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G = C62.65C23order 288 = 25·32

60th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: Dic3.11D12, C62.65C23, C4⋊Dic37S3, C6.19(S3×D4), C6.11(S3×Q8), (C2×C12).23D6, C6.20(C2×D12), C2.21(S3×D12), Dic3⋊C419S3, C32(D6⋊Q8), C31(C4.D12), (C3×Dic3).9D4, C6.55(C4○D12), (C2×Dic3).26D6, Dic3⋊Dic316C2, C6.42(D42S3), C3211(C22⋊Q8), (C6×C12).185C22, C6.11D12.6C2, C6.D12.2C2, C2.18(D6.3D6), (C6×Dic3).39C22, C2.12(Dic3.D6), (C2×C4).28S32, (C2×C3⋊S3)⋊2Q8, (C3×C6).52(C2×D4), (C3×C6).31(C2×Q8), (C3×C4⋊Dic3)⋊14C2, C22.109(C2×S32), (C3×Dic3⋊C4)⋊18C2, (C3×C6).67(C4○D4), (C2×C322Q8)⋊12C2, (C2×C6).84(C22×S3), (C2×C6.D6).4C2, (C22×C3⋊S3).18C22, (C2×C3⋊Dic3).48C22, SmallGroup(288,543)

Series: Derived Chief Lower central Upper central

C1C62 — C62.65C23
C1C3C32C3×C6C62C6×Dic3C2×C6.D6 — C62.65C23
C32C62 — C62.65C23
C1C22C2×C4

Generators and relations for C62.65C23
 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=a3d >

Subgroups: 698 in 165 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×4], S3 [×6], C6 [×6], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×2], Dic3 [×6], C12 [×8], D6 [×12], 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 [×4], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C22⋊Q8, C3×Dic3 [×2], C3×Dic3 [×3], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4 [×5], C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4 [×2], C6.D6 [×2], C322Q8 [×2], C6×Dic3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊Q8, C4.D12, C6.D12, Dic3⋊Dic3, C3×Dic3⋊C4, C3×C4⋊Dic3, C6.11D12, C2×C6.D6, C2×C322Q8, C62.65C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, D12 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8 [×2], C2×S32, D6⋊Q8, C4.D12, Dic3.D6, S3×D12, D6.3D6, C62.65C23

Smallest permutation representation of C62.65C23
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 41 16 36)(2 42 17 31)(3 37 18 32)(4 38 13 33)(5 39 14 34)(6 40 15 35)(7 24 45 27)(8 19 46 28)(9 20 47 29)(10 21 48 30)(11 22 43 25)(12 23 44 26)
(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 41 10 38)(8 42 11 39)(9 37 12 40)(13 27 16 30)(14 28 17 25)(15 29 18 26)(31 43 34 46)(32 44 35 47)(33 45 36 48)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I12A···12H12I···12P
order122222333444444446···666612···1212···12
size11111818224466661212362···24444···412···12

42 irreducible representations

dim1111111122222222244444444
type+++++++++++-+++++--++
imageC1C2C2C2C2C2C2C2S3S3D4Q8D6D6C4○D4D12C4○D12S32S3×D4D42S3S3×Q8C2×S32Dic3.D6S3×D12D6.3D6
kernelC62.65C23C6.D12Dic3⋊Dic3C3×Dic3⋊C4C3×C4⋊Dic3C6.11D12C2×C6.D6C2×C322Q8Dic3⋊C4C4⋊Dic3C3×Dic3C2×C3⋊S3C2×Dic3C2×C12C3×C6Dic3C6C2×C4C6C6C6C22C2C2C2
# reps1111111111224224411121222

Matrix representation of C62.65C23 in GL8(𝔽13)

10000000
01000000
00010000
0012120000
000012000
000001200
000000120
000000012
,
01000000
1212000000
00100000
00010000
00001000
00000100
000000120
000000012
,
120000000
11000000
001200000
000120000
000012000
000001200
00000019
000000712
,
120000000
012000000
001200000
00110000
000012000
00000100
00000019
000000712
,
120000000
012000000
001200000
000120000
00000100
000012000
000000123
00000081

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

C62.65C23 in GAP, Magma, Sage, TeX

C_6^2._{65}C_2^3
% in TeX

G:=Group("C6^2.65C2^3");
// GroupNames label

G:=SmallGroup(288,543);
// by ID

G=gap.SmallGroup(288,543);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,135,58,1356,9414]);
// 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=a^3*d>;
// generators/relations

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