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## G = C22×C32⋊C6order 216 = 23·33

### Direct product of C22 and C32⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C32⋊C6
 Chief series C1 — C3 — C32 — He3 — C32⋊C6 — C2×C32⋊C6 — C22×C32⋊C6
 Lower central C32 — C22×C32⋊C6
 Upper central C1 — C22

Generators and relations for C22×C32⋊C6
G = < a,b,c,d,e | a2=b2=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 480 in 122 conjugacy classes, 47 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C3, C3 [×3], C22, C22 [×6], S3 [×8], C6 [×3], C6 [×13], C23, C32 [×2], C32, D6 [×12], C2×C6, C2×C6 [×9], C3×S3 [×4], C3⋊S3 [×4], C3×C6 [×6], C3×C6 [×3], C22×S3 [×2], C22×C6, He3, S3×C6 [×6], C2×C3⋊S3 [×6], C62 [×2], C62, C32⋊C6 [×4], C2×He3 [×3], S3×C2×C6, C22×C3⋊S3, C2×C32⋊C6 [×6], C22×He3, C22×C32⋊C6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C3×S3, C22×S3, C22×C6, S3×C6 [×3], C32⋊C6, S3×C2×C6, C2×C32⋊C6 [×3], C22×C32⋊C6

Smallest permutation representation of C22×C32⋊C6
On 36 points
Generators in S36
(1 11)(2 12)(3 7)(4 8)(5 10)(6 9)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 9)(2 10)(3 8)(4 7)(5 12)(6 11)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(19 35)(20 36)(21 31)(22 32)(23 33)(24 34)
(1 19 25)(2 28 22)(3 21 29)(4 26 24)(5 23 27)(6 30 20)(7 17 34)(8 31 14)(9 35 16)(10 13 32)(11 15 36)(12 33 18)
(1 3 5)(2 6 4)(7 10 11)(8 12 9)(13 15 17)(14 18 16)(19 21 23)(20 24 22)(25 29 27)(26 28 30)(31 33 35)(32 36 34)
(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)

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

G:=Group( (1,11)(2,12)(3,7)(4,8)(5,10)(6,9)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35), (1,9)(2,10)(3,8)(4,7)(5,12)(6,11)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(19,35)(20,36)(21,31)(22,32)(23,33)(24,34), (1,19,25)(2,28,22)(3,21,29)(4,26,24)(5,23,27)(6,30,20)(7,17,34)(8,31,14)(9,35,16)(10,13,32)(11,15,36)(12,33,18), (1,3,5)(2,6,4)(7,10,11)(8,12,9)(13,15,17)(14,18,16)(19,21,23)(20,24,22)(25,29,27)(26,28,30)(31,33,35)(32,36,34), (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) );

G=PermutationGroup([(1,11),(2,12),(3,7),(4,8),(5,10),(6,9),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,9),(2,10),(3,8),(4,7),(5,12),(6,11),(13,28),(14,29),(15,30),(16,25),(17,26),(18,27),(19,35),(20,36),(21,31),(22,32),(23,33),(24,34)], [(1,19,25),(2,28,22),(3,21,29),(4,26,24),(5,23,27),(6,30,20),(7,17,34),(8,31,14),(9,35,16),(10,13,32),(11,15,36),(12,33,18)], [(1,3,5),(2,6,4),(7,10,11),(8,12,9),(13,15,17),(14,18,16),(19,21,23),(20,24,22),(25,29,27),(26,28,30),(31,33,35),(32,36,34)], [(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)])

C22×C32⋊C6 is a maximal subgroup of   C62.4D6  C62.21D6  C62⋊D6
C22×C32⋊C6 is a maximal quotient of   C62.36D6  C62.13D6  (Q8×He3)⋊C2

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D ··· 6I 6J ··· 6R 6S ··· 6Z order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 6 6 6 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 9 9 9 9 2 3 3 6 6 6 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 C32⋊C6 C2×C32⋊C6 kernel C22×C32⋊C6 C2×C32⋊C6 C22×He3 C22×C3⋊S3 C2×C3⋊S3 C62 C62 C3×C6 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 1 3 2 6 1 3

Matrix representation of C22×C32⋊C6 in GL8(𝔽7)

 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6
,
 6 1 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 1 6 0 0 0 6 0 0 0 0 0 0 1 6 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 1 6
,
 0 3 0 0 0 0 0 0 3 0 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 0 0 6 0 0 0 0 0 0 0 6 1 0 0 0 0 1 6 0 0 0 0 0 0 0 6 0 0

G:=sub<GL(8,GF(7))| [6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6],[6,6,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6],[0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,6,0,0,0,0,6,6,0,0,0,0,0,0,0,1,0,0] >;

C22×C32⋊C6 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes C_6
% in TeX

G:=Group("C2^2xC3^2:C6");
// GroupNames label

G:=SmallGroup(216,110);
// by ID

G=gap.SmallGroup(216,110);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,1444,382,5189]);
// Polycyclic

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

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