Copied to
clipboard

?

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), (C2×C6)⋊3(C22×C6), (C22×C6)⋊7(C2×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

Subgroups: 1068 in 420 conjugacy classes, 128 normal (10 characteristic)
C1, C2 [×7], C2 [×8], C3, C3 [×3], C22, C22 [×7], C22 [×49], C6 [×7], C6 [×29], C23, C23 [×7], C23 [×49], C32, A4 [×3], C2×C6, C2×C6 [×7], C2×C6 [×70], C24 [×7], C24 [×8], C3×C6 [×7], C2×A4 [×21], C22×C6, C22×C6 [×7], C22×C6 [×52], C25, C3×A4, C62 [×7], C22×A4 [×21], C23×C6 [×7], C23×C6 [×8], C6×A4 [×7], C2×C62, C23×A4 [×3], C24×C6, A4×C2×C6 [×7], A4×C22×C6

Quotients:
C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], C23, C32, A4, C2×C6 [×28], C3×C6 [×7], C2×A4 [×7], C22×C6 [×4], C3×A4, C62 [×7], C22×A4 [×7], C6×A4 [×7], C2×C62, C23×A4, A4×C2×C6 [×7], A4×C22×C6

Generators and relations
 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 >

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

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

(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 32)(8 33)(9 34)(10 35)(11 36)(12 31)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(55 71)(56 72)(57 67)(58 68)(59 69)(60 70)

(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(49 63)(50 64)(51 65)(52 66)(53 61)(54 62)(55 71)(56 72)(57 67)(58 68)(59 69)(60 70)

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

Matrix representation ►G ⊆ GL5(𝔽7)

10000
01000
00600
00060
00006
,
10000
06000
00600
00060
00006
,
50000
01000
00100
00010
00001
,
10000
01000
00100
00160
00106
,
10000
01000
00600
00060
00601
,
10000
01000
00150
00061
00060

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] >;
 

96 conjugacy classes

class 1 2A···2G2H···2O3A3B3C···3H6A···6N6O···6AD6AE···6BT
order12···22···2333···36···66···66···6
size11···13···3114···41···13···34···4

96 irreducible representations

dim1111113333
type++++
imageC1C2C3C3C6C6A4C2×A4C3×A4C6×A4
kernelA4×C22×C6A4×C2×C6C23×A4C24×C6C22×A4C23×C6C22×C6C2×C6C23C22
# reps1762421417214

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
 

׿
×
⋊
ℤ
𝔽
○
≀