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G = C62.112C23order 288 = 25·32

107th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.112C23, (S3×C6)⋊5D4, C23.15S32, D63(C3⋊D4), (C3×Dic3)⋊4D4, C6.168(S3×D4), C34(D63D4), D6⋊Dic334C2, C625C48C2, (C22×C6).72D6, Dic33(C3⋊D4), C3212(C4⋊D4), (C2×Dic3).83D6, (C22×S3).48D6, Dic3⋊Dic327C2, C35(C23.14D6), C6.68(D42S3), (C2×C62).31C22, C2.17(D6.4D6), (C6×Dic3).85C22, (C6×C3⋊D4)⋊2C2, (C2×C3⋊D4)⋊2S3, (C2×S3×Dic3)⋊2C2, C2.40(S3×C3⋊D4), C6.64(C2×C3⋊D4), (C2×D6⋊S3)⋊7C2, C22.138(C2×S32), (C3×C6).158(C2×D4), (S3×C2×C6).44C22, (C3×C6).83(C4○D4), (C2×C6).131(C22×S3), (C2×C3⋊Dic3).68C22, SmallGroup(288,618)

Series: Derived Chief Lower central Upper central

C1C62 — C62.112C23
C1C3C32C3×C6C62C6×Dic3C2×S3×Dic3 — C62.112C23
C32C62 — C62.112C23
C1C22C23

Generators and relations for C62.112C23
 G = < a,b,c,d,e | a6=b6=d2=e2=1, c2=b3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=a3d >

Subgroups: 762 in 205 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×3], C6 [×6], C6 [×10], C2×C4 [×6], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], Dic3 [×7], C12 [×3], D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×18], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×3], C3×C6 [×3], C3×C6, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×2], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3 [×2], S3×C6 [×2], S3×C6 [×5], C62, C62 [×3], Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4 [×4], S3×C2×C4, C22×Dic3, C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], S3×Dic3 [×2], D6⋊S3 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×4], C2×C3⋊Dic3 [×2], S3×C2×C6 [×2], C2×C62, D63D4, C23.14D6, D6⋊Dic3, Dic3⋊Dic3, C625C4, C2×S3×Dic3, C2×D6⋊S3, C6×C3⋊D4 [×2], C62.112C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, C3⋊D4 [×4], C22×S3 [×2], C4⋊D4, S32, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C2×S32, D63D4, C23.14D6, D6.4D6, S3×C3⋊D4 [×2], C62.112C23

Smallest permutation representation of C62.112C23
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 17 5 15 3 13)(2 18 6 16 4 14)(7 47 11 45 9 43)(8 48 12 46 10 44)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 38 33 40 35 42)(32 39 34 41 36 37)
(1 39 15 36)(2 40 16 31)(3 41 17 32)(4 42 18 33)(5 37 13 34)(6 38 14 35)(7 27 45 21)(8 28 46 22)(9 29 47 23)(10 30 48 24)(11 25 43 19)(12 26 44 20)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 26)(14 25)(15 30)(16 29)(17 28)(18 27)(31 47)(32 46)(33 45)(34 44)(35 43)(36 48)
(7 48)(8 43)(9 44)(10 45)(11 46)(12 47)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222223334444446···66···6666612121212
size11114661222466121818362···24···41212121212121212

42 irreducible representations

dim1111111222222222444444
type+++++++++++++++-+-
imageC1C2C2C2C2C2C2S3D4D4D6D6D6C4○D4C3⋊D4C3⋊D4S32S3×D4D42S3C2×S32D6.4D6S3×C3⋊D4
kernelC62.112C23D6⋊Dic3Dic3⋊Dic3C625C4C2×S3×Dic3C2×D6⋊S3C6×C3⋊D4C2×C3⋊D4C3×Dic3S3×C6C2×Dic3C22×S3C22×C6C3×C6Dic3D6C23C6C6C22C2C2
# reps1111112222222244122124

Matrix representation of C62.112C23 in GL8(𝔽13)

120000000
012000000
001200000
000120000
00001000
00000100
00000001
0000001212
,
120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
08000000
80000000
00100000
00010000
000001200
000012000
00000010
00000001
,
01000000
10000000
00010000
00100000
000012000
000001200
00000010
0000001212
,
10000000
012000000
00100000
000120000
000012000
000001200
00000010
00000001

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

C62.112C23 in GAP, Magma, Sage, TeX

C_6^2._{112}C_2^3
% in TeX

G:=Group("C6^2.112C2^3");
// GroupNames label

G:=SmallGroup(288,618);
// by ID

G=gap.SmallGroup(288,618);
# by ID

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

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

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