Copied to
clipboard

G = C62.38C23  order 288 = 25·32

33rd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.38C23, Dic3⋊C4⋊3S3, (C4×Dic3)⋊12S3, (C2×C12).259D6, Dic3⋊Dic3⋊4C2, C3⋊1(C42⋊3S3), (Dic3×C12)⋊21C2, C6.27(C4○D12), (C2×Dic3).16D6, C6.38(D4⋊2S3), (C6×C12).220C22, C6.10(Q8⋊3S3), C32⋊6(C42⋊2C2), C62.C22⋊23C2, C6.D12.1C2, C6.11D12.8C2, C2.13(D6.6D6), C2.14(D6.D6), C2.16(D6.3D6), (C6×Dic3).10C22, (C2×C4).44S32, C22.95(C2×S32), C3⋊1(C4⋊C4⋊S3), (C3×Dic3⋊C4)⋊9C2, (C3×C6).62(C4○D4), (C2×C6).57(C22×S3), (C22×C3⋊S3).12C22, (C2×C3⋊Dic3).32C22, SmallGroup(288,516)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.38C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C62.38C23
Lower central
C32 — C62 — C62.38C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.38C23
 G = < a,b,c,d,e | a6=b6=1, c2=e2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ece-1=a3b3c, de=ed >

Subgroups: 570 in 137 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C3⋊S3, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊2C2, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C6×Dic3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C42⋊3S3, C4⋊C4⋊S3, C6.D12, Dic3⋊Dic3, C62.C22, Dic3×C12, C3×Dic3⋊C4, C6.11D12, C62.38C23
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, C42⋊2C2, S32, C4○D12, D4⋊2S3, Q8⋊3S3, C2×S32, C42⋊3S3, C4⋊C4⋊S3, D6.D6, D6.6D6, D6.3D6, C62.38C23

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

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

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

48 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E4F4G4H4I6A···6F6G6H6I12A12B12C12D12E···12J12K···12R12S12T12U12V
order122223334444444446···66661212121212···1212···1212121212
size1111362242266661212362···244422224···46···612121212

48 irreducible representations

dim11111112222224444444
type++++++++++++-+++
imageC1C2C2C2C2C2C2S3S3D6D6C4○D4C4○D12S32D4⋊2S3Q8⋊3S3C2×S32D6.D6D6.6D6D6.3D6
kernelC62.38C23C6.D12Dic3⋊Dic3C62.C22Dic3×C12C3×Dic3⋊C4C6.11D12C4×Dic3Dic3⋊C4C2×Dic3C2×C12C3×C6C6C2×C4C6C6C22C2C2C2
# reps121111111426161111222

Matrix representation of C62.38C23 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
120000000
012000000
001200000
000120000
00000100
0000121200
00000010
00000001
,
08000000
80000000
00510000
00080000
000012000
00001100
00000010
00000001
,
01000000
120000000
00100000
003120000
00001000
00000100
00000001
00000010
,
50000000
05000000
00800000
001150000
000012000
000001200
00000010
00000001

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

C62.38C23 in GAP, Magma, Sage, TeX 

C_6^2._{38}C_2^3
 
% in TeX
 
G:=Group("C6^2.38C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,516);
 
// by ID
 
G=gap.SmallGroup(288,516);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,590,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=e^2=b^3,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^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,e*c*e^-1=a^3*b^3*c,d*e=e*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁