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G = C4.Dic9order 144 = 24·32

The non-split extension by C4 of Dic9 acting via Dic9/C18=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Dic9, C36.1C4, C92M4(2), C12.53D6, C4.15D18, C22.Dic9, C12.1Dic3, C36.15C22, C9⋊C85C2, (C2×C4).2D9, (C2×C18).3C4, (C2×C36).4C2, C18.7(C2×C4), (C2×C12).10S3, C6.7(C2×Dic3), (C2×C6).4Dic3, C2.3(C2×Dic9), C3.(C4.Dic3), SmallGroup(144,10)

Series: Derived Chief Lower central Upper central

C1C18 — C4.Dic9
C1C3C9C18C36C9⋊C8 — C4.Dic9
C9C18 — C4.Dic9
C1C4C2×C4

Generators and relations for C4.Dic9
 G = < a,b,c | a4=1, b18=a2, c2=b9, ab=ba, cac-1=a-1, cbc-1=b17 >

2C2
2C6
9C8
9C8
2C18
9M4(2)
3C3⋊C8
3C3⋊C8
3C4.Dic3

Smallest permutation representation of C4.Dic9
On 72 points
Generators in S72
(1 10 19 28)(2 11 20 29)(3 12 21 30)(4 13 22 31)(5 14 23 32)(6 15 24 33)(7 16 25 34)(8 17 26 35)(9 18 27 36)(37 64 55 46)(38 65 56 47)(39 66 57 48)(40 67 58 49)(41 68 59 50)(42 69 60 51)(43 70 61 52)(44 71 62 53)(45 72 63 54)
(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 61 10 70 19 43 28 52)(2 42 11 51 20 60 29 69)(3 59 12 68 21 41 30 50)(4 40 13 49 22 58 31 67)(5 57 14 66 23 39 32 48)(6 38 15 47 24 56 33 65)(7 55 16 64 25 37 34 46)(8 72 17 45 26 54 35 63)(9 53 18 62 27 71 36 44)

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

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

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

C4.Dic9 is a maximal subgroup of
C424D9  C72.C4  C36.53D4  C4.D36  C36.48D4  C36.D4  C36.9D4  Q83Dic9  D36.2C4  M4(2)×D9  D366C22  C36.C23  D4.Dic9  D4.D18  D4⋊D18  C4.Dic27  D6.Dic9  C36.C12  C36.69D6
C4.Dic9 is a maximal quotient of
C42.D9  C36⋊C8  C36.55D4  C4.Dic27  D6.Dic9  C36.69D6

42 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I36A···36L
order122344466688889991212121218···1836···36
size11221122221818181822222222···22···2

42 irreducible representations

dim1111122222222222
type++++-+-+-+-
imageC1C2C2C4C4S3Dic3D6Dic3M4(2)D9Dic9D18Dic9C4.Dic3C4.Dic9
kernelC4.Dic9C9⋊C8C2×C36C36C2×C18C2×C12C12C12C2×C6C9C2×C4C4C4C22C3C1
# reps12122111123333412

Matrix representation of C4.Dic9 in GL2(𝔽37) generated by

310
06
,
50
022
,
06
10
G:=sub<GL(2,GF(37))| [31,0,0,6],[5,0,0,22],[0,1,6,0] >;

C4.Dic9 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_9
% in TeX

G:=Group("C4.Dic9");
// GroupNames label

G:=SmallGroup(144,10);
// by ID

G=gap.SmallGroup(144,10);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,50,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^18=a^2,c^2=b^9,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^17>;
// generators/relations

Export

Subgroup lattice of C4.Dic9 in TeX

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