metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊4D4, C22⋊3D36, C23.20D18, C9⋊1C22≀C2, (C2×C4)⋊1D18, (C2×C18)⋊1D4, C2.7(D4×D9), D18⋊C4⋊4C2, (C2×D36)⋊2C2, C22⋊C4⋊2D9, C18.5(C2×D4), C6.79(S3×D4), (C2×C12).2D6, C2.7(C2×D36), (C2×C6).4D12, (C2×C36)⋊1C22, C3.(D6⋊D4), C6.34(C2×D12), (C23×D9)⋊1C2, (C22×C6).42D6, (C2×C18).23C23, (C2×Dic9)⋊1C22, (C22×D9)⋊1C22, C22.41(C22×D9), (C22×C18).12C22, (C2×C9⋊D4)⋊1C2, (C9×C22⋊C4)⋊3C2, (C3×C22⋊C4).3S3, (C2×C6).180(C22×S3), SmallGroup(288,92)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊3D36
G = < a,b,c,d | a2=b2=c36=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1148 in 195 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×5], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], C9, Dic3, C12 [×2], D6 [×19], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, D9 [×5], C18, C18 [×2], C18 [×2], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C22×S3 [×9], C22×C6, C22≀C2, Dic9, C36 [×2], D18 [×4], D18 [×15], C2×C18, C2×C18 [×2], C2×C18 [×2], D6⋊C4 [×2], C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, D36 [×4], C2×Dic9, C9⋊D4 [×2], C2×C36 [×2], C22×D9, C22×D9 [×2], C22×D9 [×6], C22×C18, D6⋊D4, D18⋊C4 [×2], C9×C22⋊C4, C2×D36 [×2], C2×C9⋊D4, C23×D9, C22⋊3D36
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D9, D12 [×2], C22×S3, C22≀C2, D18 [×3], C2×D12, S3×D4 [×2], D36 [×2], C22×D9, D6⋊D4, C2×D36, D4×D9 [×2], C22⋊3D36
(1 43)(3 45)(5 47)(7 49)(9 51)(11 53)(13 55)(15 57)(17 59)(19 61)(21 63)(23 65)(25 67)(27 69)(29 71)(31 37)(33 39)(35 41)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)
G:=sub<Sym(72)| (1,43)(3,45)(5,47)(7,49)(9,51)(11,53)(13,55)(15,57)(17,59)(19,61)(21,63)(23,65)(25,67)(27,69)(29,71)(31,37)(33,39)(35,41), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)>;
G:=Group( (1,43)(3,45)(5,47)(7,49)(9,51)(11,53)(13,55)(15,57)(17,59)(19,61)(21,63)(23,65)(25,67)(27,69)(29,71)(31,37)(33,39)(35,41), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43) );
G=PermutationGroup([(1,43),(3,45),(5,47),(7,49),(9,51),(11,53),(13,55),(15,57),(17,59),(19,61),(21,63),(23,65),(25,67),(27,69),(29,71),(31,37),(33,39),(35,41)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,60),(20,59),(21,58),(22,57),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43)])
54 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 18 | 18 | 36 | 2 | 4 | 4 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | D12 | D18 | D18 | D36 | S3×D4 | D4×D9 |
kernel | C22⋊3D36 | D18⋊C4 | C9×C22⋊C4 | C2×D36 | C2×C9⋊D4 | C23×D9 | C3×C22⋊C4 | D18 | C2×C18 | C2×C12 | C22×C6 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
# reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 12 | 2 | 6 |
Matrix representation of C22⋊3D36 ►in GL4(𝔽37) generated by
36 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
36 | 0 | 0 | 0 |
0 | 36 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 |
36 | 0 | 0 | 0 |
0 | 0 | 4 | 8 |
0 | 0 | 29 | 12 |
0 | 36 | 0 | 0 |
36 | 0 | 0 | 0 |
0 | 0 | 32 | 10 |
0 | 0 | 5 | 5 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,1,0,0,0,0,0,4,29,0,0,8,12],[0,36,0,0,36,0,0,0,0,0,32,5,0,0,10,5] >;
C22⋊3D36 in GAP, Magma, Sage, TeX
C_2^2\rtimes_3D_{36}
% in TeX
G:=Group("C2^2:3D36");
// GroupNames label
G:=SmallGroup(288,92);
// by ID
G=gap.SmallGroup(288,92);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^36=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations