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## G = C4.Dic9order 144 = 24·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C4.Dic9
 Chief series C1 — C3 — C9 — C18 — C36 — C9⋊C8 — C4.Dic9
 Lower central C9 — C18 — C4.Dic9
 Upper central C1 — C4 — C2×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 >

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 2A 2B 3 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 36A ··· 36L order 1 2 2 3 4 4 4 6 6 6 8 8 8 8 9 9 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 2 2 1 1 2 2 2 2 18 18 18 18 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + - image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) D9 Dic9 D18 Dic9 C4.Dic3 C4.Dic9 kernel C4.Dic9 C9⋊C8 C2×C36 C36 C2×C18 C2×C12 C12 C12 C2×C6 C9 C2×C4 C4 C4 C22 C3 C1 # reps 1 2 1 2 2 1 1 1 1 2 3 3 3 3 4 12

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

 31 0 0 6
,
 5 0 0 22
,
 0 6 1 0
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

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