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G = C62.117C23order 288 = 25·32

112nd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.117C23, C23.17S32, C6.71(S3×D4), (C22×C6).77D6, C6.69(C4○D12), C34(C23.9D6), C6.D412S3, C6.D126C2, (C2×Dic3).46D6, Dic3⋊Dic320C2, C2.30(Dic3⋊D6), C6.56(D42S3), (C2×C62).36C22, C2.29(D6.3D6), (C6×Dic3).27C22, C3214(C22.D4), (C2×C3⋊S3).27D4, C22.140(C2×S32), (C3×C6).163(C2×D4), (C3×C6).87(C4○D4), (C2×C6.D6)⋊15C2, (C2×C327D4).8C2, (C3×C6.D4)⋊16C2, (C2×C6).136(C22×S3), (C22×C3⋊S3).33C22, (C2×C3⋊Dic3).71C22, SmallGroup(288,623)

Series: Derived Chief Lower central Upper central

C1C62 — C62.117C23
C1C3C32C3×C6C62C6×Dic3C2×C6.D6 — C62.117C23
C32C62 — C62.117C23
C1C22C23

Generators and relations for C62.117C23
 G = < a,b,c,d,e | a6=b6=e2=1, c2=d2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=a3b3d >

Subgroups: 770 in 183 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×6], C6 [×6], C6 [×7], C2×C4 [×7], D4 [×2], C23, C23, C32, Dic3 [×8], C12 [×4], D6 [×12], C2×C6 [×2], C2×C6 [×11], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6, C4×S3 [×4], C2×Dic3 [×4], C2×Dic3 [×3], C3⋊D4 [×8], C2×C12 [×4], C22×S3 [×3], C22×C6 [×2], C22×C6, C22.D4, C3×Dic3 [×4], C3⋊Dic3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×3], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C6.D4 [×2], C3×C22⋊C4 [×2], S3×C2×C4 [×2], C2×C3⋊D4 [×3], C6.D6 [×2], C6×Dic3 [×4], C2×C3⋊Dic3, C327D4 [×2], C22×C3⋊S3, C2×C62, C23.9D6 [×2], C6.D12, Dic3⋊Dic3 [×2], C3×C6.D4 [×2], C2×C6.D6, C2×C327D4, C62.117C23
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 [×2], C2×S32, C23.9D6 [×2], D6.3D6 [×2], Dic3⋊D6, C62.117C23

Smallest permutation representation of C62.117C23
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 47 9 43 11 45)(8 48 10 44 12 46)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 42 33 38 35 40)(32 37 34 39 36 41)
(1 31 16 38)(2 32 17 39)(3 33 18 40)(4 34 13 41)(5 35 14 42)(6 36 15 37)(7 27 43 20)(8 28 44 21)(9 29 45 22)(10 30 46 23)(11 25 47 24)(12 26 48 19)
(1 41 16 34)(2 40 17 33)(3 39 18 32)(4 38 13 31)(5 37 14 36)(6 42 15 35)(7 30 43 23)(8 29 44 22)(9 28 45 21)(10 27 46 20)(11 26 47 19)(12 25 48 24)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G···6Q12A···12H
order122222233344444446···66···612···12
size11114181822466661212362···24···412···12

42 irreducible representations

dim111111222222444444
type++++++++++++-++
imageC1C2C2C2C2C2S3D4D6D6C4○D4C4○D12S32S3×D4D42S3C2×S32D6.3D6Dic3⋊D6
kernelC62.117C23C6.D12Dic3⋊Dic3C3×C6.D4C2×C6.D6C2×C327D4C6.D4C2×C3⋊S3C2×Dic3C22×C6C3×C6C6C23C6C6C22C2C2
# reps112211224248122142

Matrix representation of C62.117C23 in GL8(𝔽13)

120000000
012000000
00100000
00010000
00000100
0000121200
00000010
00000001
,
10000000
01000000
001210000
001200000
00001000
00000100
000000120
000000012
,
120000000
01000000
000120000
001200000
000012000
000001200
00000080
00000008
,
10000000
012000000
001200000
000120000
000012000
00001100
00000050
00000058
,
01000000
10000000
00100000
00010000
00001000
00000100
000000111
000000012

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

C62.117C23 in GAP, Magma, Sage, TeX

C_6^2._{117}C_2^3
% in TeX

G:=Group("C6^2.117C2^3");
// GroupNames label

G:=SmallGroup(288,623);
// by ID

G=gap.SmallGroup(288,623);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,303,1356,9414]);
// Polycyclic

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

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