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G = C336D4order 216 = 23·33

3rd semidirect product of C33 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C336D4, C6.12S32, (S3×C6)⋊3S3, D61(C3⋊S3), (C3×C6).31D6, C335C43C2, C32(D6⋊S3), C326(C3⋊D4), C32(C327D4), (C32×C6).9C22, (S3×C3×C6)⋊3C2, (C6×C3⋊S3)⋊2C2, (C2×C3⋊S3)⋊4S3, C2.4(S3×C3⋊S3), C6.4(C2×C3⋊S3), SmallGroup(216,127)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C336D4
C1C3C32C33C32×C6S3×C3×C6 — C336D4
C33C32×C6 — C336D4
C1C2

Generators and relations for C336D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 532 in 120 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×5], C6, C6 [×4], C6 [×9], D4, C32, C32 [×4], C32 [×4], Dic3 [×9], D6, D6 [×4], C2×C6 [×5], C3×S3 [×8], C3⋊S3, C3×C6, C3×C6 [×4], C3×C6 [×5], C3⋊D4 [×5], C33, C3⋊Dic3 [×9], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C32×C6, D6⋊S3 [×4], C327D4, C335C4, S3×C3×C6, C6×C3⋊S3, C336D4
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], C3⋊S3, C3⋊D4 [×5], S32 [×4], C2×C3⋊S3, D6⋊S3 [×4], C327D4, S3×C3⋊S3, C336D4

Smallest permutation representation of C336D4
On 72 points
Generators in S72
(1 54 63)(2 64 55)(3 56 61)(4 62 53)(5 16 59)(6 60 13)(7 14 57)(8 58 15)(9 23 40)(10 37 24)(11 21 38)(12 39 22)(17 71 30)(18 31 72)(19 69 32)(20 29 70)(25 52 34)(26 35 49)(27 50 36)(28 33 51)(41 46 65)(42 66 47)(43 48 67)(44 68 45)
(1 32 40)(2 37 29)(3 30 38)(4 39 31)(5 35 47)(6 48 36)(7 33 45)(8 46 34)(9 54 19)(10 20 55)(11 56 17)(12 18 53)(13 43 50)(14 51 44)(15 41 52)(16 49 42)(21 61 71)(22 72 62)(23 63 69)(24 70 64)(25 58 65)(26 66 59)(27 60 67)(28 68 57)
(1 23 19)(2 20 24)(3 21 17)(4 18 22)(5 49 66)(6 67 50)(7 51 68)(8 65 52)(9 32 63)(10 64 29)(11 30 61)(12 62 31)(13 48 27)(14 28 45)(15 46 25)(16 26 47)(33 44 57)(34 58 41)(35 42 59)(36 60 43)(37 55 70)(38 71 56)(39 53 72)(40 69 54)
(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 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)
(1 26)(2 25)(3 28)(4 27)(5 9)(6 12)(7 11)(8 10)(13 22)(14 21)(15 24)(16 23)(17 45)(18 48)(19 47)(20 46)(29 65)(30 68)(31 67)(32 66)(33 56)(34 55)(35 54)(36 53)(37 58)(38 57)(39 60)(40 59)(41 70)(42 69)(43 72)(44 71)(49 63)(50 62)(51 61)(52 64)

G:=sub<Sym(72)| (1,54,63)(2,64,55)(3,56,61)(4,62,53)(5,16,59)(6,60,13)(7,14,57)(8,58,15)(9,23,40)(10,37,24)(11,21,38)(12,39,22)(17,71,30)(18,31,72)(19,69,32)(20,29,70)(25,52,34)(26,35,49)(27,50,36)(28,33,51)(41,46,65)(42,66,47)(43,48,67)(44,68,45), (1,32,40)(2,37,29)(3,30,38)(4,39,31)(5,35,47)(6,48,36)(7,33,45)(8,46,34)(9,54,19)(10,20,55)(11,56,17)(12,18,53)(13,43,50)(14,51,44)(15,41,52)(16,49,42)(21,61,71)(22,72,62)(23,63,69)(24,70,64)(25,58,65)(26,66,59)(27,60,67)(28,68,57), (1,23,19)(2,20,24)(3,21,17)(4,18,22)(5,49,66)(6,67,50)(7,51,68)(8,65,52)(9,32,63)(10,64,29)(11,30,61)(12,62,31)(13,48,27)(14,28,45)(15,46,25)(16,26,47)(33,44,57)(34,58,41)(35,42,59)(36,60,43)(37,55,70)(38,71,56)(39,53,72)(40,69,54), (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)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,26)(2,25)(3,28)(4,27)(5,9)(6,12)(7,11)(8,10)(13,22)(14,21)(15,24)(16,23)(17,45)(18,48)(19,47)(20,46)(29,65)(30,68)(31,67)(32,66)(33,56)(34,55)(35,54)(36,53)(37,58)(38,57)(39,60)(40,59)(41,70)(42,69)(43,72)(44,71)(49,63)(50,62)(51,61)(52,64)>;

G:=Group( (1,54,63)(2,64,55)(3,56,61)(4,62,53)(5,16,59)(6,60,13)(7,14,57)(8,58,15)(9,23,40)(10,37,24)(11,21,38)(12,39,22)(17,71,30)(18,31,72)(19,69,32)(20,29,70)(25,52,34)(26,35,49)(27,50,36)(28,33,51)(41,46,65)(42,66,47)(43,48,67)(44,68,45), (1,32,40)(2,37,29)(3,30,38)(4,39,31)(5,35,47)(6,48,36)(7,33,45)(8,46,34)(9,54,19)(10,20,55)(11,56,17)(12,18,53)(13,43,50)(14,51,44)(15,41,52)(16,49,42)(21,61,71)(22,72,62)(23,63,69)(24,70,64)(25,58,65)(26,66,59)(27,60,67)(28,68,57), (1,23,19)(2,20,24)(3,21,17)(4,18,22)(5,49,66)(6,67,50)(7,51,68)(8,65,52)(9,32,63)(10,64,29)(11,30,61)(12,62,31)(13,48,27)(14,28,45)(15,46,25)(16,26,47)(33,44,57)(34,58,41)(35,42,59)(36,60,43)(37,55,70)(38,71,56)(39,53,72)(40,69,54), (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)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,26)(2,25)(3,28)(4,27)(5,9)(6,12)(7,11)(8,10)(13,22)(14,21)(15,24)(16,23)(17,45)(18,48)(19,47)(20,46)(29,65)(30,68)(31,67)(32,66)(33,56)(34,55)(35,54)(36,53)(37,58)(38,57)(39,60)(40,59)(41,70)(42,69)(43,72)(44,71)(49,63)(50,62)(51,61)(52,64) );

G=PermutationGroup([(1,54,63),(2,64,55),(3,56,61),(4,62,53),(5,16,59),(6,60,13),(7,14,57),(8,58,15),(9,23,40),(10,37,24),(11,21,38),(12,39,22),(17,71,30),(18,31,72),(19,69,32),(20,29,70),(25,52,34),(26,35,49),(27,50,36),(28,33,51),(41,46,65),(42,66,47),(43,48,67),(44,68,45)], [(1,32,40),(2,37,29),(3,30,38),(4,39,31),(5,35,47),(6,48,36),(7,33,45),(8,46,34),(9,54,19),(10,20,55),(11,56,17),(12,18,53),(13,43,50),(14,51,44),(15,41,52),(16,49,42),(21,61,71),(22,72,62),(23,63,69),(24,70,64),(25,58,65),(26,66,59),(27,60,67),(28,68,57)], [(1,23,19),(2,20,24),(3,21,17),(4,18,22),(5,49,66),(6,67,50),(7,51,68),(8,65,52),(9,32,63),(10,64,29),(11,30,61),(12,62,31),(13,48,27),(14,28,45),(15,46,25),(16,26,47),(33,44,57),(34,58,41),(35,42,59),(36,60,43),(37,55,70),(38,71,56),(39,53,72),(40,69,54)], [(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),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72)], [(1,26),(2,25),(3,28),(4,27),(5,9),(6,12),(7,11),(8,10),(13,22),(14,21),(15,24),(16,23),(17,45),(18,48),(19,47),(20,46),(29,65),(30,68),(31,67),(32,66),(33,56),(34,55),(35,54),(36,53),(37,58),(38,57),(39,60),(40,59),(41,70),(42,69),(43,72),(44,71),(49,63),(50,62),(51,61),(52,64)])

C336D4 is a maximal subgroup of
S3×D6⋊S3  D6⋊S32  D6.S32  D6.4S32  (C3×D12)⋊S3  C12.73S32  C12.57S32  C12⋊S32  C62.91D6  S3×C327D4  C3⋊S3×C3⋊D4
C336D4 is a maximal quotient of
C336D8  C3312SD16  C3313SD16  C336Q16  C62.77D6  C62.78D6  C62.81D6

33 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I 4 6A···6E6F6G6H6I6J···6Q6R6S
order12223···3333346···666666···666
size116182···24444542···244446···61818

33 irreducible representations

dim11112222244
type+++++++++-
imageC1C2C2C2S3S3D4D6C3⋊D4S32D6⋊S3
kernelC336D4C335C4S3×C3×C6C6×C3⋊S3S3×C6C2×C3⋊S3C33C3×C6C32C6C3
# reps111141151044

Matrix representation of C336D4 in GL6(𝔽13)

12120000
100000
001000
000100
0000012
0000112
,
010000
12120000
001000
000100
000010
000001
,
100000
010000
0001200
0011200
000010
000001
,
1190000
1120000
000100
001000
000001
000010
,
1190000
420000
0001200
0012000
000010
000001

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,11,0,0,0,0,9,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C336D4 in GAP, Magma, Sage, TeX

C_3^3\rtimes_6D_4
% in TeX

G:=Group("C3^3:6D4");
// GroupNames label

G:=SmallGroup(216,127);
// by ID

G=gap.SmallGroup(216,127);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,201,730,5189]);
// Polycyclic

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

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