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## G = C22×C33⋊C2order 216 = 23·33

### Direct product of C22 and C33⋊C2

Aliases: C22×C33⋊C2, C6214S3, C337C23, (C3×C6)⋊8D6, (C3×C62)⋊7C2, (C32×C6)⋊6C22, C329(C22×S3), C62(C2×C3⋊S3), (C2×C6)⋊5(C3⋊S3), C32(C22×C3⋊S3), SmallGroup(216,176)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×C33⋊C2
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 2164 in 448 conjugacy classes, 151 normal (5 characteristic)
C1, C2 [×3], C2 [×4], C3 [×13], C22, C22 [×6], S3 [×52], C6 [×39], C23, C32 [×13], D6 [×78], C2×C6 [×13], C3⋊S3 [×52], C3×C6 [×39], C22×S3 [×13], C33, C2×C3⋊S3 [×78], C62 [×13], C33⋊C2 [×4], C32×C6 [×3], C22×C3⋊S3 [×13], C2×C33⋊C2 [×6], C3×C62, C22×C33⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×13], C23, D6 [×39], C3⋊S3 [×13], C22×S3 [×13], C2×C3⋊S3 [×39], C33⋊C2, C22×C3⋊S3 [×13], C2×C33⋊C2 [×3], C22×C33⋊C2

Smallest permutation representation of C22×C33⋊C2
On 108 points
Generators in S108
(1 59)(2 60)(3 58)(4 74)(5 75)(6 73)(7 61)(8 62)(9 63)(10 57)(11 55)(12 56)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 66)(20 64)(21 65)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(1 32)(2 33)(3 31)(4 47)(5 48)(6 46)(7 34)(8 35)(9 36)(10 84)(11 82)(12 83)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 93)(20 91)(21 92)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(37 64)(38 65)(39 66)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)
(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)(97 98 99)(100 101 102)(103 104 105)(106 107 108)
(1 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 108 19)(11 106 20)(12 107 21)(25 37 28)(26 38 29)(27 39 30)(31 48 40)(32 46 41)(33 47 42)(34 51 43)(35 49 44)(36 50 45)(52 64 55)(53 65 56)(54 66 57)(58 75 67)(59 73 68)(60 74 69)(61 78 70)(62 76 71)(63 77 72)(79 91 82)(80 92 83)(81 93 84)(85 102 94)(86 100 95)(87 101 96)(88 105 97)(89 103 98)(90 104 99)
(1 26 8)(2 27 9)(3 25 7)(4 39 23)(5 37 24)(6 38 22)(10 99 96)(11 97 94)(12 98 95)(13 28 16)(14 29 17)(15 30 18)(19 104 101)(20 105 102)(21 103 100)(31 52 34)(32 53 35)(33 54 36)(40 55 43)(41 56 44)(42 57 45)(46 65 49)(47 66 50)(48 64 51)(58 79 61)(59 80 62)(60 81 63)(67 82 70)(68 83 71)(69 84 72)(73 92 76)(74 93 77)(75 91 78)(85 106 88)(86 107 89)(87 108 90)
(1 86)(2 85)(3 87)(4 94)(5 96)(6 95)(7 108)(8 107)(9 106)(10 24)(11 23)(12 22)(13 101)(14 100)(15 102)(16 19)(17 21)(18 20)(25 90)(26 89)(27 88)(28 104)(29 103)(30 105)(31 60)(32 59)(33 58)(34 81)(35 80)(36 79)(37 99)(38 98)(39 97)(40 74)(41 73)(42 75)(43 93)(44 92)(45 91)(46 68)(47 67)(48 69)(49 83)(50 82)(51 84)(52 63)(53 62)(54 61)(55 77)(56 76)(57 78)(64 72)(65 71)(66 70)

G:=sub<Sym(108)| (1,59)(2,60)(3,58)(4,74)(5,75)(6,73)(7,61)(8,62)(9,63)(10,57)(11,55)(12,56)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,66)(20,64)(21,65)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,32)(2,33)(3,31)(4,47)(5,48)(6,46)(7,34)(8,35)(9,36)(10,84)(11,82)(12,83)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,93)(20,91)(21,92)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(37,64)(38,65)(39,66)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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)(97,98,99)(100,101,102)(103,104,105)(106,107,108), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,108,19)(11,106,20)(12,107,21)(25,37,28)(26,38,29)(27,39,30)(31,48,40)(32,46,41)(33,47,42)(34,51,43)(35,49,44)(36,50,45)(52,64,55)(53,65,56)(54,66,57)(58,75,67)(59,73,68)(60,74,69)(61,78,70)(62,76,71)(63,77,72)(79,91,82)(80,92,83)(81,93,84)(85,102,94)(86,100,95)(87,101,96)(88,105,97)(89,103,98)(90,104,99), (1,26,8)(2,27,9)(3,25,7)(4,39,23)(5,37,24)(6,38,22)(10,99,96)(11,97,94)(12,98,95)(13,28,16)(14,29,17)(15,30,18)(19,104,101)(20,105,102)(21,103,100)(31,52,34)(32,53,35)(33,54,36)(40,55,43)(41,56,44)(42,57,45)(46,65,49)(47,66,50)(48,64,51)(58,79,61)(59,80,62)(60,81,63)(67,82,70)(68,83,71)(69,84,72)(73,92,76)(74,93,77)(75,91,78)(85,106,88)(86,107,89)(87,108,90), (1,86)(2,85)(3,87)(4,94)(5,96)(6,95)(7,108)(8,107)(9,106)(10,24)(11,23)(12,22)(13,101)(14,100)(15,102)(16,19)(17,21)(18,20)(25,90)(26,89)(27,88)(28,104)(29,103)(30,105)(31,60)(32,59)(33,58)(34,81)(35,80)(36,79)(37,99)(38,98)(39,97)(40,74)(41,73)(42,75)(43,93)(44,92)(45,91)(46,68)(47,67)(48,69)(49,83)(50,82)(51,84)(52,63)(53,62)(54,61)(55,77)(56,76)(57,78)(64,72)(65,71)(66,70)>;

G:=Group( (1,59)(2,60)(3,58)(4,74)(5,75)(6,73)(7,61)(8,62)(9,63)(10,57)(11,55)(12,56)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,66)(20,64)(21,65)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,32)(2,33)(3,31)(4,47)(5,48)(6,46)(7,34)(8,35)(9,36)(10,84)(11,82)(12,83)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,93)(20,91)(21,92)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(37,64)(38,65)(39,66)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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)(97,98,99)(100,101,102)(103,104,105)(106,107,108), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,108,19)(11,106,20)(12,107,21)(25,37,28)(26,38,29)(27,39,30)(31,48,40)(32,46,41)(33,47,42)(34,51,43)(35,49,44)(36,50,45)(52,64,55)(53,65,56)(54,66,57)(58,75,67)(59,73,68)(60,74,69)(61,78,70)(62,76,71)(63,77,72)(79,91,82)(80,92,83)(81,93,84)(85,102,94)(86,100,95)(87,101,96)(88,105,97)(89,103,98)(90,104,99), (1,26,8)(2,27,9)(3,25,7)(4,39,23)(5,37,24)(6,38,22)(10,99,96)(11,97,94)(12,98,95)(13,28,16)(14,29,17)(15,30,18)(19,104,101)(20,105,102)(21,103,100)(31,52,34)(32,53,35)(33,54,36)(40,55,43)(41,56,44)(42,57,45)(46,65,49)(47,66,50)(48,64,51)(58,79,61)(59,80,62)(60,81,63)(67,82,70)(68,83,71)(69,84,72)(73,92,76)(74,93,77)(75,91,78)(85,106,88)(86,107,89)(87,108,90), (1,86)(2,85)(3,87)(4,94)(5,96)(6,95)(7,108)(8,107)(9,106)(10,24)(11,23)(12,22)(13,101)(14,100)(15,102)(16,19)(17,21)(18,20)(25,90)(26,89)(27,88)(28,104)(29,103)(30,105)(31,60)(32,59)(33,58)(34,81)(35,80)(36,79)(37,99)(38,98)(39,97)(40,74)(41,73)(42,75)(43,93)(44,92)(45,91)(46,68)(47,67)(48,69)(49,83)(50,82)(51,84)(52,63)(53,62)(54,61)(55,77)(56,76)(57,78)(64,72)(65,71)(66,70) );

G=PermutationGroup([(1,59),(2,60),(3,58),(4,74),(5,75),(6,73),(7,61),(8,62),(9,63),(10,57),(11,55),(12,56),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,66),(20,64),(21,65),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108)], [(1,32),(2,33),(3,31),(4,47),(5,48),(6,46),(7,34),(8,35),(9,36),(10,84),(11,82),(12,83),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,93),(20,91),(21,92),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(37,64),(38,65),(39,66),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108)], [(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),(97,98,99),(100,101,102),(103,104,105),(106,107,108)], [(1,6,14),(2,4,15),(3,5,13),(7,24,16),(8,22,17),(9,23,18),(10,108,19),(11,106,20),(12,107,21),(25,37,28),(26,38,29),(27,39,30),(31,48,40),(32,46,41),(33,47,42),(34,51,43),(35,49,44),(36,50,45),(52,64,55),(53,65,56),(54,66,57),(58,75,67),(59,73,68),(60,74,69),(61,78,70),(62,76,71),(63,77,72),(79,91,82),(80,92,83),(81,93,84),(85,102,94),(86,100,95),(87,101,96),(88,105,97),(89,103,98),(90,104,99)], [(1,26,8),(2,27,9),(3,25,7),(4,39,23),(5,37,24),(6,38,22),(10,99,96),(11,97,94),(12,98,95),(13,28,16),(14,29,17),(15,30,18),(19,104,101),(20,105,102),(21,103,100),(31,52,34),(32,53,35),(33,54,36),(40,55,43),(41,56,44),(42,57,45),(46,65,49),(47,66,50),(48,64,51),(58,79,61),(59,80,62),(60,81,63),(67,82,70),(68,83,71),(69,84,72),(73,92,76),(74,93,77),(75,91,78),(85,106,88),(86,107,89),(87,108,90)], [(1,86),(2,85),(3,87),(4,94),(5,96),(6,95),(7,108),(8,107),(9,106),(10,24),(11,23),(12,22),(13,101),(14,100),(15,102),(16,19),(17,21),(18,20),(25,90),(26,89),(27,88),(28,104),(29,103),(30,105),(31,60),(32,59),(33,58),(34,81),(35,80),(36,79),(37,99),(38,98),(39,97),(40,74),(41,73),(42,75),(43,93),(44,92),(45,91),(46,68),(47,67),(48,69),(49,83),(50,82),(51,84),(52,63),(53,62),(54,61),(55,77),(56,76),(57,78),(64,72),(65,71),(66,70)])

C22×C33⋊C2 is a maximal subgroup of   C62.79D6  C62.148D6  C6223D6  C22×S3×C3⋊S3
C22×C33⋊C2 is a maximal quotient of   C62.160D6  C62.100D6  (Q8×C33)⋊C2

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3M 6A ··· 6AM order 1 2 2 2 2 2 2 2 3 ··· 3 6 ··· 6 size 1 1 1 1 27 27 27 27 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 S3 D6 kernel C22×C33⋊C2 C2×C33⋊C2 C3×C62 C62 C3×C6 # reps 1 6 1 13 39

Matrix representation of C22×C33⋊C2 in GL6(ℤ)

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

G:=sub<GL(6,Integers())| [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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,-1] >;

C22×C33⋊C2 in GAP, Magma, Sage, TeX

C_2^2\times C_3^3\rtimes C_2
% in TeX

G:=Group("C2^2xC3^3:C2");
// GroupNames label

G:=SmallGroup(216,176);
// by ID

G=gap.SmallGroup(216,176);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,387,1444,5189]);
// Polycyclic

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

׿
×
𝔽