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([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,55),(7,56),(8,57),(9,58),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,19),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,64),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,46)], [(1,69,54,27),(2,70,46,19),(3,71,47,20),(4,72,48,21),(5,64,49,22),(6,65,50,23),(7,66,51,24),(8,67,52,25),(9,68,53,26),(10,61,34,37),(11,62,35,38),(12,63,36,39),(13,55,28,40),(14,56,29,41),(15,57,30,42),(16,58,31,43),(17,59,32,44),(18,60,33,45)], [(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)], [(1,54),(2,60),(3,37),(4,48),(5,63),(6,40),(7,51),(8,57),(9,43),(10,71),(11,35),(12,22),(13,65),(14,29),(15,25),(16,68),(17,32),(18,19),(20,34),(21,72),(23,28),(24,66),(26,31),(27,69),(30,67),(33,70),(36,64),(38,62),(39,49),(41,56),(42,52),(44,59),(45,46),(47,61),(50,55),(53,58)], [(1,44),(2,46),(3,61),(4,38),(5,49),(6,55),(7,41),(8,52),(9,58),(10,20),(11,72),(12,36),(13,23),(14,66),(15,30),(16,26),(17,69),(18,33),(19,70),(21,35),(22,64),(24,29),(25,67),(27,32),(28,65),(31,68),(34,71),(37,47),(39,63),(40,50),(42,57),(43,53),(45,60),(48,62),(51,56),(54,59)], [(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)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6N | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12P | 18A | ··· | 18R | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | A4 | C2×A4 | C2×A4 | C3.A4 | C4×A4 | C2×C3.A4 | C2×C3.A4 | C4×C3.A4 |
| kernel | C2×C4×C3.A4 | C4×C3.A4 | C22×C3.A4 | C23×C12 | C2×C3.A4 | C22×C12 | C23×C6 | C23×C4 | C22×C6 | C22×C4 | C24 | C23 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 24 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C2×C4×C3.A4 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 26 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 4 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 25 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 36 |
| 9 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 17 | 28 | 3 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[26,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,36,0,0,0,0,36,0,0,4,0,1],[1,0,0,0,0,36,0,0,0,25,1,0,0,0,0,36],[9,0,0,0,0,34,0,17,0,0,0,28,0,0,1,3] >;
C2×C4×C3.A4 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_3.A_4
% in TeX
G:=Group("C2xC4xC3.A4"); // GroupNames label
G:=SmallGroup(288,343);
// by ID
G=gap.SmallGroup(288,343);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,92,142,1531,2666]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=c^3=d^2=e^2=1,f^3=c,a*b=b*a,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