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## G = A4×C22×C6order 288 = 25·32

### Direct product of C22×C6 and A4

Aliases: A4×C22×C6, C252C32, C233C62, C244(C3×C6), (C23×C6)⋊7C6, (C24×C6)⋊1C3, C22⋊(C2×C62), (C22×C6)⋊7(C2×C6), (C2×C6)⋊3(C22×C6), SmallGroup(288,1041)

Series: Derived Chief Lower central Upper central

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

Generators and relations for A4×C22×C6
G = < a,b,c,d,e,f | a2=b2=c6=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 1068 in 420 conjugacy classes, 128 normal (10 characteristic)
C1, C2, C2, C3, C3, C22, C22, C22, C6, C6, C23, C23, C23, C32, A4, C2×C6, C2×C6, C2×C6, C24, C24, C3×C6, C2×A4, C22×C6, C22×C6, C22×C6, C25, C3×A4, C62, C22×A4, C23×C6, C23×C6, C6×A4, C2×C62, C23×A4, C24×C6, A4×C2×C6, A4×C22×C6
Quotients: C1, C2, C3, C22, C6, C23, C32, A4, C2×C6, C3×C6, C2×A4, C22×C6, C3×A4, C62, C22×A4, C6×A4, C2×C62, C23×A4, A4×C2×C6, A4×C22×C6

Smallest permutation representation of A4×C22×C6
On 72 points
Generators in S72
(1 51)(2 52)(3 53)(4 54)(5 49)(6 50)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)(13 47)(14 48)(15 43)(16 44)(17 45)(18 46)(25 69)(26 70)(27 71)(28 72)(29 67)(30 68)(31 65)(32 66)(33 61)(34 62)(35 63)(36 64)(37 60)(38 55)(39 56)(40 57)(41 58)(42 59)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 40)(8 41)(9 42)(10 37)(11 38)(12 39)(19 56)(20 57)(21 58)(22 59)(23 60)(24 55)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)(61 72)(62 67)(63 68)(64 69)(65 70)(66 71)
(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)
(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(19 59)(20 60)(21 55)(22 56)(23 57)(24 58)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)(43 51)(44 52)(45 53)(46 54)(47 49)(48 50)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)
(1 11 27)(2 12 28)(3 7 29)(4 8 30)(5 9 25)(6 10 26)(13 39 33)(14 40 34)(15 41 35)(16 42 36)(17 37 31)(18 38 32)(19 72 52)(20 67 53)(21 68 54)(22 69 49)(23 70 50)(24 71 51)(43 58 63)(44 59 64)(45 60 65)(46 55 66)(47 56 61)(48 57 62)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C ··· 3H 6A ··· 6N 6O ··· 6AD 6AE ··· 6BT order 1 2 ··· 2 2 ··· 2 3 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 ··· 1 3 ··· 3 1 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

96 irreducible representations

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

Matrix representation of A4×C22×C6 in GL5(𝔽7)

 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 6 0 0 0 1 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 6 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 5 0 0 0 0 6 1 0 0 0 6 0

G:=sub<GL(5,GF(7))| [1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,6,0,0,0,6,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,5,6,6,0,0,0,1,0] >;

A4×C22×C6 in GAP, Magma, Sage, TeX

A_4\times C_2^2\times C_6
% in TeX

G:=Group("A4xC2^2xC6");
// GroupNames label

G:=SmallGroup(288,1041);
// by ID

G=gap.SmallGroup(288,1041);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^2=c^6=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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