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G = C62.112C23  order 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, D6⋊3(C3⋊D4), (C3×Dic3)⋊4D4, C6.168(S3×D4), C3⋊4(D6⋊3D4), D6⋊Dic3⋊34C2, C62⋊5C4⋊8C2, (C22×C6).72D6, Dic3⋊3(C3⋊D4), C32⋊12(C4⋊D4), (C2×Dic3).83D6, (C22×S3).48D6, Dic3⋊Dic3⋊27C2, C3⋊5(C23.14D6), C6.68(D4⋊2S3), (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

Derived series
C1 — C62 — C62.112C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×S3×Dic3 — C62.112C23
Lower central
C32 — C62 — C62.112C23
Upper central
C1 — C22 — C23

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, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×Dic3, D6⋊S3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, C2×C62, D6⋊3D4, C23.14D6, D6⋊Dic3, Dic3⋊Dic3, C62⋊5C4, C2×S3×Dic3, C2×D6⋊S3, C6×C3⋊D4, C62.112C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S32, S3×D4, D4⋊2S3, C2×C3⋊D4, C2×S32, D6⋊3D4, C23.14D6, D6.4D6, S3×C3⋊D4, 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 45 11 43 9 47)(8 46 12 44 10 48)(19 25 21 27 23 29)(20 26 22 28 24 30)(31 42 33 38 35 40)(32 37 34 39 36 41)

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

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

42 conjugacy classes

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

42 irreducible representations

dim1111111222222222444444
type+++++++++++++++-+-
imageC1C2C2C2C2C2C2S3D4D4D6D6D6C4○D4C3⋊D4C3⋊D4S32S3×D4D4⋊2S3C2×S32D6.4D6S3×C3⋊D4
kernelC62.112C23D6⋊Dic3Dic3⋊Dic3C62⋊5C4C2×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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