direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊C12, C62.2C6, C62.1S3, (C3×C6)⋊C12, He3⋊6(C2×C4), C6.17(S3×C6), (C2×He3)⋊2C4, C3⋊Dic3⋊3C6, (C3×C6)⋊1Dic3, (C3×C6).12D6, C32⋊2(C2×C12), C3.2(C6×Dic3), C6.5(C3×Dic3), C22.(C32⋊C6), C32⋊2(C2×Dic3), (C22×He3).1C2, (C2×He3).9C22, (C2×C3⋊Dic3)⋊C3, (C3×C6).4(C2×C6), (C2×C6).12(C3×S3), C2.2(C2×C32⋊C6), SmallGroup(216,59)
Series: Derived ►Chief ►Lower central ►Upper central
C32 — C2×C32⋊C12 |
Generators and relations for C2×C32⋊C12
G = < a,b,c,d | a2=b3=c3=d12=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=c-1 >
Subgroups: 200 in 66 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×2], C22, C6, C6 [×2], C6 [×9], C2×C4, C32 [×2], C32, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C3×C6 [×2], C3×C6 [×4], C3×C6 [×3], C2×Dic3 [×2], C2×C12, He3, C3×Dic3 [×2], C3⋊Dic3 [×2], C62 [×2], C62, C2×He3, C2×He3 [×2], C6×Dic3, C2×C3⋊Dic3, C32⋊C12 [×2], C22×He3, C2×C32⋊C12
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C3×S3, C2×Dic3, C2×C12, C3×Dic3 [×2], S3×C6, C32⋊C6, C6×Dic3, C32⋊C12 [×2], C2×C32⋊C6, C2×C32⋊C12
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)
(1 64 72)(2 6 33)(3 11 30)(4 63 67)(5 36 9)(7 70 66)(8 12 27)(10 69 61)(13 55 17)(14 52 22)(15 43 39)(16 20 58)(18 42 46)(19 49 23)(21 37 45)(24 48 40)(25 29 65)(26 34 62)(28 68 32)(31 35 71)(38 56 60)(41 51 59)(44 50 54)(47 57 53)
(1 36 68)(2 69 25)(3 26 70)(4 71 27)(5 28 72)(6 61 29)(7 30 62)(8 63 31)(9 32 64)(10 65 33)(11 34 66)(12 67 35)(13 59 45)(14 46 60)(15 49 47)(16 48 50)(17 51 37)(18 38 52)(19 53 39)(20 40 54)(21 55 41)(22 42 56)(23 57 43)(24 44 58)
(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)| (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63), (1,64,72)(2,6,33)(3,11,30)(4,63,67)(5,36,9)(7,70,66)(8,12,27)(10,69,61)(13,55,17)(14,52,22)(15,43,39)(16,20,58)(18,42,46)(19,49,23)(21,37,45)(24,48,40)(25,29,65)(26,34,62)(28,68,32)(31,35,71)(38,56,60)(41,51,59)(44,50,54)(47,57,53), (1,36,68)(2,69,25)(3,26,70)(4,71,27)(5,28,72)(6,61,29)(7,30,62)(8,63,31)(9,32,64)(10,65,33)(11,34,66)(12,67,35)(13,59,45)(14,46,60)(15,49,47)(16,48,50)(17,51,37)(18,38,52)(19,53,39)(20,40,54)(21,55,41)(22,42,56)(23,57,43)(24,44,58), (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( (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63), (1,64,72)(2,6,33)(3,11,30)(4,63,67)(5,36,9)(7,70,66)(8,12,27)(10,69,61)(13,55,17)(14,52,22)(15,43,39)(16,20,58)(18,42,46)(19,49,23)(21,37,45)(24,48,40)(25,29,65)(26,34,62)(28,68,32)(31,35,71)(38,56,60)(41,51,59)(44,50,54)(47,57,53), (1,36,68)(2,69,25)(3,26,70)(4,71,27)(5,28,72)(6,61,29)(7,30,62)(8,63,31)(9,32,64)(10,65,33)(11,34,66)(12,67,35)(13,59,45)(14,46,60)(15,49,47)(16,48,50)(17,51,37)(18,38,52)(19,53,39)(20,40,54)(21,55,41)(22,42,56)(23,57,43)(24,44,58), (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([(1,21),(2,22),(3,23),(4,24),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20),(25,56),(26,57),(27,58),(28,59),(29,60),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,61),(47,62),(48,63)], [(1,64,72),(2,6,33),(3,11,30),(4,63,67),(5,36,9),(7,70,66),(8,12,27),(10,69,61),(13,55,17),(14,52,22),(15,43,39),(16,20,58),(18,42,46),(19,49,23),(21,37,45),(24,48,40),(25,29,65),(26,34,62),(28,68,32),(31,35,71),(38,56,60),(41,51,59),(44,50,54),(47,57,53)], [(1,36,68),(2,69,25),(3,26,70),(4,71,27),(5,28,72),(6,61,29),(7,30,62),(8,63,31),(9,32,64),(10,65,33),(11,34,66),(12,67,35),(13,59,45),(14,46,60),(15,49,47),(16,48,50),(17,51,37),(18,38,52),(19,53,39),(20,40,54),(21,55,41),(22,42,56),(23,57,43),(24,44,58)], [(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)])
C2×C32⋊C12 is a maximal subgroup of
He3⋊C42 C62.D6 C62.3D6 C62.4D6 C62.5D6 C62.19D6 C62.20D6 C62.21D6 C62⋊3C12 C62.8D6 C2×C4×C32⋊C6 C62.13D6
C2×C32⋊C12 is a maximal quotient of
He3⋊7M4(2) C62.20D6 C62⋊3C12
40 conjugacy classes
class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | ··· | 6I | 6J | ··· | 6R | 12A | ··· | 12H |
order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
40 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
type | + | + | + | + | - | + | + | - | + | ||||||||
image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 | C32⋊C6 | C32⋊C12 | C2×C32⋊C6 |
kernel | C2×C32⋊C12 | C32⋊C12 | C22×He3 | C2×C3⋊Dic3 | C2×He3 | C3⋊Dic3 | C62 | C3×C6 | C62 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
# reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 1 |
Matrix representation of C2×C32⋊C12 ►in GL10(𝔽13)
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 2 | 1 |
0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 1 | 2 |
0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 3 | 3 |
0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 | 3 | 3 |
0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 | 9 | 9 |
0 | 0 | 0 | 0 | 10 | 10 | 1 | 0 | 9 | 9 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 10 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 3 | 12 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 12 | 12 |
0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 6 | 6 | 3 | 6 | 3 |
0 | 0 | 0 | 0 | 0 | 3 | 3 | 10 | 3 | 10 |
0 | 0 | 0 | 0 | 4 | 10 | 12 | 3 | 9 | 10 |
0 | 0 | 0 | 0 | 10 | 9 | 9 | 10 | 6 | 0 |
0 | 0 | 0 | 0 | 10 | 1 | 11 | 0 | 1 | 10 |
0 | 0 | 0 | 0 | 4 | 5 | 4 | 3 | 11 | 0 |
G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4,4,12,12,0,10,0,0,0,0,4,4,0,12,0,10,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,2,1,3,3,9,9,0,0,0,0,1,2,3,3,9,9],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,10,0,4,0,0,0,0,0,1,0,0,3,0,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,4,10,10,4,0,0,0,0,6,3,10,9,1,5,0,0,0,0,6,3,12,9,11,4,0,0,0,0,3,10,3,10,0,3,0,0,0,0,6,3,9,6,1,11,0,0,0,0,3,10,10,0,10,0] >;
C2×C32⋊C12 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes C_{12}
% in TeX
G:=Group("C2xC3^2:C12");
// GroupNames label
G:=SmallGroup(216,59);
// by ID
G=gap.SmallGroup(216,59);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,1444,736,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;
// generators/relations