A4xC2^2xC6 - GroupNames

G = A₄×C₂²×C₆ order 288 = 2⁵·3²

Direct product of C₂²×C₆ and A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄×C₂²×C₆, C₂⁵⋊₂C₃², C₂³⋊₃C₆², C₂⁴⋊₄(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

G = A4×C22×C6 order 288 = 25·32

Direct product of C22×C6 and A4

G = A₄×C₂²×C₆ order 288 = 2⁵·3²

Direct product of C₂²×C₆ and A₄