Copied to
clipboard

G = C62.82C23order 288 = 25·32

77th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.82C23, D6⋊C411S3, C6.52(S3×D4), C3⋊Dic312D4, C32(Dic3⋊D4), (C2×C12).202D6, C329(C4⋊D4), C6.34(C4○D12), (C2×Dic3).33D6, C62.C224C2, (C22×S3).17D6, C2.15(D6⋊D6), C2.12(Dic3⋊D6), (C6×C12).238C22, C2.18(D6.D6), (C6×Dic3).17C22, (C2×C4).97S32, (C2×C3⋊S3)⋊9D4, (C3×D6⋊C4)⋊8C2, (C2×D6⋊S3)⋊4C2, (C2×C3⋊D12)⋊5C2, C22.120(C2×S32), (C3×C6).108(C2×D4), (S3×C2×C6).32C22, (C3×C6).51(C4○D4), (C2×C6).101(C22×S3), (C22×C3⋊S3).73C22, (C2×C3⋊Dic3).136C22, (C2×C4×C3⋊S3)⋊14C2, SmallGroup(288,560)

Series: Derived Chief Lower central Upper central

C1C62 — C62.82C23
C1C3C32C3×C6C62S3×C2×C6C2×D6⋊S3 — C62.82C23
C32C62 — C62.82C23
C1C22C2×C4

Generators and relations for C62.82C23
 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, dcd=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 978 in 215 conjugacy classes, 48 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×10], C6 [×6], C6 [×5], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×8], C12 [×6], D6 [×20], C2×C6 [×2], C2×C6 [×7], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×3], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12, S3×C6 [×6], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, Dic3⋊C4 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], S3×C2×C4 [×3], C2×D12 [×2], C2×C3⋊D4 [×4], D6⋊S3 [×2], C3⋊D12 [×4], C6×Dic3 [×2], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, Dic3⋊D4 [×2], C62.C22, C3×D6⋊C4 [×2], C2×D6⋊S3, C2×C3⋊D12 [×2], C2×C4×C3⋊S3, C62.82C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, C22×S3 [×2], C4⋊D4, S32, C4○D12 [×2], S3×D4 [×4], C2×S32, Dic3⋊D4 [×2], D6.D6, D6⋊D6, Dic3⋊D6, C62.82C23

Smallest permutation representation of C62.82C23
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 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(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), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (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), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (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)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,45),(14,46),(15,47),(16,48),(17,43),(18,44),(19,39),(20,40),(21,41),(22,42),(23,37),(24,38),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(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)])

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222223334444446···6666666612···1212121212
size11111212181822422121218182···2444121212124···412121212

42 irreducible representations

dim11111122222222444444
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6C4○D4C4○D12S32S3×D4C2×S32D6.D6D6⋊D6Dic3⋊D6
kernelC62.82C23C62.C22C3×D6⋊C4C2×D6⋊S3C2×C3⋊D12C2×C4×C3⋊S3D6⋊C4C3⋊Dic3C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6C6C2×C4C6C22C2C2C2
# reps11212122222228141222

Matrix representation of C62.82C23 in GL8(ℤ)

-10000000
0-1000000
00100000
00010000
0000-1100
0000-1000
00000010
00000001
,
-10000000
0-1000000
00010000
00-1-10000
00001000
00000100
000000-10
0000000-1
,
0-1000000
-10000000
00-100000
00110000
0000-1000
00000-100
000000-12
00000001
,
-10000000
01000000
00100000
00010000
00000100
00001000
0000001-2
0000000-1
,
01000000
-10000000
00-100000
000-10000
0000-1000
00000-100
00000010
0000001-1

G:=sub<GL(8,Integers())| [-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,-1,-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],[-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,-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,-1],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,2,1],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,-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,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1] >;

C62.82C23 in GAP, Magma, Sage, TeX

C_6^2._{82}C_2^3
% in TeX

G:=Group("C6^2.82C2^3");
// GroupNames label

G:=SmallGroup(288,560);
// by ID

G=gap.SmallGroup(288,560);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,422,219,58,1356,9414]);
// 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,d*c*d=a^3*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
// generators/relations

׿
×
𝔽