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G = C22⋊C4×C3⋊S3  order 288 = 25·32

Direct product of C22⋊C4 and C3⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C22⋊C4×C3⋊S3, C62.224C23, (C2×C12)⋊12D6, C62⋊12(C2×C4), C6.106(S3×D4), (C6×C12)⋊22C22, (C22×C6).86D6, C62⋊5C4⋊11C2, C6.11D12⋊19C2, (C2×C62).63C22, (C2×C6)⋊9(C4×S3), C6.66(S3×C2×C4), C2.1(D4×C3⋊S3), C22⋊4(C4×C3⋊S3), C3⋊3(S3×C22⋊C4), (C22×C3⋊S3)⋊6C4, (C3×C22⋊C4)⋊8S3, (C2×C3⋊S3).70D4, (C3×C6).229(C2×D4), (C23×C3⋊S3).3C2, C23.18(C2×C3⋊S3), C32⋊10(C2×C22⋊C4), (C3×C6).97(C22×C4), (C32×C22⋊C4)⋊16C2, (C2×C6).241(C22×S3), (C2×C3⋊Dic3)⋊17C22, C22.13(C22×C3⋊S3), (C22×C3⋊S3).81C22, C2.8(C2×C4×C3⋊S3), (C2×C4×C3⋊S3)⋊17C2, (C2×C4)⋊5(C2×C3⋊S3), (C2×C3⋊S3)⋊16(C2×C4), SmallGroup(288,737)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C22⋊C4×C3⋊S3
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C23×C3⋊S3 — C22⋊C4×C3⋊S3
Lower central
C32 — C3×C6 — C22⋊C4×C3⋊S3
Upper central
C1 — C22 — C22⋊C4

Generators and relations for C22⋊C4×C3⋊S3
 G = < a,b,c,d,e,f | a2=b2=c4=d3=e3=f2=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1700 in 396 conjugacy classes, 93 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C24, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C2×C22⋊C4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C22×C3⋊S3, C22×C3⋊S3, C2×C62, S3×C22⋊C4, C6.11D12, C62⋊5C4, C32×C22⋊C4, C2×C4×C3⋊S3, C23×C3⋊S3, C22⋊C4×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C3⋊S3, C4×S3, C22×S3, C2×C22⋊C4, C2×C3⋊S3, S3×C2×C4, S3×D4, C4×C3⋊S3, C22×C3⋊S3, S3×C22⋊C4, C2×C4×C3⋊S3, D4×C3⋊S3, C22⋊C4×C3⋊S3

Smallest permutation representation of C22⋊C4×C3⋊S3
►On 72 points
Generators in S72
(1 3)(2 39)(4 37)(5 7)(6 30)(8 32)(9 11)(10 50)(12 52)(13 15)(14 64)(16 62)(17 46)(18 20)(19 48)(21 23)(22 34)(24 36)(25 27)(26 55)(28 53)(29 31)(33 35)(38 40)(41 43)(42 66)(44 68)(45 47)(49 51)(54 56)(57 59)(58 70)(60 72)(61 63)(65 67)(69 71)

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

(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)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B3C3D4A4B4C4D4E4F4G4H6A···6L6M···6T12A···12P
order1222222222223333444444446···66···612···12
size1111229999181822222222181818182···24···44···4

60 irreducible representations

dim1111111222224
type+++++++++++
imageC1C2C2C2C2C2C4S3D4D6D6C4×S3S3×D4
kernelC22⋊C4×C3⋊S3C6.11D12C62⋊5C4C32×C22⋊C4C2×C4×C3⋊S3C23×C3⋊S3C22×C3⋊S3C3×C22⋊C4C2×C3⋊S3C2×C12C22×C6C2×C6C6
# reps12112184484168

Matrix representation of C22⋊C4×C3⋊S3 ►in GL6(𝔽13)

100000
010000
0012000
0001200
000010
0000012
,
100000
010000
001000
000100
0000120
0000012
,
1200000
0120000
008000
000800
000001
000010
,
100000
010000
000100
00121200
000010
000001
,
0120000
1120000
001000
000100
000010
000001
,
010000
100000
001000
00121200
000010
000001

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C22⋊C4×C3⋊S3 in GAP, Magma, Sage, TeX 

C_2^2\rtimes C_4\times C_3\rtimes S_3
 
% in TeX
 
G:=Group("C2^2:C4xC3:S3");
 
// GroupNames label
 
G:=SmallGroup(288,737);
 
// by ID
 
G=gap.SmallGroup(288,737);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^3=e^3=f^2=1,c*a*c^-1=a*b=b*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,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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