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## G = C22×D6⋊S3order 288 = 25·32

### Direct product of C22 and D6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C22×D6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — D6⋊S3 — C2×D6⋊S3 — C22×D6⋊S3
 Lower central C32 — C3×C6 — C22×D6⋊S3
 Upper central C1 — C23

Generators and relations for C22×D6⋊S3
G = < a,b,c,d,e,f | a2=b2=c6=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, fdf=c3d, fef=e-1 >

Subgroups: 1618 in 499 conjugacy classes, 148 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×2], C3, C4 [×4], C22 [×7], C22 [×32], S3 [×8], C6 [×14], C6 [×15], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, Dic3 [×12], D6 [×8], D6 [×24], C2×C6 [×14], C2×C6 [×39], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×8], C3×C6, C3×C6 [×6], C2×Dic3 [×18], C3⋊D4 [×32], C22×S3 [×12], C22×S3 [×8], C22×C6 [×2], C22×C6 [×21], C22×D4, C3⋊Dic3 [×4], S3×C6 [×8], S3×C6 [×24], C62 [×7], C22×Dic3 [×3], C2×C3⋊D4 [×24], S3×C23 [×2], C23×C6 [×2], D6⋊S3 [×16], C2×C3⋊Dic3 [×6], S3×C2×C6 [×12], S3×C2×C6 [×8], C2×C62, C22×C3⋊D4 [×2], C2×D6⋊S3 [×12], C22×C3⋊Dic3, S3×C22×C6 [×2], C22×D6⋊S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, C3⋊D4 [×8], C22×S3 [×14], C22×D4, S32, C2×C3⋊D4 [×12], S3×C23 [×2], D6⋊S3 [×4], C2×S32 [×3], C22×C3⋊D4 [×2], C2×D6⋊S3 [×6], C22×S32, C22×D6⋊S3

Smallest permutation representation of C22×D6⋊S3
On 96 points
Generators in S96
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 64)(50 65)(51 66)(52 61)(53 62)(54 63)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 46)(8 45)(9 44)(10 43)(11 48)(12 47)(13 28)(14 27)(15 26)(16 25)(17 30)(18 29)(19 34)(20 33)(21 32)(22 31)(23 36)(24 35)(49 85)(50 90)(51 89)(52 88)(53 87)(54 86)(55 91)(56 96)(57 95)(58 94)(59 93)(60 92)(61 73)(62 78)(63 77)(64 76)(65 75)(66 74)(67 79)(68 84)(69 83)(70 82)(71 81)(72 80)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)

G:=sub<Sym(96)| (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,64)(50,65)(51,66)(52,61)(53,62)(54,63)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,88)(74,89)(75,90)(76,85)(77,86)(78,87)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,46)(8,45)(9,44)(10,43)(11,48)(12,47)(13,28)(14,27)(15,26)(16,25)(17,30)(18,29)(19,34)(20,33)(21,32)(22,31)(23,36)(24,35)(49,85)(50,90)(51,89)(52,88)(53,87)(54,86)(55,91)(56,96)(57,95)(58,94)(59,93)(60,92)(61,73)(62,78)(63,77)(64,76)(65,75)(66,74)(67,79)(68,84)(69,83)(70,82)(71,81)(72,80), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)>;

G:=Group( (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,64)(50,65)(51,66)(52,61)(53,62)(54,63)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,88)(74,89)(75,90)(76,85)(77,86)(78,87)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,46)(8,45)(9,44)(10,43)(11,48)(12,47)(13,28)(14,27)(15,26)(16,25)(17,30)(18,29)(19,34)(20,33)(21,32)(22,31)(23,36)(24,35)(49,85)(50,90)(51,89)(52,88)(53,87)(54,86)(55,91)(56,96)(57,95)(58,94)(59,93)(60,92)(61,73)(62,78)(63,77)(64,76)(65,75)(66,74)(67,79)(68,84)(69,83)(70,82)(71,81)(72,80), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78) );

G=PermutationGroup([(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,22),(8,23),(9,24),(10,19),(11,20),(12,21),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(49,64),(50,65),(51,66),(52,61),(53,62),(54,63),(55,70),(56,71),(57,72),(58,67),(59,68),(60,69),(73,88),(74,89),(75,90),(76,85),(77,86),(78,87),(79,94),(80,95),(81,96),(82,91),(83,92),(84,93)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,40),(2,39),(3,38),(4,37),(5,42),(6,41),(7,46),(8,45),(9,44),(10,43),(11,48),(12,47),(13,28),(14,27),(15,26),(16,25),(17,30),(18,29),(19,34),(20,33),(21,32),(22,31),(23,36),(24,35),(49,85),(50,90),(51,89),(52,88),(53,87),(54,86),(55,91),(56,96),(57,95),(58,94),(59,93),(60,92),(61,73),(62,78),(63,77),(64,76),(65,75),(66,74),(67,79),(68,84),(69,83),(70,82),(71,81),(72,80)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48),(49,51,53),(50,52,54),(55,57,59),(56,58,60),(61,63,65),(62,64,66),(67,69,71),(68,70,72),(73,77,75),(74,78,76),(79,83,81),(80,84,82),(85,89,87),(86,90,88),(91,95,93),(92,96,94)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78)])

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 4A 4B 4C 4D 6A ··· 6N 6O ··· 6U 6V ··· 6AK order 1 2 ··· 2 2 ··· 2 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 ··· 1 6 ··· 6 2 2 4 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + - + image C1 C2 C2 C2 S3 D4 D6 D6 C3⋊D4 S32 D6⋊S3 C2×S32 kernel C22×D6⋊S3 C2×D6⋊S3 C22×C3⋊Dic3 S3×C22×C6 S3×C23 C62 C22×S3 C22×C6 C2×C6 C23 C22 C22 # reps 1 12 1 2 2 4 12 2 16 1 4 3

Matrix representation of C22×D6⋊S3 in GL8(𝔽13)

 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 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 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
,
 12 0 0 0 0 0 0 0 0 12 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 12 12 0 0 0 0 0 0 1 0
,
 1 6 0 0 0 0 0 0 0 12 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 12 12 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 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 4 8 0 0 0 0 0 0 3 9 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 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,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,6,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,3,0,0,0,0,0,0,8,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C22×D6⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times D_6\rtimes S_3
% in TeX

G:=Group("C2^2xD6:S3");
// GroupNames label

G:=SmallGroup(288,973);
// by ID

G=gap.SmallGroup(288,973);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,1356,9414]);
// Polycyclic

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

׿
×
𝔽