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G = C62.42C23order 288 = 25·32

37th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C12.12Dic6, C62.42C23, (C3×C12).14Q8, (C2×C12).134D6, C4⋊Dic3.12S3, C6.21(C2×Dic6), C6.6(D42S3), (C6×C12).98C22, (C2×Dic3).20D6, C34(C4.Dic6), C4.6(C322Q8), C328(C42.C2), Dic3⋊Dic3.2C2, C6.28(Q83S3), C2.13(D12⋊S3), (C6×Dic3).12C22, (C2×C4).115S32, C22.99(C2×S32), (C3×C6).37(C2×Q8), (C4×C3⋊Dic3).2C2, C2.5(C2×C322Q8), (C3×C6).26(C4○D4), (C3×C4⋊Dic3).15C2, (C2×C6).61(C22×S3), (C2×C3⋊Dic3).123C22, SmallGroup(288,520)

Series: Derived Chief Lower central Upper central

C1C62 — C62.42C23
C1C3C32C3×C6C62C6×Dic3Dic3⋊Dic3 — C62.42C23
C32C62 — C62.42C23
C1C22C2×C4

Generators and relations for C62.42C23
 G = < a,b,c,d,e | a6=b6=1, c2=a3, d2=a3b3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 394 in 127 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×6], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], C32, Dic3 [×12], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×6], C3×C6, C3×C6 [×2], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C42.C2, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C62, C4×Dic3 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4⋊Dic3 [×4], C3×C4⋊C4 [×2], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C4.Dic6 [×2], Dic3⋊Dic3 [×4], C3×C4⋊Dic3 [×2], C4×C3⋊Dic3, C62.42C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], Q8 [×2], C23, D6 [×6], C2×Q8, C4○D4 [×2], Dic6 [×4], C22×S3 [×2], C42.C2, S32, C2×Dic6 [×2], D42S3 [×2], Q83S3 [×2], C322Q8 [×2], C2×S32, C4.Dic6 [×2], D12⋊S3 [×2], C2×C322Q8, C62.42C23

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

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

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

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

42 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H4I4J6A···6F6G6H6I12A···12H12I···12P
order122233344444444446···666612···1212···12
size11112242212121212181818182···24444···412···12

42 irreducible representations

dim1111222222444444
type+++++-++-+-+-+
imageC1C2C2C2S3Q8D6D6C4○D4Dic6S32D42S3Q83S3C322Q8C2×S32D12⋊S3
kernelC62.42C23Dic3⋊Dic3C3×C4⋊Dic3C4×C3⋊Dic3C4⋊Dic3C3×C12C2×Dic3C2×C12C3×C6C12C2×C4C6C6C4C22C2
# reps1421224248122214

Matrix representation of C62.42C23 in GL6(𝔽13)

100000
010000
0012100
0012000
0000120
0000012
,
1200000
0120000
001000
000100
0000121
0000120
,
080000
500000
0012000
0001200
000022
0000411
,
010000
1200000
0001200
0012000
0000106
000073
,
800000
050000
0012000
0001200
0000120
0000012

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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,4,0,0,0,0,2,11],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C62.42C23 in GAP, Magma, Sage, TeX

C_6^2._{42}C_2^3
% in TeX

G:=Group("C6^2.42C2^3");
// GroupNames label

G:=SmallGroup(288,520);
// by ID

G=gap.SmallGroup(288,520);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,176,422,219,100,1356,9414]);
// Polycyclic

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

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