Copied to
clipboard

G = D366C22order 288 = 25·32

4th semidirect product of D36 and C22 acting via C22/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.7D18, C36.16D4, D366C22, C36.13C23, Dic186C22, D4⋊D95C2, (C2×D4)⋊2D9, C9⋊C83C22, (D4×C18)⋊2C2, C94(C8⋊C22), D4.D95C2, (C6×D4).5S3, (C3×D4).31D6, (C2×C4).14D18, C18.47(C2×D4), (C2×C12).57D6, (C2×C18).40D4, C4.Dic96C2, D365C23C2, C4.16(C9⋊D4), C3.(D126C22), (D4×C9).7C22, C4.13(C22×D9), C12.11(C3⋊D4), C12.52(C22×S3), (C2×C36).35C22, C22.10(C9⋊D4), C2.10(C2×C9⋊D4), C6.94(C2×C3⋊D4), (C2×C6).79(C3⋊D4), SmallGroup(288,143)

Series: Derived Chief Lower central Upper central

C1C36 — D366C22
C1C3C9C18C36D36D365C2 — D366C22
C9C18C36 — D366C22
C1C2C2×C4C2×D4

Generators and relations for D366C22
 G = < a,b,c,d | a36=b2=c2=d2=1, bab=a-1, ac=ca, dad=a19, cbc=a18b, dbd=a27b, cd=dc >

Subgroups: 404 in 102 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, C9, Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, D9, C18, C18 [×3], C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C8⋊C22, Dic9, C36 [×2], D18, C2×C18, C2×C18 [×4], C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C4○D12, C6×D4, C9⋊C8 [×2], Dic18, C4×D9, D36, C9⋊D4, C2×C36, D4×C9 [×2], D4×C9, C22×C18, D126C22, C4.Dic9, D4.D9 [×2], D4⋊D9 [×2], D365C2, D4×C18, D366C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C8⋊C22, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, D126C22, C2×C9⋊D4, D366C22

Smallest permutation representation of D366C22
On 72 points
Generators in S72
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)
(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(2 20)(4 22)(6 24)(8 26)(10 28)(12 30)(14 32)(16 34)(18 36)(37 46)(38 65)(39 48)(40 67)(41 50)(42 69)(43 52)(44 71)(45 54)(47 56)(49 58)(51 60)(53 62)(55 64)(57 66)(59 68)(61 70)(63 72)

G:=sub<Sym(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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48), (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (2,20)(4,22)(6,24)(8,26)(10,28)(12,30)(14,32)(16,34)(18,36)(37,46)(38,65)(39,48)(40,67)(41,50)(42,69)(43,52)(44,71)(45,54)(47,56)(49,58)(51,60)(53,62)(55,64)(57,66)(59,68)(61,70)(63,72)>;

G:=Group( (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48), (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (2,20)(4,22)(6,24)(8,26)(10,28)(12,30)(14,32)(16,34)(18,36)(37,46)(38,65)(39,48)(40,67)(41,50)(42,69)(43,52)(44,71)(45,54)(47,56)(49,58)(51,60)(53,62)(55,64)(57,66)(59,68)(61,70)(63,72) );

G=PermutationGroup([(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,72),(13,71),(14,70),(15,69),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48)], [(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(2,20),(4,22),(6,24),(8,26),(10,28),(12,30),(14,32),(16,34),(18,36),(37,46),(38,65),(39,48),(40,67),(41,50),(42,69),(43,52),(44,71),(45,54),(47,56),(49,58),(51,60),(53,62),(55,64),(57,66),(59,68),(61,70),(63,72)])

51 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C6A6B6C6D6E6F6G8A8B9A9B9C12A12B18A···18I18J···18U36A···36F
order1222223444666666688999121218···1818···1836···36
size11244362223622244443636222442···24···44···4

51 irreducible representations

dim111111222222222222444
type+++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D9C3⋊D4C3⋊D4D18D18C9⋊D4C9⋊D4C8⋊C22D126C22D366C22
kernelD366C22C4.Dic9D4.D9D4⋊D9D365C2D4×C18C6×D4C36C2×C18C2×C12C3×D4C2×D4C12C2×C6C2×C4D4C4C22C9C3C1
# reps112211111123223666126

Matrix representation of D366C22 in GL6(𝔽73)

200000
56370000
001300
00487200
005749072
0004910
,
6860000
6950000
00490070
0016011
00221024
00700024
,
7200000
0720000
001000
000100
00570720
00570072
,
100000
010000
001000
00487200
00570072
00570720

G:=sub<GL(6,GF(73))| [2,56,0,0,0,0,0,37,0,0,0,0,0,0,1,48,57,0,0,0,3,72,49,49,0,0,0,0,0,1,0,0,0,0,72,0],[68,69,0,0,0,0,6,5,0,0,0,0,0,0,49,16,22,70,0,0,0,0,1,0,0,0,0,1,0,0,0,0,70,1,24,24],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,57,57,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,48,57,57,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

D366C22 in GAP, Magma, Sage, TeX

D_{36}\rtimes_6C_2^2
% in TeX

G:=Group("D36:6C2^2");
// GroupNames label

G:=SmallGroup(288,143);
// by ID

G=gap.SmallGroup(288,143);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,675,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^36=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^19,c*b*c=a^18*b,d*b*d=a^27*b,c*d=d*c>;
// generators/relations

׿
×
𝔽