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## G = C23⋊2Dic9order 288 = 25·32

### The semidirect product of C23 and Dic9 acting via Dic9/C9=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C23⋊2Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×C18 — C18.D4 — C23⋊2Dic9
 Lower central C9 — C18 — C2×C18 — C23⋊2Dic9
 Upper central C1 — C2 — C23 — C2×D4

Generators and relations for C232Dic9
G = < a,b,c,d,e | a2=b2=c2=d18=1, e2=d9, ab=ba, dad-1=ac=ca, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 300 in 78 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C18, C18, C2×Dic3, C2×C12, C3×D4, C22×C6, C23⋊C4, Dic9, C36, C2×C18, C2×C18, C2×C18, C6.D4, C6×D4, C2×Dic9, C2×C36, D4×C9, C22×C18, C23.7D6, C18.D4, D4×C18, C232Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D9, C2×Dic3, C3⋊D4, C23⋊C4, Dic9, D18, C6.D4, C2×Dic9, C9⋊D4, C23.7D6, C18.D4, C232Dic9

Smallest permutation representation of C232Dic9
On 72 points
Generators in S72
(1 45)(2 37)(3 47)(4 39)(5 49)(6 41)(7 51)(8 43)(9 53)(10 38)(11 48)(12 40)(13 50)(14 42)(15 52)(16 44)(17 54)(18 46)(19 56)(20 66)(21 58)(22 68)(23 60)(24 70)(25 62)(26 72)(27 64)(28 61)(29 71)(30 63)(31 55)(32 65)(33 57)(34 67)(35 59)(36 69)
(1 27)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 33)(11 34)(12 35)(13 36)(14 28)(15 29)(16 30)(17 31)(18 32)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 55)
(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(19 32)(20 33)(21 34)(22 35)(23 36)(24 28)(25 29)(26 30)(27 31)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 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 4)(2 3)(5 9)(6 8)(10 18)(11 17)(12 16)(13 15)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 36)(26 35)(27 34)(37 66 46 57)(38 65 47 56)(39 64 48 55)(40 63 49 72)(41 62 50 71)(42 61 51 70)(43 60 52 69)(44 59 53 68)(45 58 54 67)

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 3 4 4 4 4 4 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 2 4 2 4 36 36 36 36 2 2 2 4 4 4 4 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + - - + + - - + + image C1 C2 C2 C4 C4 S3 D4 Dic3 Dic3 D6 D9 C3⋊D4 Dic9 Dic9 D18 C9⋊D4 C23⋊C4 C23.7D6 C23⋊2Dic9 kernel C23⋊2Dic9 C18.D4 D4×C18 C2×C36 C22×C18 C6×D4 C2×C18 C2×C12 C22×C6 C22×C6 C2×D4 C2×C6 C2×C4 C23 C23 C22 C9 C3 C1 # reps 1 2 1 2 2 1 2 1 1 1 3 4 3 3 3 12 1 2 6

Matrix representation of C232Dic9 in GL4(𝔽37) generated by

 36 0 9 32 0 36 5 14 0 0 1 0 0 0 0 1
,
 7 14 0 0 23 30 0 0 0 0 7 14 0 0 23 30
,
 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 26 6 0 0 31 20 0 0 13 24 11 31 13 26 6 17
,
 6 26 0 0 20 31 0 0 5 18 24 26 13 32 2 13
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,9,5,1,0,32,14,0,1],[7,23,0,0,14,30,0,0,0,0,7,23,0,0,14,30],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[26,31,13,13,6,20,24,26,0,0,11,6,0,0,31,17],[6,20,5,13,26,31,18,32,0,0,24,2,0,0,26,13] >;

C232Dic9 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2{\rm Dic}_9
% in TeX

G:=Group("C2^3:2Dic9");
// GroupNames label

G:=SmallGroup(288,41);
// by ID

G=gap.SmallGroup(288,41);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,675,6725,292,9414]);
// Polycyclic

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

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