direct product, metabelian, nilpotent (class 2), monomial
Aliases: C22⋊C4×3- 1+2, C62.1C12, (C2×C36)⋊7C6, (C2×C18)⋊5C12, (C6×C12).2C6, C6.23(C6×C12), C18.12(C3×D4), (C2×C62).2C6, C18.11(C2×C12), C62.34(C2×C6), (C22×C18).5C6, (C2×C6).28C62, C6.14(D4×C32), C2.1(D4×3- 1+2), C22⋊3(C4×3- 1+2), (C2×3- 1+2).12D4, (C22×3- 1+2)⋊3C4, C23.3(C2×3- 1+2), (C23×3- 1+2).3C2, C22.3(C22×3- 1+2), (C22×3- 1+2).14C22, (C9×C22⋊C4)⋊C3, C9⋊3(C3×C22⋊C4), (C2×C6).8(C3×C12), (C2×C12).3(C3×C6), (C3×C6).29(C3×D4), C32.(C3×C22⋊C4), (C2×C18).16(C2×C6), (C3×C6).29(C2×C12), (C32×C22⋊C4).C3, (C22×C6).15(C3×C6), C3.3(C32×C22⋊C4), (C2×C4×3- 1+2)⋊7C2, C2.3(C2×C4×3- 1+2), (C3×C22⋊C4).3C32, (C2×C4)⋊1(C2×3- 1+2), (C2×3- 1+2).11(C2×C4), SmallGroup(432,205)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C6 — C2×C6 — C62 — C22×3- 1+2 — C2×C4×3- 1+2 — C22⋊C4×3- 1+2 |
Generators and relations for C22⋊C4×3- 1+2
G = < a,b,c,d,e | a2=b2=c4=d9=e3=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >
Subgroups: 230 in 136 conjugacy classes, 77 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C32, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×C6, C3×C6, C3×C6, C2×C12, C2×C12, C22×C6, C22×C6, 3- 1+2, C36, C2×C18, C2×C18, C3×C12, C62, C62, C62, C3×C22⋊C4, C3×C22⋊C4, C2×3- 1+2, C2×3- 1+2, C2×3- 1+2, C2×C36, C22×C18, C6×C12, C2×C62, C4×3- 1+2, C22×3- 1+2, C22×3- 1+2, C22×3- 1+2, C9×C22⋊C4, C32×C22⋊C4, C2×C4×3- 1+2, C23×3- 1+2, C22⋊C4×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C32, C12, C2×C6, C22⋊C4, C3×C6, C2×C12, C3×D4, 3- 1+2, C3×C12, C62, C3×C22⋊C4, C2×3- 1+2, C6×C12, D4×C32, C4×3- 1+2, C22×3- 1+2, C32×C22⋊C4, C2×C4×3- 1+2, D4×3- 1+2, C22⋊C4×3- 1+2
►On 72 points
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 69)(56 70)(57 71)(58 72)(59 64)(60 65)(61 66)(62 67)(63 68)
(1 43)(2 44)(3 45)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 68)(20 69)(21 70)(22 71)(23 72)(24 64)(25 65)(26 66)(27 67)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)
(1 70 34 11)(2 71 35 12)(3 72 36 13)(4 64 28 14)(5 65 29 15)(6 66 30 16)(7 67 31 17)(8 68 32 18)(9 69 33 10)(19 50 63 41)(20 51 55 42)(21 52 56 43)(22 53 57 44)(23 54 58 45)(24 46 59 37)(25 47 60 38)(26 48 61 39)(27 49 62 40)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(55 58 61)(57 63 60)(65 71 68)(66 69 72)
G:=sub<Sym(72)| (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,69)(56,70)(57,71)(58,72)(59,64)(60,65)(61,66)(62,67)(63,68), (1,43)(2,44)(3,45)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,68)(20,69)(21,70)(22,71)(23,72)(24,64)(25,65)(26,66)(27,67)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (1,70,34,11)(2,71,35,12)(3,72,36,13)(4,64,28,14)(5,65,29,15)(6,66,30,16)(7,67,31,17)(8,68,32,18)(9,69,33,10)(19,50,63,41)(20,51,55,42)(21,52,56,43)(22,53,57,44)(23,54,58,45)(24,46,59,37)(25,47,60,38)(26,48,61,39)(27,49,62,40), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(65,71,68)(66,69,72)>;
G:=Group( (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,69)(56,70)(57,71)(58,72)(59,64)(60,65)(61,66)(62,67)(63,68), (1,43)(2,44)(3,45)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,68)(20,69)(21,70)(22,71)(23,72)(24,64)(25,65)(26,66)(27,67)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (1,70,34,11)(2,71,35,12)(3,72,36,13)(4,64,28,14)(5,65,29,15)(6,66,30,16)(7,67,31,17)(8,68,32,18)(9,69,33,10)(19,50,63,41)(20,51,55,42)(21,52,56,43)(22,53,57,44)(23,54,58,45)(24,46,59,37)(25,47,60,38)(26,48,61,39)(27,49,62,40), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(65,71,68)(66,69,72) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,19),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,69),(56,70),(57,71),(58,72),(59,64),(60,65),(61,66),(62,67),(63,68)], [(1,43),(2,44),(3,45),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,68),(20,69),(21,70),(22,71),(23,72),(24,64),(25,65),(26,66),(27,67),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54)], [(1,70,34,11),(2,71,35,12),(3,72,36,13),(4,64,28,14),(5,65,29,15),(6,66,30,16),(7,67,31,17),(8,68,32,18),(9,69,33,10),(19,50,63,41),(20,51,55,42),(21,52,56,43),(22,53,57,44),(23,54,58,45),(24,46,59,37),(25,47,60,38),(26,48,61,39),(27,49,62,40)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(38,44,41),(39,42,45),(47,53,50),(48,51,54),(55,58,61),(57,63,60),(65,71,68),(66,69,72)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6P | 6Q | 6R | 6S | 6T | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12P | 18A | ··· | 18R | 18S | ··· | 18AD | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 3 | ··· | 3 | 2 | ··· | 2 | 6 | ··· | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | D4 | C3×D4 | C3×D4 | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C4×3- 1+2 | D4×3- 1+2 |
| kernel | C22⋊C4×3- 1+2 | C2×C4×3- 1+2 | C23×3- 1+2 | C9×C22⋊C4 | C32×C22⋊C4 | C22×3- 1+2 | C2×C36 | C22×C18 | C6×C12 | C2×C62 | C2×C18 | C62 | C2×3- 1+2 | C18 | C3×C6 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 6 | 2 | 4 | 12 | 6 | 4 | 2 | 24 | 8 | 2 | 12 | 4 | 2 | 4 | 2 | 8 | 4 |
Matrix representation of C22⋊C4×3- 1+2 ►in GL5(𝔽37)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 6 | 0 | 0 | 0 |
| 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 31 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 27 |
| 0 | 0 | 0 | 0 | 26 |
| 0 | 0 | 9 | 11 | 27 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 | 10 |
| 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 10 |
G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,31,0,0,0,6,0,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[26,0,0,0,0,0,26,0,0,0,0,0,10,0,9,0,0,0,0,11,0,0,27,26,27],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,27,26,0,0,0,10,0,10] >;
C22⋊C4×3- 1+2 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times 3_-^{1+2} % in TeX
G:=Group("C2^2:C4xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,205);
// by ID
G=gap.SmallGroup(432,205);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,772,1109]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^9=e^3=1,c*a*c^-1=a*b=b*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,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations