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G = C62.70C23order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C62.70C23, C123(C4×S3), C6.39(S3×D4), C6.12(S3×Q8), C4⋊Dic314S3, (C2×C12).138D6, C42(C6.D6), C2.2(D6⋊D6), (C2×Dic3).73D6, (C6×C12).104C22, C62.C2214C2, C2.4(Dic3.D6), (C6×Dic3).66C22, C32(S3×C4⋊C4), (C4×C3⋊S3)⋊2C4, C327(C2×C4⋊C4), C6.34(S3×C2×C4), C3⋊S34(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⋊Dic311(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

C1C3×C6 — C62.70C23
C1C3C32C3×C6C62C6×Dic3C2×C6.D6 — C62.70C23
C32C3×C6 — C62.70C23
C1C22C2×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 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×13], C23, C32, Dic3 [×10], C12 [×4], C12 [×6], D6 [×18], C2×C6 [×2], C2×C6, C4⋊C4 [×4], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×20], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C2×C4⋊C4, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, Dic3⋊C4 [×4], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×7], C6.D6 [×4], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C4⋊C4 [×2], C62.C22 [×2], C3×C4⋊Dic3 [×2], C2×C6.D6 [×2], C2×C4×C3⋊S3, C62.70C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×4], C22×S3 [×2], C2×C4⋊C4, S32, S3×C2×C4 [×2], S3×D4 [×2], S3×Q8 [×2], C6.D6 [×2], C2×S32, S3×C4⋊C4 [×2], 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 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 34 4 31)(2 35 5 32)(3 36 6 33)(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 45 22 48)(20 46 23 43)(21 47 24 44)
(1 31 4 34)(2 36 5 33)(3 35 6 32)(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 20)(2 28 17 21)(3 29 18 22)(4 30 13 23)(5 25 14 24)(6 26 15 19)(7 31 43 38)(8 32 44 39)(9 33 45 40)(10 34 46 41)(11 35 47 42)(12 36 48 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)(7,47,9,43,11,45)(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), (1,34,4,31)(2,35,5,32)(3,36,6,33)(7,27,10,30)(8,28,11,25)(9,29,12,26)(13,38,16,41)(14,39,17,42)(15,40,18,37)(19,45,22,48)(20,46,23,43)(21,47,24,44), (1,31,4,34)(2,36,5,33)(3,35,6,32)(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), (1,27,16,20)(2,28,17,21)(3,29,18,22)(4,30,13,23)(5,25,14,24)(6,26,15,19)(7,31,43,38)(8,32,44,39)(9,33,45,40)(10,34,46,41)(11,35,47,42)(12,36,48,37)>;

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

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

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

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