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G = C62.35C23order 288 = 25·32

30th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.35C23, C6.5(S3×Q8), C6.47(S3×D4), Dic3⋊C411S3, C31(D6⋊Q8), (C2×C12).191D6, C3⋊Dic3.55D4, C6.25(C4○D12), C326(C22⋊Q8), C2.8(Dic3⋊D6), (C2×Dic3).13D6, (C6×C12).218C22, C62.C2222C2, C6.D12.3C2, C2.7(Dic3.D6), C2.12(D6.D6), (C6×Dic3).77C22, (C2×C4).91S32, (C2×C3⋊S3)⋊5Q8, C22.92(C2×S32), (C3×C6).86(C2×D4), (C3×C6).19(C2×Q8), (C3×Dic3⋊C4)⋊8C2, (C2×C322Q8)⋊3C2, (C3×C6).22(C4○D4), (C2×C6).54(C22×S3), (C22×C3⋊S3).65C22, (C2×C3⋊Dic3).120C22, (C2×C4×C3⋊S3).17C2, SmallGroup(288,513)

Series: Derived Chief Lower central Upper central

C1C62 — C62.35C23
C1C3C32C3×C6C62C6×Dic3C62.C22 — C62.35C23
C32C62 — C62.35C23
C1C22C2×C4

Generators and relations for C62.35C23
 G = < a,b,c,d,e | a6=b6=1, c2=d2=b3, e2=a3, 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: 722 in 175 conjugacy classes, 48 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×10], C12 [×8], D6 [×14], C2×C6 [×2], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3⋊S3 [×2], C3×C6 [×3], Dic6 [×4], C4×S3 [×8], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C22⋊Q8, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, Dic3⋊C4 [×2], Dic3⋊C4 [×2], D6⋊C4 [×4], C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4 [×3], C322Q8 [×2], C6×Dic3 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊Q8 [×2], C6.D12 [×2], C62.C22, C3×Dic3⋊C4 [×2], C2×C322Q8, C2×C4×C3⋊S3, C62.35C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, C22×S3 [×2], C22⋊Q8, S32, C4○D12 [×2], S3×D4 [×2], S3×Q8 [×2], C2×S32, D6⋊Q8 [×2], Dic3.D6, D6.D6, Dic3⋊D6, C62.35C23

Smallest permutation representation of C62.35C23
On 48 points
Generators in S48
(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)
(1 18 5 16 3 14)(2 13 6 17 4 15)(7 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 48 16 10)(2 43 17 11)(3 44 18 12)(4 45 13 7)(5 46 14 8)(6 47 15 9)(19 34 28 39)(20 35 29 40)(21 36 30 41)(22 31 25 42)(23 32 26 37)(24 33 27 38)
(1 41 16 36)(2 40 17 35)(3 39 18 34)(4 38 13 33)(5 37 14 32)(6 42 15 31)(7 30 45 21)(8 29 46 20)(9 28 47 19)(10 27 48 24)(11 26 43 23)(12 25 44 22)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 33 10 36)(8 34 11 31)(9 35 12 32)(13 27 16 30)(14 28 17 25)(15 29 18 26)(37 47 40 44)(38 48 41 45)(39 43 42 46)

G:=sub<Sym(48)| (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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,43,9,45,11,47)(8,44,10,46,12,48)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39), (1,48,16,10)(2,43,17,11)(3,44,18,12)(4,45,13,7)(5,46,14,8)(6,47,15,9)(19,34,28,39)(20,35,29,40)(21,36,30,41)(22,31,25,42)(23,32,26,37)(24,33,27,38), (1,41,16,36)(2,40,17,35)(3,39,18,34)(4,38,13,33)(5,37,14,32)(6,42,15,31)(7,30,45,21)(8,29,46,20)(9,28,47,19)(10,27,48,24)(11,26,43,23)(12,25,44,22), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,33,10,36)(8,34,11,31)(9,35,12,32)(13,27,16,30)(14,28,17,25)(15,29,18,26)(37,47,40,44)(38,48,41,45)(39,43,42,46)>;

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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,43,9,45,11,47)(8,44,10,46,12,48)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39), (1,48,16,10)(2,43,17,11)(3,44,18,12)(4,45,13,7)(5,46,14,8)(6,47,15,9)(19,34,28,39)(20,35,29,40)(21,36,30,41)(22,31,25,42)(23,32,26,37)(24,33,27,38), (1,41,16,36)(2,40,17,35)(3,39,18,34)(4,38,13,33)(5,37,14,32)(6,42,15,31)(7,30,45,21)(8,29,46,20)(9,28,47,19)(10,27,48,24)(11,26,43,23)(12,25,44,22), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,33,10,36)(8,34,11,31)(9,35,12,32)(13,27,16,30)(14,28,17,25)(15,29,18,26)(37,47,40,44)(38,48,41,45)(39,43,42,46) );

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)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,43,9,45,11,47),(8,44,10,46,12,48),(19,30,23,28,21,26),(20,25,24,29,22,27),(31,40,33,42,35,38),(32,41,34,37,36,39)], [(1,48,16,10),(2,43,17,11),(3,44,18,12),(4,45,13,7),(5,46,14,8),(6,47,15,9),(19,34,28,39),(20,35,29,40),(21,36,30,41),(22,31,25,42),(23,32,26,37),(24,33,27,38)], [(1,41,16,36),(2,40,17,35),(3,39,18,34),(4,38,13,33),(5,37,14,32),(6,42,15,31),(7,30,45,21),(8,29,46,20),(9,28,47,19),(10,27,48,24),(11,26,43,23),(12,25,44,22)], [(1,21,4,24),(2,22,5,19),(3,23,6,20),(7,33,10,36),(8,34,11,31),(9,35,12,32),(13,27,16,30),(14,28,17,25),(15,29,18,26),(37,47,40,44),(38,48,41,45),(39,43,42,46)])

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I12A···12H12I···12P
order122222333444444446···666612···1212···12
size11111818224221212121218182···24444···412···12

42 irreducible representations

dim11111122222224444444
type++++++++-++++-++
imageC1C2C2C2C2C2S3D4Q8D6D6C4○D4C4○D12S32S3×D4S3×Q8C2×S32Dic3.D6D6.D6Dic3⋊D6
kernelC62.35C23C6.D12C62.C22C3×Dic3⋊C4C2×C322Q8C2×C4×C3⋊S3Dic3⋊C4C3⋊Dic3C2×C3⋊S3C2×Dic3C2×C12C3×C6C6C2×C4C6C6C22C2C2C2
# reps12121122242281221222

Matrix representation of C62.35C23 in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
000000120
000000012
,
120000000
012000000
00010000
0012120000
00001000
00000100
00000010
00000001
,
05000000
50000000
00100000
0012120000
00001000
00000100
00000008
00000050
,
01000000
120000000
001200000
000120000
000001200
000012000
00000001
00000010
,
80000000
05000000
001200000
000120000
000012000
000001200
00000080
00000008

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

C62.35C23 in GAP, Magma, Sage, TeX

C_6^2._{35}C_2^3
% in TeX

G:=Group("C6^2.35C2^3");
// GroupNames label

G:=SmallGroup(288,513);
// by ID

G=gap.SmallGroup(288,513);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,120,422,219,58,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=d^2=b^3,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^-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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