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G = C62.55C23order 288 = 25·32

50th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.55C23, (S3×C6)⋊1D4, D6⋊C415S3, (C2×D12)⋊2S3, C3⋊Dic31D4, C6.36(S3×D4), (C6×D12)⋊21C2, D61(C3⋊D4), C35(Dic3⋊D4), D6⋊Dic320C2, (C2×C12).226D6, C323(C4⋊D4), C6.44(C4○D12), (C2×Dic3).67D6, C6.Dic69C2, (C22×S3).10D6, C32(C23.14D6), C2.13(D6⋊D6), C6.22(D42S3), (C6×C12).180C22, C2.14(D125S3), (C6×Dic3).61C22, (C2×C4).21S32, (C3×D6⋊C4)⋊12C2, (C2×S3×Dic3)⋊11C2, (C3×C6).44(C2×D4), C2.12(S3×C3⋊D4), C6.32(C2×C3⋊D4), (C2×D6⋊S3)⋊1C2, C22.102(C2×S32), (S3×C2×C6).16C22, (C3×C6).32(C4○D4), (C2×C6).74(C22×S3), (C2×C3⋊Dic3).39C22, SmallGroup(288,533)

Series: Derived Chief Lower central Upper central

C1C62 — C62.55C23
C1C3C32C3×C6C62S3×C2×C6C2×S3×Dic3 — C62.55C23
C32C62 — C62.55C23
C1C22C2×C4

Generators and relations for C62.55C23
 G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, e2=a3b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3b3c, ede-1=b3d >

Subgroups: 810 in 201 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×10], C12 [×4], D6 [×2], D6 [×8], C2×C6 [×2], C2×C6 [×11], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×4], C3×C6 [×3], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C22×S3 [×3], C22×C6 [×3], C4⋊D4, C3×Dic3, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×8], C62, Dic3⋊C4 [×3], D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4 [×4], C6×D4, S3×Dic3 [×2], D6⋊S3 [×4], C3×D12 [×2], C6×Dic3, C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×3], Dic3⋊D4, C23.14D6, D6⋊Dic3, C3×D6⋊C4, C6.Dic6, C2×S3×Dic3, C2×D6⋊S3 [×2], C6×D12, C62.55C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C4○D12, S3×D4 [×3], D42S3, C2×C3⋊D4, C2×S32, Dic3⋊D4, C23.14D6, D125S3, D6⋊D6, S3×C3⋊D4, C62.55C23

Smallest permutation representation of C62.55C23
On 96 points
Generators in S96
(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 18 5 16 3 14)(2 13 6 17 4 15)(7 95 11 93 9 91)(8 96 12 94 10 92)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)(43 53 45 49 47 51)(44 54 46 50 48 52)(55 63 57 65 59 61)(56 64 58 66 60 62)(67 74 69 76 71 78)(68 75 70 77 72 73)(79 86 83 90 81 88)(80 87 84 85 82 89)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 33 4 36)(2 32 5 35)(3 31 6 34)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)(25 53 28 50)(26 52 29 49)(27 51 30 54)(55 80 58 83)(56 79 59 82)(57 84 60 81)(61 89 64 86)(62 88 65 85)(63 87 66 90)(67 92 70 95)(68 91 71 94)(69 96 72 93)
(1 30 13 24)(2 25 14 19)(3 26 15 20)(4 27 16 21)(5 28 17 22)(6 29 18 23)(7 89 96 81)(8 90 91 82)(9 85 92 83)(10 86 93 84)(11 87 94 79)(12 88 95 80)(31 46 39 49)(32 47 40 50)(33 48 41 51)(34 43 42 52)(35 44 37 53)(36 45 38 54)(55 67 62 73)(56 68 63 74)(57 69 64 75)(58 70 65 76)(59 71 66 77)(60 72 61 78)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J···6O12A···12H12I12J
order122222223334444446···66666···612···121212
size11116612122244661818362···244412···124···41212

42 irreducible representations

dim111111122222222224444444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4C3⋊D4C4○D12S32S3×D4D42S3C2×S32D125S3D6⋊D6S3×C3⋊D4
kernelC62.55C23D6⋊Dic3C3×D6⋊C4C6.Dic6C2×S3×Dic3C2×D6⋊S3C6×D12D6⋊C4C2×D12C3⋊Dic3S3×C6C2×Dic3C2×C12C22×S3C3×C6D6C6C2×C4C6C6C22C2C2C2
# reps111112111221232441311222

Matrix representation of C62.55C23 in GL6(𝔽13)

100000
010000
0012100
0012000
0000120
0000012
,
1200000
0120000
001000
000100
000030
000009
,
1110000
1020000
001000
000100
0000010
000040
,
2120000
3110000
0001200
0012000
000050
000005
,
460000
890000
0012000
0001200
000080
000005

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[11,10,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,10,0],[2,3,0,0,0,0,12,11,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[4,8,0,0,0,0,6,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5] >;

C62.55C23 in GAP, Magma, Sage, TeX

C_6^2._{55}C_2^3
% in TeX

G:=Group("C6^2.55C2^3");
// GroupNames label

G:=SmallGroup(288,533);
// by ID

G=gap.SmallGroup(288,533);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,142,1356,9414]);
// Polycyclic

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

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