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G = C62.53C23order 288 = 25·32

48th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.53C23, C6.9(S3×Q8), C6.49(S3×D4), Dic36(C4×S3), C6.D63C4, Dic3⋊C414S3, (C2×C12).199D6, C2.2(Dic3⋊D6), (C2×Dic3).65D6, Dic3⋊Dic324C2, (C6×C12).231C22, C2.3(Dic3.D6), (C6×Dic3).34C22, C31(S3×C4⋊C4), C2.18(C4×S32), (C2×C4).95S32, C325(C2×C4⋊C4), C6.17(S3×C2×C4), C3⋊S33(C4⋊C4), (C2×C3⋊S3).8Q8, (C2×C3⋊S3).54D4, C22.33(C2×S32), (C3×C6).93(C2×D4), (C3×C6).26(C2×Q8), (C3×Dic3)⋊3(C2×C4), (C3×Dic3⋊C4)⋊13C2, (C3×C6).16(C22×C4), (C2×C6).72(C22×S3), (C2×C6.D6).2C2, (C22×C3⋊S3).70C22, (C2×C3⋊Dic3).130C22, (C2×C4×C3⋊S3).20C2, (C2×C3⋊S3).30(C2×C4), SmallGroup(288,531)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.53C23
C1C3C32C3×C6C62C6×Dic3C2×C6.D6 — C62.53C23
C32C3×C6 — C62.53C23
C1C22C2×C4

Generators and relations for C62.53C23
 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, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 786 in 211 conjugacy classes, 66 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×8], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×13], C23, C32, Dic3 [×4], Dic3 [×6], C12 [×10], D6 [×18], C2×C6 [×2], C2×C6, C4⋊C4 [×4], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×20], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C2×C4⋊C4, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×6], C62, Dic3⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×7], C6.D6 [×4], C6.D6 [×2], C6×Dic3 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C4⋊C4 [×2], Dic3⋊Dic3 [×2], C3×Dic3⋊C4 [×2], C2×C6.D6 [×2], C2×C4×C3⋊S3, C62.53C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×4], C22×S3 [×2], C2×C4⋊C4, S32, S3×C2×C4 [×2], S3×D4 [×2], S3×Q8 [×2], C2×S32, S3×C4⋊C4 [×2], Dic3.D6, C4×S32, Dic3⋊D6, C62.53C23

Smallest permutation representation of C62.53C23
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 36 16 41)(2 31 17 42)(3 32 18 37)(4 33 13 38)(5 34 14 39)(6 35 15 40)(7 21 45 30)(8 22 46 25)(9 23 47 26)(10 24 48 27)(11 19 43 28)(12 20 44 29)
(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), (1,36,16,41)(2,31,17,42)(3,32,18,37)(4,33,13,38)(5,34,14,39)(6,35,15,40)(7,21,45,30)(8,22,46,25)(9,23,47,26)(10,24,48,27)(11,19,43,28)(12,20,44,29), (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), (1,36,16,41)(2,31,17,42)(3,32,18,37)(4,33,13,38)(5,34,14,39)(6,35,15,40)(7,21,45,30)(8,22,46,25)(9,23,47,26)(10,24,48,27)(11,19,43,28)(12,20,44,29), (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)], [(1,36,16,41),(2,31,17,42),(3,32,18,37),(4,33,13,38),(5,34,14,39),(6,35,15,40),(7,21,45,30),(8,22,46,25),(9,23,47,26),(10,24,48,27),(11,19,43,28),(12,20,44,29)], [(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)])

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C···4J4K4L6A···6F6G6H6I12A···12H12I···12P
order12222222333444···4446···666612···1212···12
size11119999224226···618182···24444···412···12

48 irreducible representations

dim1111112222224444444
type+++++++-++++-++
imageC1C2C2C2C2C4S3D4Q8D6D6C4×S3S32S3×D4S3×Q8C2×S32Dic3.D6C4×S32Dic3⋊D6
kernelC62.53C23Dic3⋊Dic3C3×Dic3⋊C4C2×C6.D6C2×C4×C3⋊S3C6.D6Dic3⋊C4C2×C3⋊S3C2×C3⋊S3C2×Dic3C2×C12Dic3C2×C4C6C6C22C2C2C2
# reps1222182224281221222

Matrix representation of C62.53C23 in GL8(𝔽13)

10000000
01000000
00100000
00010000
000012000
000001200
000000121
000000120
,
01000000
1212000000
001200000
000120000
000012000
000001200
00000010
00000001
,
10000000
1212000000
000120000
00100000
00007100
00002600
00000010
00000001
,
10000000
01000000
00010000
001200000
000061200
000011700
00000001
00000010
,
10000000
01000000
00010000
00100000
000012300
00008100
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,7,2,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,11,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C62.53C23 in GAP, Magma, Sage, TeX

C_6^2._{53}C_2^3
% in TeX

G:=Group("C6^2.53C2^3");
// GroupNames label

G:=SmallGroup(288,531);
// by ID

G=gap.SmallGroup(288,531);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,219,58,1356,9414]);
// 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,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
// generators/relations

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