direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C22⋊C4, C62⋊3C4, C22.2C62, C62.36C22, (C2×C6)⋊3C12, (C2×C12)⋊2C6, (C6×C12)⋊3C2, C2.1(C6×C12), C6.18(C3×D4), (C3×C6).39D4, C6.13(C2×C12), C22⋊2(C3×C12), (C22×C6).7C6, C23.2(C3×C6), (C2×C62).1C2, C2.1(D4×C32), (C2×C4)⋊1(C3×C6), (C2×C6).19(C2×C6), (C3×C6).35(C2×C4), SmallGroup(144,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C22⋊C4
G = < a,b,c,d,e | a3=b3=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 138 in 102 conjugacy classes, 66 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C23, C32, C12, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×C12, C22×C6, C3×C12, C62, C62, C62, C3×C22⋊C4, C6×C12, C2×C62, C32×C22⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C32, C12, C2×C6, C22⋊C4, C3×C6, C2×C12, C3×D4, C3×C12, C62, C3×C22⋊C4, C6×C12, D4×C32, C32×C22⋊C4
►On 72 points
Generators in S72
(1 6 51)(2 7 52)(3 8 49)(4 5 50)(9 53 17)(10 54 18)(11 55 19)(12 56 20)(13 45 41)(14 46 42)(15 47 43)(16 48 44)(21 39 29)(22 40 30)(23 37 31)(24 38 32)(25 69 61)(26 70 62)(27 71 63)(28 72 64)(33 65 57)(34 66 58)(35 67 59)(36 68 60)
(1 47 11)(2 48 12)(3 45 9)(4 46 10)(5 42 54)(6 43 55)(7 44 56)(8 41 53)(13 17 49)(14 18 50)(15 19 51)(16 20 52)(21 25 57)(22 26 58)(23 27 59)(24 28 60)(29 61 65)(30 62 66)(31 63 67)(32 64 68)(33 39 69)(34 40 70)(35 37 71)(36 38 72)
(1 3)(2 22)(4 24)(5 38)(6 8)(7 40)(9 11)(10 60)(12 58)(13 15)(14 64)(16 62)(17 19)(18 68)(20 66)(21 23)(25 27)(26 48)(28 46)(29 31)(30 52)(32 50)(33 35)(34 56)(36 54)(37 39)(41 43)(42 72)(44 70)(45 47)(49 51)(53 55)(57 59)(61 63)(65 67)(69 71)
(1 23)(2 24)(3 21)(4 22)(5 40)(6 37)(7 38)(8 39)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(41 69)(42 70)(43 71)(44 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)| (1,6,51)(2,7,52)(3,8,49)(4,5,50)(9,53,17)(10,54,18)(11,55,19)(12,56,20)(13,45,41)(14,46,42)(15,47,43)(16,48,44)(21,39,29)(22,40,30)(23,37,31)(24,38,32)(25,69,61)(26,70,62)(27,71,63)(28,72,64)(33,65,57)(34,66,58)(35,67,59)(36,68,60), (1,47,11)(2,48,12)(3,45,9)(4,46,10)(5,42,54)(6,43,55)(7,44,56)(8,41,53)(13,17,49)(14,18,50)(15,19,51)(16,20,52)(21,25,57)(22,26,58)(23,27,59)(24,28,60)(29,61,65)(30,62,66)(31,63,67)(32,64,68)(33,39,69)(34,40,70)(35,37,71)(36,38,72), (1,3)(2,22)(4,24)(5,38)(6,8)(7,40)(9,11)(10,60)(12,58)(13,15)(14,64)(16,62)(17,19)(18,68)(20,66)(21,23)(25,27)(26,48)(28,46)(29,31)(30,52)(32,50)(33,35)(34,56)(36,54)(37,39)(41,43)(42,72)(44,70)(45,47)(49,51)(53,55)(57,59)(61,63)(65,67)(69,71), (1,23)(2,24)(3,21)(4,22)(5,40)(6,37)(7,38)(8,39)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(41,69)(42,70)(43,71)(44,72), (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( (1,6,51)(2,7,52)(3,8,49)(4,5,50)(9,53,17)(10,54,18)(11,55,19)(12,56,20)(13,45,41)(14,46,42)(15,47,43)(16,48,44)(21,39,29)(22,40,30)(23,37,31)(24,38,32)(25,69,61)(26,70,62)(27,71,63)(28,72,64)(33,65,57)(34,66,58)(35,67,59)(36,68,60), (1,47,11)(2,48,12)(3,45,9)(4,46,10)(5,42,54)(6,43,55)(7,44,56)(8,41,53)(13,17,49)(14,18,50)(15,19,51)(16,20,52)(21,25,57)(22,26,58)(23,27,59)(24,28,60)(29,61,65)(30,62,66)(31,63,67)(32,64,68)(33,39,69)(34,40,70)(35,37,71)(36,38,72), (1,3)(2,22)(4,24)(5,38)(6,8)(7,40)(9,11)(10,60)(12,58)(13,15)(14,64)(16,62)(17,19)(18,68)(20,66)(21,23)(25,27)(26,48)(28,46)(29,31)(30,52)(32,50)(33,35)(34,56)(36,54)(37,39)(41,43)(42,72)(44,70)(45,47)(49,51)(53,55)(57,59)(61,63)(65,67)(69,71), (1,23)(2,24)(3,21)(4,22)(5,40)(6,37)(7,38)(8,39)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(41,69)(42,70)(43,71)(44,72), (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([[(1,6,51),(2,7,52),(3,8,49),(4,5,50),(9,53,17),(10,54,18),(11,55,19),(12,56,20),(13,45,41),(14,46,42),(15,47,43),(16,48,44),(21,39,29),(22,40,30),(23,37,31),(24,38,32),(25,69,61),(26,70,62),(27,71,63),(28,72,64),(33,65,57),(34,66,58),(35,67,59),(36,68,60)], [(1,47,11),(2,48,12),(3,45,9),(4,46,10),(5,42,54),(6,43,55),(7,44,56),(8,41,53),(13,17,49),(14,18,50),(15,19,51),(16,20,52),(21,25,57),(22,26,58),(23,27,59),(24,28,60),(29,61,65),(30,62,66),(31,63,67),(32,64,68),(33,39,69),(34,40,70),(35,37,71),(36,38,72)], [(1,3),(2,22),(4,24),(5,38),(6,8),(7,40),(9,11),(10,60),(12,58),(13,15),(14,64),(16,62),(17,19),(18,68),(20,66),(21,23),(25,27),(26,48),(28,46),(29,31),(30,52),(32,50),(33,35),(34,56),(36,54),(37,39),(41,43),(42,72),(44,70),(45,47),(49,51),(53,55),(57,59),(61,63),(65,67),(69,71)], [(1,23),(2,24),(3,21),(4,22),(5,40),(6,37),(7,38),(8,39),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(41,69),(42,70),(43,71),(44,72)], [(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)]])
C32×C22⋊C4 is a maximal subgroup of
C62.110D4 C62.221C23 C62⋊6Q8 C62.223C23 C62.225C23 C62⋊12D4 C62.227C23 C62.228C23 C62.229C23 C62.69D4 D4×C3×C12
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 6Y | ··· | 6AN | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | C3×D4 |
| kernel | C32×C22⋊C4 | C6×C12 | C2×C62 | C3×C22⋊C4 | C62 | C2×C12 | C22×C6 | C2×C6 | C3×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 2 | 16 |
Matrix representation of C32×C22⋊C4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0] >;
C32×C22⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times C_2^2\rtimes C_4
% in TeX
G:=Group("C3^2xC2^2:C4"); // GroupNames label
G:=SmallGroup(144,102);
// by ID
G=gap.SmallGroup(144,102);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,432,457]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations