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G = C62.98C23order 288 = 25·32

93rd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.98C23, Dic3222C2, C23.11S32, (C22×C6).63D6, C6.60(C4○D12), C625C4.2C2, (C2×Dic3).38D6, C62.C229C2, C6.D4.1S3, Dic3⋊Dic319C2, C6.47(D42S3), C327(C422C2), C35(C23.8D6), (C2×C62).17C22, C2.21(D6.3D6), C2.13(D6.4D6), (C6×Dic3).23C22, C22.130(C2×S32), (C3×C6).74(C4○D4), (C2×C6).117(C22×S3), (C3×C6.D4).4C2, (C2×C3⋊Dic3).61C22, SmallGroup(288,604)

Series: Derived Chief Lower central Upper central

C1C62 — C62.98C23
C1C3C32C3×C6C62C6×Dic3Dic32 — C62.98C23
C32C62 — C62.98C23
C1C22C23

Generators and relations for C62.98C23
 G = < a,b,c,d,e | a6=b6=e2=1, c2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, ede=b3d >

Subgroups: 442 in 137 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3 [×2], C3, C4 [×6], C22, C22 [×3], C6 [×6], C6 [×7], C2×C4 [×6], C23, C32, Dic3 [×10], C12 [×4], C2×C6 [×2], C2×C6 [×11], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×C6 [×3], C3×C6, C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×4], C22×C6 [×2], C22×C6, C422C2, C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C62 [×3], C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C6.D4 [×2], C6.D4 [×3], C3×C22⋊C4 [×2], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C2×C62, C23.8D6 [×2], Dic32, Dic3⋊Dic3 [×2], C62.C22, C3×C6.D4 [×2], C625C4, C62.98C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4 [×3], C22×S3 [×2], C422C2, S32, C4○D12 [×2], D42S3 [×4], C2×S32, C23.8D6 [×2], D6.3D6 [×2], D6.4D6, C62.98C23

Smallest permutation representation of C62.98C23
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 16 3 18 5 14)(2 17 4 13 6 15)(7 48 9 44 11 46)(8 43 10 45 12 47)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 41 35 39 33 37)(32 42 36 40 34 38)
(1 42 18 34)(2 37 13 35)(3 38 14 36)(4 39 15 31)(5 40 16 32)(6 41 17 33)(7 27 44 20)(8 28 45 21)(9 29 46 22)(10 30 47 23)(11 25 48 24)(12 26 43 19)
(1 23 4 20)(2 22 5 19)(3 21 6 24)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 29 16 26)(14 28 17 25)(15 27 18 30)(31 44 34 47)(32 43 35 46)(33 48 36 45)
(7 10)(8 11)(9 12)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 42)(32 37)(33 38)(34 39)(35 40)(36 41)(43 46)(44 47)(45 48)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E4F4G4H4I6A···6F6G···6Q12A···12H
order122223334444444446···66···612···12
size11114224666612121818362···24···412···12

42 irreducible representations

dim1111112222244444
type++++++++++-+-
imageC1C2C2C2C2C2S3D6D6C4○D4C4○D12S32D42S3C2×S32D6.3D6D6.4D6
kernelC62.98C23Dic32Dic3⋊Dic3C62.C22C3×C6.D4C625C4C6.D4C2×Dic3C22×C6C3×C6C6C23C6C22C2C2
# reps1121212426814142

Matrix representation of C62.98C23 in GL8(𝔽13)

120000000
012000000
00100000
00010000
00001000
00000100
00000001
0000001212
,
10000000
01000000
001200000
000120000
00000100
0000121200
00000010
00000001
,
120000000
111000000
00080000
00800000
00001000
0000121200
00000010
00000001
,
80000000
08000000
00010000
00100000
000012000
000001200
00000010
0000001212
,
112000000
012000000
00100000
000120000
000012000
000001200
00000010
00000001

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

C62.98C23 in GAP, Magma, Sage, TeX

C_6^2._{98}C_2^3
% in TeX

G:=Group("C6^2.98C2^3");
// GroupNames label

G:=SmallGroup(288,604);
// by ID

G=gap.SmallGroup(288,604);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,219,1356,9414]);
// Polycyclic

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

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