direct product, metabelian, nilpotent (class 3), monomial
Aliases: C32×C4≀C2, C122⋊17C2, C62.97D4, (C4×C12)⋊16C6, D4⋊2(C3×C12), (C3×D4)⋊5C12, C4.3(C6×C12), C42⋊6(C3×C6), Q8⋊3(C3×C12), (C3×Q8)⋊7C12, C12.89(C3×D4), C12.38(C2×C12), (C3×C12).184D4, (D4×C32)⋊11C4, M4(2)⋊4(C3×C6), (C2×C4).16C62, (Q8×C32)⋊11C4, C4.17(D4×C32), (C3×M4(2))⋊10C6, C22.3(D4×C32), (C6×C12).363C22, (C32×M4(2))⋊16C2, C4○D4.3(C3×C6), (C2×C6).32(C3×D4), (C3×C4○D4).16C6, C6.35(C3×C22⋊C4), (C2×C12).150(C2×C6), (C3×C12).120(C2×C4), (C32×C4○D4).7C2, C2.8(C32×C22⋊C4), (C3×C6).84(C22⋊C4), SmallGroup(288,322)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4≀C2
G = < a,b,c,d,e | a3=b3=c4=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c-1d >
Subgroups: 204 in 132 conjugacy classes, 72 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C32, C12, C12, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×C6, C3×C6, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4≀C2, C3×C12, C3×C12, C62, C62, C4×C12, C3×M4(2), C3×C4○D4, C3×C24, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, C3×C4≀C2, C122, C32×M4(2), C32×C4○D4, C32×C4≀C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C32, C12, C2×C6, C22⋊C4, C3×C6, C2×C12, C3×D4, C4≀C2, C3×C12, C62, C3×C22⋊C4, C6×C12, D4×C32, C3×C4≀C2, C32×C22⋊C4, C32×C4≀C2
►On 72 points
(1 35 19)(2 36 20)(3 33 17)(4 34 18)(5 25 21)(6 26 22)(7 27 23)(8 28 24)(9 29 13)(10 30 14)(11 31 15)(12 32 16)(37 69 53)(38 70 54)(39 71 55)(40 72 56)(41 61 57)(42 62 58)(43 63 59)(44 64 60)(45 65 49)(46 66 50)(47 67 51)(48 68 52)
(1 11 7)(2 12 8)(3 9 5)(4 10 6)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)
(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 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 44)(8 43)(9 46)(10 45)(11 48)(12 47)(13 50)(14 49)(15 52)(16 51)(17 54)(18 53)(19 56)(20 55)(21 58)(22 57)(23 60)(24 59)(25 62)(26 61)(27 64)(28 63)(29 66)(30 65)(31 68)(32 67)(33 70)(34 69)(35 72)(36 71)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 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)| (1,35,19)(2,36,20)(3,33,17)(4,34,18)(5,25,21)(6,26,22)(7,27,23)(8,28,24)(9,29,13)(10,30,14)(11,31,15)(12,32,16)(37,69,53)(38,70,54)(39,71,55)(40,72,56)(41,61,57)(42,62,58)(43,63,59)(44,64,60)(45,65,49)(46,66,50)(47,67,51)(48,68,52), (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68), (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), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,44)(8,43)(9,46)(10,45)(11,48)(12,47)(13,50)(14,49)(15,52)(16,51)(17,54)(18,53)(19,56)(20,55)(21,58)(22,57)(23,60)(24,59)(25,62)(26,61)(27,64)(28,63)(29,66)(30,65)(31,68)(32,67)(33,70)(34,69)(35,72)(36,71), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(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( (1,35,19)(2,36,20)(3,33,17)(4,34,18)(5,25,21)(6,26,22)(7,27,23)(8,28,24)(9,29,13)(10,30,14)(11,31,15)(12,32,16)(37,69,53)(38,70,54)(39,71,55)(40,72,56)(41,61,57)(42,62,58)(43,63,59)(44,64,60)(45,65,49)(46,66,50)(47,67,51)(48,68,52), (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68), (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), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,44)(8,43)(9,46)(10,45)(11,48)(12,47)(13,50)(14,49)(15,52)(16,51)(17,54)(18,53)(19,56)(20,55)(21,58)(22,57)(23,60)(24,59)(25,62)(26,61)(27,64)(28,63)(29,66)(30,65)(31,68)(32,67)(33,70)(34,69)(35,72)(36,71), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(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([[(1,35,19),(2,36,20),(3,33,17),(4,34,18),(5,25,21),(6,26,22),(7,27,23),(8,28,24),(9,29,13),(10,30,14),(11,31,15),(12,32,16),(37,69,53),(38,70,54),(39,71,55),(40,72,56),(41,61,57),(42,62,58),(43,63,59),(44,64,60),(45,65,49),(46,66,50),(47,67,51),(48,68,52)], [(1,11,7),(2,12,8),(3,9,5),(4,10,6),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,42),(6,41),(7,44),(8,43),(9,46),(10,45),(11,48),(12,47),(13,50),(14,49),(15,52),(16,51),(17,54),(18,53),(19,56),(20,55),(21,58),(22,57),(23,60),(24,59),(25,62),(26,61),(27,64),(28,63),(29,66),(30,65),(31,68),(32,67),(33,70),(34,69),(35,72),(36,71)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(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)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | ··· | 4G | 4H | 6A | ··· | 6H | 6I | ··· | 6P | 6Q | ··· | 6X | 8A | 8B | 12A | ··· | 12P | 12Q | ··· | 12BD | 12BE | ··· | 12BL | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | C3×D4 | C3×D4 | C4≀C2 | C3×C4≀C2 |
| kernel | C32×C4≀C2 | C122 | C32×M4(2) | C32×C4○D4 | C3×C4≀C2 | D4×C32 | Q8×C32 | C4×C12 | C3×M4(2) | C3×C4○D4 | C3×D4 | C3×Q8 | C3×C12 | C62 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 8 | 8 | 8 | 16 | 16 | 1 | 1 | 8 | 8 | 4 | 32 |
Matrix representation of C32×C4≀C2 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 27 | 21 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 0 | 21 | 46 |
| 0 | 19 | 52 |
| 72 | 0 | 0 |
| 0 | 72 | 2 |
| 0 | 0 | 46 |
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[8,0,0,0,8,0,0,0,8],[1,0,0,0,27,0,0,21,46],[1,0,0,0,21,19,0,46,52],[72,0,0,0,72,0,0,2,46] >;
C32×C4≀C2 in GAP, Magma, Sage, TeX
C_3^2\times C_4\wr C_2
% in TeX
G:=Group("C3^2xC4wrC2"); // GroupNames label
G:=SmallGroup(288,322);
// by ID
G=gap.SmallGroup(288,322);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,6304,3161,172,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^-1*d>;
// generators/relations