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## G = C22×S3×C3⋊S3order 432 = 24·33

### Direct product of C22, S3 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C22×S3×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — S3×C32 — S3×C3⋊S3 — C2×S3×C3⋊S3 — C22×S3×C3⋊S3
 Lower central C33 — C22×S3×C3⋊S3
 Upper central C1 — C22

Generators and relations for C22×S3×C3⋊S3
G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e3=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, dcd=c-1, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, geg=e-1, gfg=f-1 >

Subgroups: 5464 in 932 conjugacy classes, 188 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C22, C22, S3, S3, C6, C6, C23, C32, C32, C32, D6, D6, C2×C6, C2×C6, C2×C6, C24, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22×S3, C22×C6, C33, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×C23, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, C2×S32, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, C22×C3⋊S3, C2×C62, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C3×C62, C22×S32, C23×C3⋊S3, C2×S3×C3⋊S3, S3×C62, C2×C6×C3⋊S3, C22×C33⋊C2, C22×S3×C3⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C24, C3⋊S3, C22×S3, S32, C2×C3⋊S3, S3×C23, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C22×S32, C23×C3⋊S3, C2×S3×C3⋊S3, C22×S3×C3⋊S3

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A ··· 3E 3F 3G 3H 3I 6A ··· 6O 6P ··· 6AA 6AB ··· 6AQ 6AR 6AS 6AT 6AU order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 size 1 1 1 1 3 3 3 3 9 9 9 9 27 27 27 27 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D6 D6 D6 S32 C2×S32 kernel C22×S3×C3⋊S3 C2×S3×C3⋊S3 S3×C62 C2×C6×C3⋊S3 C22×C33⋊C2 S3×C2×C6 C22×C3⋊S3 S3×C6 C2×C3⋊S3 C62 C2×C6 C6 # reps 1 12 1 1 1 4 1 24 6 5 4 12

Matrix representation of C22×S3×C3⋊S3 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C22×S3×C3⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times C_3\rtimes S_3
% in TeX

G:=Group("C2^2xS3xC3:S3");
// GroupNames label

G:=SmallGroup(432,768);
// by ID

G=gap.SmallGroup(432,768);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,571,2028,14118]);
// Polycyclic

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

׿
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