Copied to
clipboard

G = C62.70C23  order 288 = 25·32

65th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.70C23, C12⋊3(C4×S3), C6.39(S3×D4), C6.12(S3×Q8), C4⋊Dic3⋊14S3, (C2×C12).138D6, C4⋊2(C6.D6), C2.2(D6⋊D6), (C2×Dic3).73D6, (C6×C12).104C22, C62.C22⋊14C2, C2.4(Dic3.D6), (C6×Dic3).66C22, C3⋊2(S3×C4⋊C4), (C4×C3⋊S3)⋊2C4, C32⋊7(C2×C4⋊C4), C6.34(S3×C2×C4), C3⋊S3⋊4(C4⋊C4), (C3×C12)⋊8(C2×C4), (C2×C4).118S32, (C2×C3⋊S3).9Q8, (C2×C3⋊S3).45D4, C22.38(C2×S32), (C3×C6).55(C2×D4), (C3×C6).34(C2×Q8), C3⋊Dic3⋊11(C2×C4), (C3×C4⋊Dic3)⋊17C2, (C3×C6).59(C22×C4), (C2×C6).89(C22×S3), (C2×C6.D6).7C2, C2.11(C2×C6.D6), (C22×C3⋊S3).71C22, (C2×C3⋊Dic3).132C22, (C2×C4×C3⋊S3).4C2, (C2×C3⋊S3).39(C2×C4), SmallGroup(288,548)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62.70C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.70C23
Lower central
C32 — C3×C6 — C62.70C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.70C23
 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, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 786 in 211 conjugacy classes, 68 normal (16 characteristic)
C1, 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, C4⋊C4, C22×C4, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×C4⋊C4, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, C6.D6, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C4⋊C4, C62.C22, C3×C4⋊Dic3, C2×C6.D6, C2×C4×C3⋊S3, C62.70C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, S32, S3×C2×C4, S3×D4, S3×Q8, C6.D6, C2×S32, S3×C4⋊C4, Dic3.D6, D6⋊D6, C2×C6.D6, C62.70C23

Smallest permutation representation of C62.70C23
►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 36 4 33)(2 31 5 34)(3 32 6 35)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 38 16 41)(14 39 17 42)(15 40 18 37)(19 43 22 46)(20 44 23 47)(21 45 24 48)

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C···4J4K4L6A···6F6G6H6I12A···12H12I···12P
order12222222333444···4446···666612···1212···12
size11119999224226···618182···24444···412···12

48 irreducible representations

dim1111112222224444444
type+++++++-++++-++
imageC1C2C2C2C2C4S3D4Q8D6D6C4×S3S32S3×D4S3×Q8C6.D6C2×S32Dic3.D6D6⋊D6
kernelC62.70C23C62.C22C3×C4⋊Dic3C2×C6.D6C2×C4×C3⋊S3C4×C3⋊S3C4⋊Dic3C2×C3⋊S3C2×C3⋊S3C2×Dic3C2×C12C12C2×C4C6C6C4C22C2C2
# reps1222182224281222122

Matrix representation of C62.70C23 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
000012000
000001200
000000121
000000120
,
121000000
120000000
001200000
000120000
000012000
000001200
00000010
00000001
,
01000000
10000000
00100000
000120000
00000800
00008000
00000010
00000001
,
10000000
01000000
00100000
000120000
00000500
00005000
00000001
00000010
,
120000000
012000000
00010000
001200000
00000100
000012000
00000010
00000001

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

C62.70C23 in GAP, Magma, Sage, TeX 

C_6^2._{70}C_2^3
 
% in TeX
 
G:=Group("C6^2.70C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,548);
 
// by ID
 
G=gap.SmallGroup(288,548);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,422,219,100,1356,9414]);
 
// 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,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁