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G = C62.44C23  order 288 = 25·32

39th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.44C23, C12.48(C4×S3), (C4×Dic3)⋊13S3, (C2×C12).304D6, C3⋊1(C42⋊2S3), (Dic3×C12)⋊22C2, C6.29(C4○D12), (C2×Dic3).96D6, (C6×C12).222C22, C62.C22⋊26C2, C32⋊6(C42⋊C2), C2.2(D6.D6), C6.D12.9C2, C4.17(C6.D6), (C6×Dic3).110C22, (C4×C3⋊S3)⋊5C4, C6.31(S3×C2×C4), (C2×C4).137S32, C22.24(C2×S32), (C3×C12).90(C2×C4), C2.9(C2×C6.D6), (C3×C6).27(C4○D4), C3⋊Dic3.44(C2×C4), (C2×C6).63(C22×S3), (C3×C6).52(C22×C4), (C22×C3⋊S3).67C22, (C2×C3⋊Dic3).125C22, (C2×C4×C3⋊S3).18C2, (C2×C3⋊S3).38(C2×C4), SmallGroup(288,522)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62.44C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C62.C22 — C62.44C23
Lower central
C32 — C3×C6 — C62.44C23
Upper central
C1 — C2×C4

Generators and relations for C62.44C23
 G = < a,b,c,d,e | a6=b6=1, c2=d2=a3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, de=ed >

Subgroups: 658 in 179 conjugacy classes, 60 normal (12 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊C2, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C42⋊2S3, C6.D12, C62.C22, Dic3×C12, C2×C4×C3⋊S3, C62.44C23
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, S32, S3×C2×C4, C4○D12, C6.D6, C2×S32, C42⋊2S3, D6.D6, C2×C6.D6, C62.44C23

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E···4L4M4N6A···6F6G6H6I12A···12H12I12J12K12L12M···12AB
order12222233344444···4446···666612···121212121212···12
size1111181822411116···618182···24442···244446···6

60 irreducible representations

dim1111112222224444
type+++++++++++
imageC1C2C2C2C2C4S3D6D6C4○D4C4×S3C4○D12S32C6.D6C2×S32D6.D6
kernelC62.44C23C6.D12C62.C22Dic3×C12C2×C4×C3⋊S3C4×C3⋊S3C4×Dic3C2×Dic3C2×C12C3×C6C12C6C2×C4C4C22C2
# reps12221824248161214

Matrix representation of C62.44C23 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
000012000
000001200
00000001
0000001212
,
121000000
120000000
001200000
000120000
000012000
000001200
00000010
00000001
,
01000000
10000000
00080000
00500000
00001300
000081200
000000120
000000012
,
120000000
012000000
00010000
00100000
000081100
00000500
00000010
0000001212
,
120000000
012000000
00800000
00080000
00005000
00000500
000000120
000000012

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

C62.44C23 in GAP, Magma, Sage, TeX 

C_6^2._{44}C_2^3
 
% in TeX
 
G:=Group("C6^2.44C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,522);
 
// by ID
 
G=gap.SmallGroup(288,522);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,64,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=d^2=a^3,e^2=b^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,d*c*d^-1=b^3*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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