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G = C62.91C23  order 288 = 25·32

86th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.91C23, D6⋊9(C4×S3), D6⋊C4⋊13S3, (C2×C12)⋊11D6, C6.54(S3×D4), D6⋊Dic3⋊24C2, (C6×C12)⋊20C22, (C2×Dic3)⋊10D6, C2.3(Dic3⋊D6), C2.4(D6⋊D6), (C22×S3).45D6, (C6×Dic3)⋊12C22, (C2×C4)⋊8S32, (C2×S32)⋊3C4, C2.25(C4×S32), C6.24(S3×C2×C4), (S3×C6)⋊8(C2×C4), C3⋊1(S3×C22⋊C4), (C3×D6⋊C4)⋊25C2, (C2×C3⋊S3).55D4, (C22×S32).2C2, C22.45(C2×S32), C32⋊3(C2×C22⋊C4), C3⋊S3⋊2(C22⋊C4), (C3×C6).113(C2×D4), (S3×C2×C6).37C22, (C2×C6.D6)⋊12C2, (C3×C6).23(C22×C4), (C2×C6).110(C22×S3), (C2×C3⋊Dic3)⋊15C22, (C22×C3⋊S3).75C22, (C2×C4×C3⋊S3)⋊15C2, (C2×C3⋊S3).31(C2×C4), SmallGroup(288,569)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62.91C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C22×S32 — C62.91C23
Lower central
C32 — C3×C6 — C62.91C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.91C23
 G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 1298 in 291 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×S3, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C6.D6, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, S3×C22⋊C4, D6⋊Dic3, C3×D6⋊C4, C2×C6.D6, C2×C4×C3⋊S3, C22×S32, C62.91C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, S32, S3×C2×C4, S3×D4, C2×S32, S3×C22⋊C4, C4×S32, D6⋊D6, Dic3⋊D6, C62.91C23

Smallest permutation representation of C62.91C23
►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 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)

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

(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), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48), (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), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48), (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)], [(1,33),(2,34),(3,35),(4,36),(5,31),(6,32),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,41),(14,42),(15,37),(16,38),(17,39),(18,40),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,47),(20,46),(21,45),(22,44),(23,43),(24,48)], [(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 2A2B2C2D2E2F2G2H2I2J2K3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222222222333444444446···6666666612···1212121212
size11116666999922422666618182···2444121212124···412121212

48 irreducible representations

dim1111111222222444444
type+++++++++++++++
imageC1C2C2C2C2C2C4S3D4D6D6D6C4×S3S32S3×D4C2×S32C4×S32D6⋊D6Dic3⋊D6
kernelC62.91C23D6⋊Dic3C3×D6⋊C4C2×C6.D6C2×C4×C3⋊S3C22×S32C2×S32D6⋊C4C2×C3⋊S3C2×Dic3C2×C12C22×S3D6C2×C4C6C22C2C2C2
# reps1221118242228141222

Matrix representation of C62.91C23 ►in GL8(𝔽13)

10000000
01000000
001200000
000120000
00001000
00000100
00000001
0000001212
,
121000000
120000000
001200000
000120000
000012000
000001200
00000010
00000001
,
012000000
120000000
007110000
001160000
000011900
00004200
00000010
00000001
,
120000000
012000000
00620000
00270000
000011900
00004200
00000010
0000001212
,
10000000
01000000
00010000
001200000
00000100
00001000
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,11,6,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C62.91C23 in GAP, Magma, Sage, TeX 

C_6^2._{91}C_2^3
 
% in TeX
 
G:=Group("C6^2.91C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,569);
 
// by ID
 
G=gap.SmallGroup(288,569);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,219,58,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=1,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d=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^-1=b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

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