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## G = C62.25C32order 324 = 22·34

### 16th non-split extension by C62 of C32 acting via C32/C3=C3

Aliases: C62.25C32, C9⋊A43C3, (C3×C9)⋊5A4, (C9×A4)⋊1C3, (C6×C18)⋊9C3, C9.6(C3×A4), C32⋊A4.4C3, (C2×C6).3C33, C3.4(C32×A4), C32.A410C3, (C3×A4).2C32, C32.14(C3×A4), (C2×C18).6C32, C221(C9○He3), C3.A4.1C32, SmallGroup(324,128)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C62.25C32
 Chief series C1 — C22 — C2×C6 — C3×A4 — C9×A4 — C62.25C32
 Lower central C22 — C2×C6 — C62.25C32
 Upper central C1 — C9 — C3×C9

Generators and relations for C62.25C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, ab=ba, cac-1=ab-1, ad=da, cbc-1=a3b4, bd=db, cd=dc >

Subgroups: 205 in 76 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C3, C3, C22, C6, C9, C9, C9, C32, C32, A4, C2×C6, C2×C6, C18, C3×C6, C3×C9, C3×C9, He3, 3- 1+2, C3.A4, C2×C18, C2×C18, C3×A4, C62, C3×C18, C9○He3, C9×A4, C9⋊A4, C32.A4, C32⋊A4, C6×C18, C62.25C32
Quotients: C1, C3, C32, A4, C33, C3×A4, C9○He3, C32×A4, C62.25C32

Smallest permutation representation of C62.25C32
On 54 points
Generators in S54
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 19)(9 20)(10 46 16 52 13 49)(11 47 17 53 14 50)(12 48 18 54 15 51)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)
(1 7 4)(2 8 5)(3 9 6)(10 49 13 52 16 46)(11 50 14 53 17 47)(12 51 15 54 18 48)(19 25 22)(20 26 23)(21 27 24)(28 42 31 45 34 39)(29 43 32 37 35 40)(30 44 33 38 36 41)
(1 49 31)(2 50 32)(3 51 33)(4 52 34)(5 53 35)(6 54 36)(7 46 28)(8 47 29)(9 48 30)(10 42 24)(11 43 25)(12 44 26)(13 45 27)(14 37 19)(15 38 20)(16 39 21)(17 40 22)(18 41 23)
(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 49 50 51 52 53 54)

G:=sub<Sym(54)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(10,46,16,52,13,49)(11,47,17,53,14,50)(12,48,18,54,15,51)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (1,7,4)(2,8,5)(3,9,6)(10,49,13,52,16,46)(11,50,14,53,17,47)(12,51,15,54,18,48)(19,25,22)(20,26,23)(21,27,24)(28,42,31,45,34,39)(29,43,32,37,35,40)(30,44,33,38,36,41), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,42,24)(11,43,25)(12,44,26)(13,45,27)(14,37,19)(15,38,20)(16,39,21)(17,40,22)(18,41,23), (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)(46,47,48,49,50,51,52,53,54)>;

G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,19)(9,20)(10,46,16,52,13,49)(11,47,17,53,14,50)(12,48,18,54,15,51)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (1,7,4)(2,8,5)(3,9,6)(10,49,13,52,16,46)(11,50,14,53,17,47)(12,51,15,54,18,48)(19,25,22)(20,26,23)(21,27,24)(28,42,31,45,34,39)(29,43,32,37,35,40)(30,44,33,38,36,41), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,42,24)(11,43,25)(12,44,26)(13,45,27)(14,37,19)(15,38,20)(16,39,21)(17,40,22)(18,41,23), (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)(46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,19),(9,20),(10,46,16,52,13,49),(11,47,17,53,14,50),(12,48,18,54,15,51),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42)], [(1,7,4),(2,8,5),(3,9,6),(10,49,13,52,16,46),(11,50,14,53,17,47),(12,51,15,54,18,48),(19,25,22),(20,26,23),(21,27,24),(28,42,31,45,34,39),(29,43,32,37,35,40),(30,44,33,38,36,41)], [(1,49,31),(2,50,32),(3,51,33),(4,52,34),(5,53,35),(6,54,36),(7,46,28),(8,47,29),(9,48,30),(10,42,24),(11,43,25),(12,44,26),(13,45,27),(14,37,19),(15,38,20),(16,39,21),(17,40,22),(18,41,23)], [(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),(46,47,48,49,50,51,52,53,54)]])

60 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3J 6A ··· 6H 9A ··· 9F 9G 9H 9I 9J 9K ··· 9V 18A ··· 18R order 1 2 3 3 3 3 3 ··· 3 6 ··· 6 9 ··· 9 9 9 9 9 9 ··· 9 18 ··· 18 size 1 3 1 1 3 3 12 ··· 12 3 ··· 3 1 ··· 1 3 3 3 3 12 ··· 12 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 type + + image C1 C3 C3 C3 C3 C3 A4 C3×A4 C3×A4 C9○He3 C62.25C32 kernel C62.25C32 C9×A4 C9⋊A4 C32.A4 C32⋊A4 C6×C18 C3×C9 C9 C32 C22 C1 # reps 1 6 12 4 2 2 1 6 2 6 18

Matrix representation of C62.25C32 in GL3(𝔽19) generated by

 18 0 0 0 7 0 0 0 8
,
 7 0 0 0 12 0 0 0 12
,
 0 1 0 0 0 1 1 0 0
,
 16 0 0 0 16 0 0 0 16
G:=sub<GL(3,GF(19))| [18,0,0,0,7,0,0,0,8],[7,0,0,0,12,0,0,0,12],[0,0,1,1,0,0,0,1,0],[16,0,0,0,16,0,0,0,16] >;

C62.25C32 in GAP, Magma, Sage, TeX

C_6^2._{25}C_3^2
% in TeX

G:=Group("C6^2.25C3^2");
// GroupNames label

G:=SmallGroup(324,128);
// by ID

G=gap.SmallGroup(324,128);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,115,650,4864,8753]);
// Polycyclic

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

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