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G = C62.115C23  order 288 = 25·32

110th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.115C23, Dic32⋊6C2, C3⋊D4⋊Dic3, C62⋊7(C2×C4), C23.32S32, C3⋊3(D4×Dic3), C6.69(S3×D4), C32⋊15(C4×D4), C3⋊Dic3⋊13D4, D6⋊4(C2×Dic3), D6⋊Dic3⋊28C2, C22⋊4(S3×Dic3), (C22×C6).75D6, Dic3⋊2(C2×Dic3), C2.4(Dic3⋊D6), C6.D4⋊10S3, (C2×Dic3).84D6, (C22×S3).51D6, Dic3⋊Dic3⋊29C2, C3⋊7(Dic3⋊4D4), C6.69(D4⋊2S3), C2.5(D6.4D6), (C2×C62).34C22, C6.20(C22×Dic3), (C6×Dic3).41C22, (C2×C6)⋊7(C4×S3), C6.99(S3×C2×C4), (C3×C3⋊D4)⋊2C4, (S3×C6)⋊10(C2×C4), C22.56(C2×S32), (C2×S3×Dic3)⋊22C2, (C2×C6)⋊5(C2×Dic3), (C6×C3⋊D4).3C2, (C2×C3⋊D4).6S3, C2.20(C2×S3×Dic3), (C3×Dic3)⋊6(C2×C4), (C3×C6).161(C2×D4), (S3×C2×C6).47C22, (C3×C6).86(C4○D4), (C3×C6).70(C22×C4), (C22×C3⋊Dic3)⋊3C2, (C3×C6.D4)⋊19C2, (C2×C6).134(C22×S3), (C2×C3⋊Dic3).147C22, SmallGroup(288,621)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62.115C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.115C23
Lower central
C32 — C3×C6 — C62.115C23
Upper central
C1 — C22 — C23

Generators and relations for C62.115C23
 G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=b3d >

Subgroups: 690 in 215 conjugacy classes, 70 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, 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, C3⋊Dic3, S3×C6, S3×C6, C62, C62, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, S3×Dic3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C2×C3⋊Dic3, S3×C2×C6, C2×C62, Dic3⋊4D4, D4×Dic3, Dic32, D6⋊Dic3, Dic3⋊Dic3, C3×C6.D4, C2×S3×Dic3, C6×C3⋊D4, C22×C3⋊Dic3, C62.115C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C4×S3, C2×Dic3, C22×S3, C4×D4, S32, S3×C2×C4, S3×D4, D4⋊2S3, C22×Dic3, S3×Dic3, C2×S32, Dic3⋊4D4, D4×Dic3, C2×S3×Dic3, D6.4D6, Dic3⋊D6, C62.115C23

Smallest permutation representation of C62.115C23
►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 36)(2 31)(3 32)(4 33)(5 34)(6 35)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A···4F4G4H4I4J4K4L6A···6F6G···6Q6R6S12A···12F
order122222223334···44444446···66···66612···12
size111122662246···6999918182···24···4121212···12

48 irreducible representations

dim1111111112222222224444444
type++++++++++++-++++--+-+
imageC1C2C2C2C2C2C2C2C4S3S3D4D6Dic3D6D6C4○D4C4×S3S32S3×D4D4⋊2S3S3×Dic3C2×S32D6.4D6Dic3⋊D6
kernelC62.115C23Dic32D6⋊Dic3Dic3⋊Dic3C3×C6.D4C2×S3×Dic3C6×C3⋊D4C22×C3⋊Dic3C3×C3⋊D4C6.D4C2×C3⋊D4C3⋊Dic3C2×Dic3C3⋊D4C22×S3C22×C6C3×C6C2×C6C23C6C6C22C22C2C2
# reps1111111181123412241222122

Matrix representation of C62.115C23 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
120000000
012000000
001200000
000120000
00000100
0000121200
00000010
00000001
,
13000000
012000000
00010000
00100000
00001000
0000121200
00000010
00000001
,
811000000
05000000
00010000
00100000
00001000
00000100
000000012
000000120
,
120000000
51000000
00100000
000120000
00001000
00000100
000000120
000000012

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
 

C62.115C23 in GAP, Magma, Sage, TeX 

C_6^2._{115}C_2^3
 
% in TeX
 
G:=Group("C6^2.115C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,621);
 
// by ID
 
G=gap.SmallGroup(288,621);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=e^2=1,d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,e*d*e=b^3*d>;
 
// generators/relations
 

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