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## G = Dic18⋊C4order 288 = 25·32

### 4th semidirect product of Dic18 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — Dic18⋊C4
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — D36⋊5C2 — Dic18⋊C4
 Lower central C9 — C18 — C36 — Dic18⋊C4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for Dic18⋊C4
G = < a,b,c | a36=c4=1, b2=a18, bab-1=a-1, cac-1=a17, cbc-1=a27b >

Subgroups: 332 in 66 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C9, Dic3 [×3], C12 [×2], D6, C2×C6, C42, M4(2), C4○D4, D9, C18, C18, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C4≀C2, Dic9 [×3], C36 [×2], D18, C2×C18, C4×Dic3, C3×M4(2), C4○D12, C72, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, D12⋊C4, C4×Dic9, C9×M4(2), D365C2, Dic18⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4≀C2, D18, D6⋊C4, C4×D9, D36, C9⋊D4, D12⋊C4, D18⋊C4, Dic18⋊C4

Smallest permutation representation of Dic18⋊C4
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 46 19 64)(2 45 20 63)(3 44 21 62)(4 43 22 61)(5 42 23 60)(6 41 24 59)(7 40 25 58)(8 39 26 57)(9 38 27 56)(10 37 28 55)(11 72 29 54)(12 71 30 53)(13 70 31 52)(14 69 32 51)(15 68 33 50)(16 67 34 49)(17 66 35 48)(18 65 36 47)
(1 28 19 10)(2 9 20 27)(3 26 21 8)(4 7 22 25)(5 24 23 6)(11 18 29 36)(12 35 30 17)(13 16 31 34)(14 33 32 15)(38 54)(39 71)(40 52)(41 69)(42 50)(43 67)(44 48)(45 65)(47 63)(49 61)(51 59)(53 57)(56 72)(58 70)(60 68)(62 66)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 8A 8B 9A 9B 9C 12A 12B 12C 18A 18B 18C 18D 18E 18F 24A 24B 24C 24D 36A ··· 36F 36G 36H 36I 72A ··· 72L order 1 2 2 2 3 4 4 4 4 4 4 4 4 6 6 8 8 9 9 9 12 12 12 18 18 18 18 18 18 24 24 24 24 36 ··· 36 36 36 36 72 ··· 72 size 1 1 2 36 2 1 1 2 18 18 18 18 36 2 4 4 4 2 2 2 2 2 4 2 2 2 4 4 4 4 4 4 4 2 ··· 2 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 D9 C4×S3 C3⋊D4 D12 C4≀C2 D18 C4×D9 C9⋊D4 D36 D12⋊C4 Dic18⋊C4 kernel Dic18⋊C4 C4×Dic9 C9×M4(2) D36⋊5C2 Dic18 D36 C3×M4(2) C36 C2×C18 C2×C12 M4(2) C12 C12 C2×C6 C9 C2×C4 C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 3 2 2 2 4 3 6 6 6 2 6

Matrix representation of Dic18⋊C4 in GL4(𝔽73) generated by

 28 70 0 0 3 31 0 0 0 0 27 0 0 0 0 46
,
 1 0 0 0 72 72 0 0 0 0 0 46 0 0 46 0
,
 1 0 0 0 72 72 0 0 0 0 46 0 0 0 0 1
G:=sub<GL(4,GF(73))| [28,3,0,0,70,31,0,0,0,0,27,0,0,0,0,46],[1,72,0,0,0,72,0,0,0,0,0,46,0,0,46,0],[1,72,0,0,0,72,0,0,0,0,46,0,0,0,0,1] >;

Dic18⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{18}\rtimes C_4
% in TeX

G:=Group("Dic18:C4");
// GroupNames label

G:=SmallGroup(288,32);
// by ID

G=gap.SmallGroup(288,32);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=c^4=1,b^2=a^18,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^27*b>;
// generators/relations

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