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G = C62.49C23order 288 = 25·32

44th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.49C23, D64(C4×S3), C322(C4×D4), D6⋊C419S3, Dic3217C2, D6⋊S35C4, C6.137(S3×D4), D6⋊Dic319C2, (C3×Dic3)⋊13D4, (C2×C12).196D6, C6.42(C4○D12), Dic36(C3⋊D4), (C2×Dic3).63D6, (C22×S3).31D6, C35(Dic34D4), C2.4(D125S3), C6.20(D42S3), (C6×C12).227C22, C6.Dic616C2, (C6×Dic3).152C22, (S3×C2×C4)⋊8S3, C2.15(C4×S32), (C2×C4).47S32, C33(C4×C3⋊D4), C6.14(S3×C2×C4), (S3×C6)⋊1(C2×C4), (S3×C2×C12)⋊16C2, (C2×S3×Dic3)⋊7C2, C2.2(S3×C3⋊D4), (C3×D6⋊C4)⋊20C2, C3⋊Dic32(C2×C4), C22.29(C2×S32), (C3×C6).91(C2×D4), C6.30(C2×C3⋊D4), (S3×C2×C6).12C22, (C2×D6⋊S3).2C2, (C3×C6).29(C4○D4), (C3×C6).13(C22×C4), (C2×C6).68(C22×S3), (C2×C3⋊Dic3).36C22, SmallGroup(288,527)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.49C23
C1C3C32C3×C6C62S3×C2×C6C2×S3×Dic3 — C62.49C23
C32C3×C6 — C62.49C23
C1C22C2×C4

Generators and relations for C62.49C23
 G = < a,b,c,d,e | a6=b6=c2=1, d2=e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3b3c, de=ed >

Subgroups: 682 in 201 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×7], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×2], Dic3 [×10], C12 [×6], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], C2×Dic3 [×2], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C22×S3 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×4], S3×C6 [×4], C62, C4×Dic3 [×2], Dic3⋊C4 [×3], D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4 [×2], C22×C12, S3×Dic3 [×2], D6⋊S3 [×4], S3×C12 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], Dic34D4, C4×C3⋊D4, Dic32, D6⋊Dic3, C3×D6⋊C4, C6.Dic6, C2×S3×Dic3, C2×D6⋊S3, S3×C2×C12, C62.49C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×4], C3⋊D4 [×2], C22×S3 [×2], C4×D4, S32, S3×C2×C4 [×2], C4○D12, S3×D4, D42S3, C2×C3⋊D4, C2×S32, Dic34D4, C4×C3⋊D4, D125S3, C4×S32, S3×C3⋊D4, C62.49C23

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G4H4I4J4K4L6A···6F6G6H6I6J6K6L6M6N6O12A12B12C12D12E···12J12K12L12M12N12O12P
order122222223334444444444446···66666666661212121212···12121212121212
size1111666622422333366181818182···24446666121222224···466661212

54 irreducible representations

dim11111111122222222224444444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C4S3S3D4D6D6D6C4○D4C3⋊D4C4×S3C4○D12S32S3×D4D42S3C2×S32D125S3C4×S32S3×C3⋊D4
kernelC62.49C23Dic32D6⋊Dic3C3×D6⋊C4C6.Dic6C2×S3×Dic3C2×D6⋊S3S3×C2×C12D6⋊S3D6⋊C4S3×C2×C4C3×Dic3C2×Dic3C2×C12C22×S3C3×C6Dic3D6C6C2×C4C6C6C22C2C2C2
# reps11111111811222224841111222

Matrix representation of C62.49C23 in GL6(𝔽13)

1200000
0120000
0012000
0001200
0000121
0000120
,
100000
010000
0011200
001000
000010
000001
,
0120000
1200000
0011200
0001200
000010
000001
,
800000
080000
005000
000500
000001
000010
,
010000
1200000
005000
000500
000010
000001

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C62.49C23 in GAP, Magma, Sage, TeX

C_6^2._{49}C_2^3
% in TeX

G:=Group("C6^2.49C2^3");
// GroupNames label

G:=SmallGroup(288,527);
// by ID

G=gap.SmallGroup(288,527);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,58,1356,9414]);
// Polycyclic

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

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