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G = C18.D4order 144 = 24·32

7th non-split extension by C18 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C18.11D4, C23.2D9, C222Dic9, C22.7D18, (C2×C18)⋊2C4, C92(C22⋊C4), C18.9(C2×C4), (C2×C6).23D6, (C2×Dic9)⋊2C2, C2.3(C9⋊D4), (C22×C6).6S3, C2.5(C2×Dic9), (C2×C6).5Dic3, C6.18(C3⋊D4), (C2×C18).7C22, (C22×C18).2C2, C3.(C6.D4), C6.10(C2×Dic3), SmallGroup(144,19)

Series: Derived Chief Lower central Upper central

C1C18 — C18.D4
C1C3C9C18C2×C18C2×Dic9 — C18.D4
C9C18 — C18.D4
C1C22C23

Generators and relations for C18.D4
 G = < a,b,c | a18=b4=1, c2=a9, bab-1=cac-1=a-1, cbc-1=a9b-1 >

2C2
2C2
2C22
2C22
18C4
18C4
2C6
2C6
9C2×C4
9C2×C4
2C2×C6
2C2×C6
6Dic3
6Dic3
2C18
2C18
9C22⋊C4
3C2×Dic3
3C2×Dic3
2Dic9
2Dic9
2C2×C18
2C2×C18
3C6.D4

Smallest permutation representation of C18.D4
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 59 45 19)(2 58 46 36)(3 57 47 35)(4 56 48 34)(5 55 49 33)(6 72 50 32)(7 71 51 31)(8 70 52 30)(9 69 53 29)(10 68 54 28)(11 67 37 27)(12 66 38 26)(13 65 39 25)(14 64 40 24)(15 63 41 23)(16 62 42 22)(17 61 43 21)(18 60 44 20)
(1 28 10 19)(2 27 11 36)(3 26 12 35)(4 25 13 34)(5 24 14 33)(6 23 15 32)(7 22 16 31)(8 21 17 30)(9 20 18 29)(37 58 46 67)(38 57 47 66)(39 56 48 65)(40 55 49 64)(41 72 50 63)(42 71 51 62)(43 70 52 61)(44 69 53 60)(45 68 54 59)

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

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

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

C18.D4 is a maximal subgroup of
C22.D36  C232Dic9  C23.16D18  C222Dic18  C23.8D18  C22⋊C4×D9  C23.9D18  Dic9.D4  C36.49D4  C23.26D18  C4×C9⋊D4  C23.28D18  D4×Dic9  C23.23D18  C36.17D4  C232D18  C362D4  Dic9⋊D4  C244D9  C54.D4  C18.S4  D6⋊Dic9  C62.27D6  C62.127D6  A4⋊Dic9
C18.D4 is a maximal quotient of
C36.55D4  C18.C42  C36.D4  D4⋊Dic9  C232Dic9  C36.9D4  Q82Dic9  Q83Dic9  C54.D4  D6⋊Dic9  C62.127D6

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A···6G9A9B9C18A···18U
order122222344446···699918···18
size1111222181818182···22222···2

42 irreducible representations

dim1111222222222
type+++++-++-+
imageC1C2C2C4S3D4Dic3D6D9C3⋊D4Dic9D18C9⋊D4
kernelC18.D4C2×Dic9C22×C18C2×C18C22×C6C18C2×C6C2×C6C23C6C22C22C2
# reps12141221346312

Matrix representation of C18.D4 in GL3(𝔽37) generated by

3600
030
0025
,
3100
001
010
,
600
001
0360
G:=sub<GL(3,GF(37))| [36,0,0,0,3,0,0,0,25],[31,0,0,0,0,1,0,1,0],[6,0,0,0,0,36,0,1,0] >;

C18.D4 in GAP, Magma, Sage, TeX

C_{18}.D_4
% in TeX

G:=Group("C18.D4");
// GroupNames label

G:=SmallGroup(144,19);
// by ID

G=gap.SmallGroup(144,19);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^18=b^4=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;
// generators/relations

Export

Subgroup lattice of C18.D4 in TeX

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