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G = C62.24C23order 288 = 25·32

19th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.24C23, D6⋊C414S3, C4⋊Dic33S3, C6.35(S3×D4), (C2×C12).16D6, D6⋊Dic318C2, C6.6(C4○D12), (C22×S3).3D6, C6.11D128C2, C32(C23.9D6), C33(D6.D4), C6.6(Q83S3), (C2×Dic3).57D6, C62.C227C2, C2.12(D6⋊D6), C6.35(D42S3), (C6×C12).177C22, C2.9(D6.6D6), C2.12(D6.3D6), (C6×Dic3).55C22, C324(C22.D4), (C2×C4).17S32, (C3×D6⋊C4)⋊11C2, (C3×C4⋊Dic3)⋊7C2, (C2×C3⋊S3).19D4, C22.82(C2×S32), (C3×C6).42(C2×D4), (S3×C2×C6).3C22, (C2×C6.D6)⋊7C2, (C2×C3⋊D12).4C2, (C3×C6).12(C4○D4), (C2×C6).43(C22×S3), (C22×C3⋊S3).10C22, (C2×C3⋊Dic3).23C22, SmallGroup(288,502)

Series: Derived Chief Lower central Upper central

C1C62 — C62.24C23
C1C3C32C3×C6C62S3×C2×C6D6⋊Dic3 — C62.24C23
C32C62 — C62.24C23
C1C22C2×C4

Generators and relations for C62.24C23
 G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ece-1=a3b3c, ede-1=b3d >

Subgroups: 762 in 173 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×7], C6 [×6], C6 [×4], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×6], C12 [×6], D6 [×15], C2×C6 [×2], C2×C6 [×4], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3, C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×4], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×S3 [×3], C22×C6, C22.D4, C3×Dic3 [×3], C3⋊Dic3, C3×C12, S3×C6 [×3], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4, D6⋊C4 [×4], C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C2×C3⋊D4, C6.D6 [×2], C3⋊D12 [×2], C6×Dic3 [×3], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C23.9D6, D6.D4, D6⋊Dic3, C62.C22, C3×C4⋊Dic3, C3×D6⋊C4, C6.11D12, C2×C6.D6, C2×C3⋊D12, C62.24C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C22×S3 [×2], C22.D4, S32, C4○D12 [×2], S3×D4 [×2], D42S3, Q83S3, C2×S32, C23.9D6, D6.D4, D6.6D6, D6⋊D6, D6.3D6, C62.24C23

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G6H6I6J6K12A···12H12I···12N
order122222233344444446···66666612···1212···12
size11111218182244666612362···244412124···412···12

42 irreducible representations

dim111111112222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4○D4C4○D12S32S3×D4D42S3Q83S3C2×S32D6.6D6D6⋊D6D6.3D6
kernelC62.24C23D6⋊Dic3C62.C22C3×C4⋊Dic3C3×D6⋊C4C6.11D12C2×C6.D6C2×C3⋊D12C4⋊Dic3D6⋊C4C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6C6C2×C4C6C6C6C22C2C2C2
# reps111111111123214812111222

Matrix representation of C62.24C23 in GL8(𝔽13)

10000000
01000000
00100000
00010000
00000100
0000121200
000000120
000000012
,
120000000
012000000
001210000
001200000
00001000
00000100
000000120
000000012
,
98000000
34000000
00010000
00100000
00001000
00000100
00000042
000000129
,
126000000
01000000
001200000
000120000
000012000
00001100
00000050
00000068
,
93000000
34000000
001200000
000120000
000012000
000001200
00000063
00000057

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[9,3,0,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,12,0,0,0,0,0,0,2,9],[12,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,6,0,0,0,0,0,0,0,8],[9,3,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,6,5,0,0,0,0,0,0,3,7] >;

C62.24C23 in GAP, Magma, Sage, TeX

C_6^2._{24}C_2^3
% in TeX

G:=Group("C6^2.24C2^3");
// GroupNames label

G:=SmallGroup(288,502);
// by ID

G=gap.SmallGroup(288,502);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,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=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,e*d*e^-1=b^3*d>;
// generators/relations

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