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G = C62.28C23order 288 = 25·32

23rd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.28C23, Dic3213C2, D6⋊C4.7S3, C4⋊Dic35S3, (C2×C12).17D6, D6⋊Dic3.4C2, (C22×S3).4D6, C6.40(C4○D12), (C2×Dic3).58D6, C6.Dic67C2, Dic3⋊Dic312C2, C6.36(D42S3), C2.8(D125S3), C2.9(D12⋊S3), (C6×C12).178C22, C6.24(Q83S3), C322(C422C2), C34(C23.8D6), (C6×Dic3).7C22, C2.13(D6.3D6), (C2×C4).18S32, (C3×D6⋊C4).6C2, (C3×C4⋊Dic3)⋊9C2, C22.85(C2×S32), C33(C4⋊C4⋊S3), (S3×C2×C6).4C22, (C3×C6).60(C4○D4), (C2×C6).47(C22×S3), (C2×C3⋊Dic3).26C22, SmallGroup(288,506)

Series: Derived Chief Lower central Upper central

C1C62 — C62.28C23
C1C3C32C3×C6C62S3×C2×C6D6⋊Dic3 — C62.28C23
C32C62 — C62.28C23
C1C22C2×C4

Generators and relations for C62.28C23
 G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ece-1=a3c, ede-1=b3d >

Subgroups: 458 in 133 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3 [×2], C3, C4 [×6], C22, C22 [×3], S3, C6 [×6], C6 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×10], C12 [×6], D6 [×3], C2×C6 [×2], C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×S3, C3×C6 [×3], C2×Dic3 [×3], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C422C2, C3×Dic3 [×3], C3⋊Dic3 [×2], C3×C12, S3×C6 [×3], C62, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4 [×2], C6.D4 [×2], C3×C22⋊C4, C3×C4⋊C4, C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, C23.8D6, C4⋊C4⋊S3, Dic32, D6⋊Dic3 [×2], Dic3⋊Dic3, C3×C4⋊Dic3, C3×D6⋊C4, C6.Dic6, C62.28C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4 [×3], C22×S3 [×2], C422C2, S32, C4○D12 [×2], D42S3 [×3], Q83S3, C2×S32, C23.8D6, C4⋊C4⋊S3, D125S3, D12⋊S3, D6.3D6, C62.28C23

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E4F4G4H4I6A···6F6G6H6I6J6K12A···12H12I···12N
order122223334444444446···66666612···1212···12
size11111222446666121818362···244412124···412···12

42 irreducible representations

dim111111122222224444444
type+++++++++++++-++-
imageC1C2C2C2C2C2C2S3S3D6D6D6C4○D4C4○D12S32D42S3Q83S3C2×S32D125S3D12⋊S3D6.3D6
kernelC62.28C23Dic32D6⋊Dic3Dic3⋊Dic3C3×C4⋊Dic3C3×D6⋊C4C6.Dic6C4⋊Dic3D6⋊C4C2×Dic3C2×C12C22×S3C3×C6C6C2×C4C6C6C22C2C2C2
# reps112111111321681311222

Matrix representation of C62.28C23 in GL6(𝔽13)

1200000
0120000
000100
00121200
000010
000001
,
1200000
0120000
0012000
0001200
0000121
0000120
,
100000
0120000
0011900
004200
000001
000010
,
050000
500000
0031000
0071000
000010
000001
,
010000
1200000
0010700
006300
0000120
0000012

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C62.28C23 in GAP, Magma, Sage, TeX

C_6^2._{28}C_2^3
% in TeX

G:=Group("C6^2.28C2^3");
// GroupNames label

G:=SmallGroup(288,506);
// by ID

G=gap.SmallGroup(288,506);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,219,100,1356,9414]);
// Polycyclic

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

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