Copied to
clipboard

## G = C33⋊8(C2×C4)  order 216 = 23·33

### 5th semidirect product of C33 and C2×C4 acting via C2×C4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊8(C2×C4)
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊8(C2×C4)
 Lower central C33 — C33⋊8(C2×C4)
 Upper central C1 — C2

Generators and relations for C338(C2×C4)
G = < a,b,c,d,e | a3=b3=c3=d2=e4=1, ab=ba, ac=ca, dad=eae-1=a-1, bc=cb, dbd=ebe-1=b-1, dcd=c-1, ce=ec, de=ed >

Subgroups: 764 in 136 conjugacy classes, 36 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, S3 [×18], C6, C6 [×4], C6 [×4], C2×C4, C32, C32 [×4], C32 [×4], Dic3, Dic3 [×4], C12 [×5], D6 [×9], C3⋊S3 [×18], C3×C6, C3×C6 [×4], C3×C6 [×4], C4×S3 [×5], C33, C3×Dic3 [×4], C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×9], C33⋊C2 [×2], C32×C6, C6.D6 [×4], C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C338(C2×C4)
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, D6 [×5], C3⋊S3, C4×S3 [×5], S32 [×4], C2×C3⋊S3, C6.D6 [×4], C4×C3⋊S3, S3×C3⋊S3, C338(C2×C4)

Smallest permutation representation of C338(C2×C4)
On 36 points
Generators in S36
(1 9 27)(2 28 10)(3 11 25)(4 26 12)(5 20 23)(6 24 17)(7 18 21)(8 22 19)(13 34 30)(14 31 35)(15 36 32)(16 29 33)
(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(5 14)(6 15)(7 16)(8 13)(9 27)(10 28)(11 25)(12 26)(17 36)(18 33)(19 34)(20 35)(21 29)(22 30)(23 31)(24 32)
(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)| (1,9,27)(2,28,10)(3,11,25)(4,26,12)(5,20,23)(6,24,17)(7,18,21)(8,22,19)(13,34,30)(14,31,35)(15,36,32)(16,29,33), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32), (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( (1,9,27)(2,28,10)(3,11,25)(4,26,12)(5,20,23)(6,24,17)(7,18,21)(8,22,19)(13,34,30)(14,31,35)(15,36,32)(16,29,33), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32), (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([(1,9,27),(2,28,10),(3,11,25),(4,26,12),(5,20,23),(6,24,17),(7,18,21),(8,22,19),(13,34,30),(14,31,35),(15,36,32),(16,29,33)], [(1,16,7),(2,8,13),(3,14,5),(4,6,15),(9,29,18),(10,19,30),(11,31,20),(12,17,32),(21,27,33),(22,34,28),(23,25,35),(24,36,26)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(5,14),(6,15),(7,16),(8,13),(9,27),(10,28),(11,25),(12,26),(17,36),(18,33),(19,34),(20,35),(21,29),(22,30),(23,31),(24,32)], [(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)])

C338(C2×C4) is a maximal subgroup of
C335(C2×C8)  C332M4(2)  S3×C6.D6  (S3×C6)⋊D6  C336(C2×Q8)  (S3×C6).D6  D6.3S32  Dic3.S32  C12.40S32  C329(S3×Q8)  C12.58S32  C4×S3×C3⋊S3  C62.90D6  C62.93D6  C6223D6
C338(C2×C4) is a maximal quotient of
C12.69S32  C339M4(2)  Dic3×C3⋊Dic3  C62.79D6  C62.81D6

36 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6E 6F 6G 6H 6I 12A ··· 12H 12I 12J order 1 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 12 12 size 1 1 27 27 2 ··· 2 4 4 4 4 3 3 9 9 2 ··· 2 4 4 4 4 6 ··· 6 18 18

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 S3 S3 D6 C4×S3 S32 C6.D6 kernel C33⋊8(C2×C4) C32×Dic3 C3×C3⋊Dic3 C2×C33⋊C2 C33⋊C2 C3×Dic3 C3⋊Dic3 C3×C6 C32 C6 C3 # reps 1 1 1 1 4 4 1 5 10 4 4

Matrix representation of C338(C2×C4) in GL6(𝔽13)

 11 3 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 10 0 0 0 0 1 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 12 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8

G:=sub<GL(6,GF(13))| [11,12,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,10,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[12,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

C338(C2×C4) in GAP, Magma, Sage, TeX

C_3^3\rtimes_8(C_2\times C_4)
% in TeX

G:=Group("C3^3:8(C2xC4)");
// GroupNames label

G:=SmallGroup(216,126);
// by ID

G=gap.SmallGroup(216,126);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,31,201,730,5189]);
// Polycyclic

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

׿
×
𝔽