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G = C62.5Q8order 288 = 25·32

2nd non-split extension by C62 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.5Q8, C12.23(C4×S3), (C2×C12).84D6, (C2×C6).4Dic6, (C3×C12).115D4, C324C8.2C4, C4.Dic3.4S3, C12.97(C3⋊D4), C6.8(Dic3⋊C4), (C6×C12).38C22, C326(C8.C4), C4.8(C6.D6), C4.19(D6⋊S3), C32(C12.53D4), C2.5(C62.C22), C22.2(C322Q8), (C2×C4).104S32, (C3×C6).25(C4⋊C4), (C3×C12).36(C2×C4), (C2×C324C8).4C2, (C3×C4.Dic3).4C2, SmallGroup(288,226)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C62.5Q8
C1C3C32C3×C6C3×C12C6×C12C3×C4.Dic3 — C62.5Q8
C32C3×C6C3×C12 — C62.5Q8
C1C4C2×C4

Generators and relations for C62.5Q8
 G = < a,b,c,d | a6=b6=1, c4=b3, d2=c2, ab=ba, cac-1=a-1b3, dad-1=ab3, bc=cb, dbd-1=b-1, dcd-1=a3c3 >

Subgroups: 178 in 71 conjugacy classes, 32 normal (16 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×5], C8 [×4], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C2×C8, M4(2) [×2], C3×C6, C3×C6, C3⋊C8 [×8], C24 [×2], C2×C12 [×2], C2×C12, C8.C4, C3×C12 [×2], C62, C2×C3⋊C8 [×3], C4.Dic3 [×2], C3×M4(2) [×2], C3×C3⋊C8 [×2], C324C8 [×2], C6×C12, C12.53D4 [×2], C3×C4.Dic3 [×2], C2×C324C8, C62.5Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, D6 [×2], C4⋊C4, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C8.C4, S32, Dic3⋊C4 [×2], C6.D6, D6⋊S3, C322Q8, C12.53D4 [×2], C62.C22, C62.5Q8

Smallest permutation representation of C62.5Q8
On 48 points
Generators in S48
(1 33 25)(2 30 34 6 26 38)(3 35 27)(4 32 36 8 28 40)(5 37 29)(7 39 31)(9 44 18 13 48 22)(10 19 41)(11 46 20 15 42 24)(12 21 43)(14 23 45)(16 17 47)
(1 37 25 5 33 29)(2 38 26 6 34 30)(3 39 27 7 35 31)(4 40 28 8 36 32)(9 22 48 13 18 44)(10 23 41 14 19 45)(11 24 42 15 20 46)(12 17 43 16 21 47)
(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 9 3 11 5 13 7 15)(2 16 4 10 6 12 8 14)(17 28 19 30 21 32 23 26)(18 27 20 29 22 31 24 25)(33 48 35 42 37 44 39 46)(34 47 36 41 38 43 40 45)

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

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

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

42 conjugacy classes

class 1 2A2B3A3B3C4A4B4C6A6B6C···6G8A8B8C8D8E8F8G8H12A12B12C12D12E···12J24A···24H
order122333444666···6888888881212121212···1224···24
size112224112224···4121212121818181822224···412···12

42 irreducible representations

dim111122222222444444
type+++++-+-++--
imageC1C2C2C4S3D4Q8D6C4×S3C3⋊D4Dic6C8.C4S32C6.D6D6⋊S3C322Q8C12.53D4C62.5Q8
kernelC62.5Q8C3×C4.Dic3C2×C324C8C324C8C4.Dic3C3×C12C62C2×C12C12C12C2×C6C32C2×C4C4C4C22C3C1
# reps121421124444111144

Matrix representation of C62.5Q8 in GL4(𝔽5) generated by

1300
4300
0023
0044
,
4200
1200
0042
0012
,
0030
0003
1000
0100
,
0002
0040
0200
4000
G:=sub<GL(4,GF(5))| [1,4,0,0,3,3,0,0,0,0,2,4,0,0,3,4],[4,1,0,0,2,2,0,0,0,0,4,1,0,0,2,2],[0,0,1,0,0,0,0,1,3,0,0,0,0,3,0,0],[0,0,0,4,0,0,2,0,0,4,0,0,2,0,0,0] >;

C62.5Q8 in GAP, Magma, Sage, TeX

C_6^2._5Q_8
% in TeX

G:=Group("C6^2.5Q8");
// GroupNames label

G:=SmallGroup(288,226);
// by ID

G=gap.SmallGroup(288,226);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,36,100,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a*b^3,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^3*c^3>;
// generators/relations

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