direct product, metabelian, nilpotent (class 3), monomial
Aliases: C32×C23⋊C4, C62.96D4, C23.1C62, (C6×C12)⋊8C4, C23⋊(C3×C12), (C2×C12)⋊2C12, (C2×C62)⋊1C4, (C22×C6)⋊2C12, (C6×D4).18C6, C62.88(C2×C4), C22.2(C6×C12), (C2×C62).1C22, C22.2(D4×C32), (C2×C4)⋊(C3×C12), (D4×C3×C6).13C2, C22⋊C4⋊1(C3×C6), (C3×C22⋊C4)⋊2C6, (C2×D4).1(C3×C6), (C2×C6).31(C3×D4), (C2×C6).29(C2×C12), C6.30(C3×C22⋊C4), (C32×C22⋊C4)⋊3C2, (C22×C6).10(C2×C6), C2.3(C32×C22⋊C4), (C3×C6).79(C22⋊C4), SmallGroup(288,317)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C23⋊C4
G = < a,b,c,d,e,f | a3=b3=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >
Subgroups: 300 in 156 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C32, C12, C2×C6, C2×C6, C22⋊C4, C2×D4, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C22×C6, C23⋊C4, C3×C12, C62, C62, C62, C3×C22⋊C4, C6×D4, C6×C12, C6×C12, D4×C32, C2×C62, C3×C23⋊C4, C32×C22⋊C4, D4×C3×C6, C32×C23⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C32, C12, C2×C6, C22⋊C4, C3×C6, C2×C12, C3×D4, C23⋊C4, C3×C12, C62, C3×C22⋊C4, C6×C12, D4×C32, C3×C23⋊C4, C32×C22⋊C4, C32×C23⋊C4
►On 72 points
Generators in S72
(1 18 8)(2 17 7)(3 23 16)(4 24 15)(5 14 10)(6 13 9)(11 36 20)(12 35 19)(21 27 34)(22 28 33)(25 31 29)(26 32 30)(37 62 49)(38 63 50)(39 64 51)(40 61 52)(41 54 70)(42 55 71)(43 56 72)(44 53 69)(45 66 58)(46 67 59)(47 68 60)(48 65 57)
(1 4 5)(2 3 6)(7 16 9)(8 15 10)(11 26 22)(12 25 21)(13 17 23)(14 18 24)(19 29 34)(20 30 33)(27 35 31)(28 36 32)(37 45 54)(38 46 55)(39 47 56)(40 48 53)(41 49 58)(42 50 59)(43 51 60)(44 52 57)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(1 47)(2 48)(3 53)(4 56)(5 39)(6 40)(7 57)(8 60)(9 52)(10 51)(11 45)(12 46)(13 61)(14 64)(15 43)(16 44)(17 65)(18 68)(19 59)(20 58)(21 38)(22 37)(23 69)(24 72)(25 55)(26 54)(27 63)(28 62)(29 42)(30 41)(31 71)(32 70)(33 49)(34 50)(35 67)(36 66)
(1 11)(4 26)(5 22)(8 20)(10 33)(14 28)(15 30)(18 36)(24 32)(37 39)(41 43)(45 47)(49 51)(54 56)(58 60)(62 64)(66 68)(70 72)
(1 11)(2 12)(3 25)(4 26)(5 22)(6 21)(7 19)(8 20)(9 34)(10 33)(13 27)(14 28)(15 30)(16 29)(17 35)(18 36)(23 31)(24 32)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 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)
G:=sub<Sym(72)| (1,18,8)(2,17,7)(3,23,16)(4,24,15)(5,14,10)(6,13,9)(11,36,20)(12,35,19)(21,27,34)(22,28,33)(25,31,29)(26,32,30)(37,62,49)(38,63,50)(39,64,51)(40,61,52)(41,54,70)(42,55,71)(43,56,72)(44,53,69)(45,66,58)(46,67,59)(47,68,60)(48,65,57), (1,4,5)(2,3,6)(7,16,9)(8,15,10)(11,26,22)(12,25,21)(13,17,23)(14,18,24)(19,29,34)(20,30,33)(27,35,31)(28,36,32)(37,45,54)(38,46,55)(39,47,56)(40,48,53)(41,49,58)(42,50,59)(43,51,60)(44,52,57)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,47)(2,48)(3,53)(4,56)(5,39)(6,40)(7,57)(8,60)(9,52)(10,51)(11,45)(12,46)(13,61)(14,64)(15,43)(16,44)(17,65)(18,68)(19,59)(20,58)(21,38)(22,37)(23,69)(24,72)(25,55)(26,54)(27,63)(28,62)(29,42)(30,41)(31,71)(32,70)(33,49)(34,50)(35,67)(36,66), (1,11)(4,26)(5,22)(8,20)(10,33)(14,28)(15,30)(18,36)(24,32)(37,39)(41,43)(45,47)(49,51)(54,56)(58,60)(62,64)(66,68)(70,72), (1,11)(2,12)(3,25)(4,26)(5,22)(6,21)(7,19)(8,20)(9,34)(10,33)(13,27)(14,28)(15,30)(16,29)(17,35)(18,36)(23,31)(24,32)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72), (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)>;
G:=Group( (1,18,8)(2,17,7)(3,23,16)(4,24,15)(5,14,10)(6,13,9)(11,36,20)(12,35,19)(21,27,34)(22,28,33)(25,31,29)(26,32,30)(37,62,49)(38,63,50)(39,64,51)(40,61,52)(41,54,70)(42,55,71)(43,56,72)(44,53,69)(45,66,58)(46,67,59)(47,68,60)(48,65,57), (1,4,5)(2,3,6)(7,16,9)(8,15,10)(11,26,22)(12,25,21)(13,17,23)(14,18,24)(19,29,34)(20,30,33)(27,35,31)(28,36,32)(37,45,54)(38,46,55)(39,47,56)(40,48,53)(41,49,58)(42,50,59)(43,51,60)(44,52,57)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,47)(2,48)(3,53)(4,56)(5,39)(6,40)(7,57)(8,60)(9,52)(10,51)(11,45)(12,46)(13,61)(14,64)(15,43)(16,44)(17,65)(18,68)(19,59)(20,58)(21,38)(22,37)(23,69)(24,72)(25,55)(26,54)(27,63)(28,62)(29,42)(30,41)(31,71)(32,70)(33,49)(34,50)(35,67)(36,66), (1,11)(4,26)(5,22)(8,20)(10,33)(14,28)(15,30)(18,36)(24,32)(37,39)(41,43)(45,47)(49,51)(54,56)(58,60)(62,64)(66,68)(70,72), (1,11)(2,12)(3,25)(4,26)(5,22)(6,21)(7,19)(8,20)(9,34)(10,33)(13,27)(14,28)(15,30)(16,29)(17,35)(18,36)(23,31)(24,32)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72), (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) );
G=PermutationGroup([[(1,18,8),(2,17,7),(3,23,16),(4,24,15),(5,14,10),(6,13,9),(11,36,20),(12,35,19),(21,27,34),(22,28,33),(25,31,29),(26,32,30),(37,62,49),(38,63,50),(39,64,51),(40,61,52),(41,54,70),(42,55,71),(43,56,72),(44,53,69),(45,66,58),(46,67,59),(47,68,60),(48,65,57)], [(1,4,5),(2,3,6),(7,16,9),(8,15,10),(11,26,22),(12,25,21),(13,17,23),(14,18,24),(19,29,34),(20,30,33),(27,35,31),(28,36,32),(37,45,54),(38,46,55),(39,47,56),(40,48,53),(41,49,58),(42,50,59),(43,51,60),(44,52,57),(61,65,69),(62,66,70),(63,67,71),(64,68,72)], [(1,47),(2,48),(3,53),(4,56),(5,39),(6,40),(7,57),(8,60),(9,52),(10,51),(11,45),(12,46),(13,61),(14,64),(15,43),(16,44),(17,65),(18,68),(19,59),(20,58),(21,38),(22,37),(23,69),(24,72),(25,55),(26,54),(27,63),(28,62),(29,42),(30,41),(31,71),(32,70),(33,49),(34,50),(35,67),(36,66)], [(1,11),(4,26),(5,22),(8,20),(10,33),(14,28),(15,30),(18,36),(24,32),(37,39),(41,43),(45,47),(49,51),(54,56),(58,60),(62,64),(66,68),(70,72)], [(1,11),(2,12),(3,25),(4,26),(5,22),(6,21),(7,19),(8,20),(9,34),(10,33),(13,27),(14,28),(15,30),(16,29),(17,35),(18,36),(23,31),(24,32),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72)], [(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)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | ··· | 4E | 6A | ··· | 6H | 6I | ··· | 6AF | 6AG | ··· | 6AN | 12A | ··· | 12AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | C3×D4 | C23⋊C4 | C3×C23⋊C4 |
| kernel | C32×C23⋊C4 | C32×C22⋊C4 | D4×C3×C6 | C3×C23⋊C4 | C6×C12 | C2×C62 | C3×C22⋊C4 | C6×D4 | C2×C12 | C22×C6 | C62 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 16 | 16 | 2 | 16 | 1 | 8 |
Matrix representation of C32×C23⋊C4 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0] >;
C32×C23⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times C_2^3\rtimes C_4
% in TeX
G:=Group("C3^2xC2^3:C4"); // GroupNames label
G:=SmallGroup(288,317);
// by ID
G=gap.SmallGroup(288,317);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,6304,4548]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^2=e^2=f^4=1,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,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations