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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 [×4], C3, C4 [×3], C22, C22 [×2], C22 [×3], C6, C6 [×4], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C9, Dic3 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×3], C22⋊C4 [×2], C2×D4, C18, C18 [×4], C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C23⋊C4, Dic9 [×2], C36, C2×C18, C2×C18 [×2], C2×C18 [×3], C6.D4 [×2], C6×D4, C2×Dic9 [×2], C2×C36, D4×C9 [×2], C22×C18 [×2], C23.7D6, C18.D4 [×2], D4×C18, C232Dic9
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D9, C2×Dic3, C3⋊D4 [×2], C23⋊C4, Dic9 [×2], D18, C6.D4, C2×Dic9, C9⋊D4 [×2], C23.7D6, C18.D4, C232Dic9

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

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

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

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

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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