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G = C62.10C23order 288 = 25·32

5th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: Dic32Dic6, C62.10C23, C32(C12⋊Q8), C6.45(S3×D4), Dic32.3C2, C324(C4⋊Q8), C6.15(S3×Q8), (C3×Dic3)⋊2Q8, (C2×C12).12D6, C2.7(S3×Dic6), C6.3(C2×Dic6), Dic3⋊C4.2S3, C3⋊Dic3.15D4, (C6×C12).1C22, (C2×Dic3).2D6, C2.6(Dic3⋊D6), Dic3⋊Dic3.7C2, (C6×Dic3).32C22, (C2×C4).12S32, (C3×C6).9(C2×Q8), C22.73(C2×S32), (C3×C6).77(C2×D4), (C3×Dic3⋊C4).4C2, (C2×C6).29(C22×S3), (C2×C322Q8).2C2, (C2×C324Q8).3C2, (C2×C3⋊Dic3).12C22, SmallGroup(288,488)

Series: Derived Chief Lower central Upper central

C1C62 — C62.10C23
C1C3C32C3×C6C62C6×Dic3Dic32 — C62.10C23
C32C62 — C62.10C23
C1C22C2×C4

Generators and relations for C62.10C23
 G = < a,b,c,d,e | a6=b6=1, c2=b3, d2=a3, e2=a3b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=a3d >

Subgroups: 546 in 155 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×10], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×4], Dic3 [×12], C12 [×10], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], Dic6 [×12], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C4⋊Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, C62, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×5], C322Q8 [×2], C6×Dic3 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C12⋊Q8 [×2], Dic32, Dic3⋊Dic3 [×2], C3×Dic3⋊C4 [×2], C2×C322Q8, C2×C324Q8, C62.10C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×4], C23, D6 [×6], C2×D4, C2×Q8 [×2], Dic6 [×4], C22×S3 [×2], C4⋊Q8, S32, C2×Dic6 [×2], S3×D4 [×2], S3×Q8 [×2], C2×S32, C12⋊Q8 [×2], S3×Dic6 [×2], Dic3⋊D6, C62.10C23

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

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

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

42 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H4I4J6A···6F6G6H6I12A···12H12I···12P
order122233344444444446···666612···1212···12
size11112244666612121818362···24444···412···12

42 irreducible representations

dim111111222222444444
type+++++++-+++-++-+-+
imageC1C2C2C2C2C2S3Q8D4D6D6Dic6S32S3×D4S3×Q8C2×S32S3×Dic6Dic3⋊D6
kernelC62.10C23Dic32Dic3⋊Dic3C3×Dic3⋊C4C2×C322Q8C2×C324Q8Dic3⋊C4C3×Dic3C3⋊Dic3C2×Dic3C2×C12Dic3C2×C4C6C6C22C2C2
# reps112211242428122142

Matrix representation of C62.10C23 in GL6(𝔽13)

1200000
0120000
0012100
0012000
000010
000001
,
1200000
0120000
001000
000100
0000112
000010
,
450000
790000
001000
000100
000050
000058
,
450000
790000
0001200
0012000
0000120
0000012
,
1100000
0120000
0012000
0001200
000037
0000610

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[4,7,0,0,0,0,5,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,0,8],[4,7,0,0,0,0,5,9,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,10,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10] >;

C62.10C23 in GAP, Magma, Sage, TeX

C_6^2._{10}C_2^3
% in TeX

G:=Group("C6^2.10C2^3");
// GroupNames label

G:=SmallGroup(288,488);
// by ID

G=gap.SmallGroup(288,488);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,422,135,142,1356,9414]);
// Polycyclic

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

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