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## G = C23×C3.A4order 288 = 25·32

### Direct product of C23 and C3.A4

Aliases: C23×C3.A4, C251C9, C243C18, C3.(C23×A4), C233(C2×C18), C22⋊(C22×C18), (C24×C6).1C3, (C23×C6).5C6, C6.19(C22×A4), (C22×C6).15A4, (C2×C6).27(C2×A4), (C2×C6).6(C22×C6), (C22×C6).38(C2×C6), SmallGroup(288,837)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C23×C3.A4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C22×C3.A4 — C23×C3.A4
 Lower central C22 — C23×C3.A4
 Upper central C1 — C22×C6

Generators and relations for C23×C3.A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d3=e2=f2=1, g3=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 828 in 324 conjugacy classes, 80 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C22, C6, C6, C23, C23, C23, C9, C2×C6, C2×C6, C2×C6, C24, C24, C18, C22×C6, C22×C6, C22×C6, C25, C3.A4, C2×C18, C23×C6, C23×C6, C2×C3.A4, C22×C18, C24×C6, C22×C3.A4, C23×C3.A4
Quotients: C1, C2, C3, C22, C6, C23, C9, A4, C2×C6, C18, C2×A4, C22×C6, C3.A4, C2×C18, C22×A4, C2×C3.A4, C22×C18, C23×A4, C22×C3.A4, C23×C3.A4

Smallest permutation representation of C23×C3.A4
On 72 points
Generators in S72
(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 64)(28 63)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 46)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 37)(9 38)(10 52)(11 53)(12 54)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 28)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 64)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 19)(9 20)(10 70)(11 71)(12 72)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(28 45)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(46 63)(47 55)(48 56)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(2 22)(3 23)(5 25)(6 26)(8 19)(9 20)(11 71)(12 72)(14 65)(15 66)(17 68)(18 69)(29 37)(30 38)(32 40)(33 41)(35 43)(36 44)(47 55)(48 56)(50 58)(51 59)(53 61)(54 62)
(1 21)(3 23)(4 24)(6 26)(7 27)(9 20)(10 70)(12 72)(13 64)(15 66)(16 67)(18 69)(28 45)(30 38)(31 39)(33 41)(34 42)(36 44)(46 63)(48 56)(49 57)(51 59)(52 60)(54 62)
(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)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 6A ··· 6N 6O ··· 6AD 9A ··· 9F 18A ··· 18AP order 1 2 ··· 2 2 ··· 2 3 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 ··· 1 3 ··· 3 1 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

96 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C6 C9 C18 A4 C2×A4 C3.A4 C2×C3.A4 kernel C23×C3.A4 C22×C3.A4 C24×C6 C23×C6 C25 C24 C22×C6 C2×C6 C23 C22 # reps 1 7 2 14 6 42 1 7 2 14

Matrix representation of C23×C3.A4 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 7 0 0

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,1,0] >;

C23×C3.A4 in GAP, Magma, Sage, TeX

C_2^3\times C_3.A_4
% in TeX

G:=Group("C2^3xC3.A4");
// GroupNames label

G:=SmallGroup(288,837);
// by ID

G=gap.SmallGroup(288,837);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,108,782,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^3=e^2=f^2=1,g^3=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations

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