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G = C3315D4order 216 = 23·33

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

metabelian, supersoluble, monomial

Aliases: C3315D4, C6212S3, (C3×C62)⋊4C2, (C3×C6).66D6, C335C44C2, C33(C327D4), C3213(C3⋊D4), C222(C33⋊C2), (C32×C6).30C22, (C2×C6)⋊4(C3⋊S3), C6.18(C2×C3⋊S3), (C2×C33⋊C2)⋊4C2, C2.5(C2×C33⋊C2), SmallGroup(216,149)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C3315D4
C1C3C32C33C32×C6C2×C33⋊C2 — C3315D4
C33C32×C6 — C3315D4
C1C2C22

Generators and relations for C3315D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1060 in 224 conjugacy classes, 87 normal (9 characteristic)
C1, C2, C2 [×2], C3 [×13], C4, C22, C22, S3 [×13], C6 [×13], C6 [×13], D4, C32 [×13], Dic3 [×13], D6 [×13], C2×C6 [×13], C3⋊S3 [×13], C3×C6 [×13], C3×C6 [×13], C3⋊D4 [×13], C33, C3⋊Dic3 [×13], C2×C3⋊S3 [×13], C62 [×13], C33⋊C2, C32×C6, C32×C6, C327D4 [×13], C335C4, C2×C33⋊C2, C3×C62, C3315D4
Quotients: C1, C2 [×3], C22, S3 [×13], D4, D6 [×13], C3⋊S3 [×13], C3⋊D4 [×13], C2×C3⋊S3 [×13], C33⋊C2, C327D4 [×13], C2×C33⋊C2, C3315D4

Smallest permutation representation of C3315D4
On 108 points
Generators in S108
(1 9 83)(2 84 10)(3 11 81)(4 82 12)(5 31 17)(6 18 32)(7 29 19)(8 20 30)(13 62 38)(14 39 63)(15 64 40)(16 37 61)(21 44 36)(22 33 41)(23 42 34)(24 35 43)(25 67 99)(26 100 68)(27 65 97)(28 98 66)(45 87 53)(46 54 88)(47 85 55)(48 56 86)(49 103 59)(50 60 104)(51 101 57)(52 58 102)(69 79 93)(70 94 80)(71 77 95)(72 96 78)(73 91 106)(74 107 92)(75 89 108)(76 105 90)
(1 70 42)(2 43 71)(3 72 44)(4 41 69)(5 40 67)(6 68 37)(7 38 65)(8 66 39)(9 94 34)(10 35 95)(11 96 36)(12 33 93)(13 97 29)(14 30 98)(15 99 31)(16 32 100)(17 64 25)(18 26 61)(19 62 27)(20 28 63)(21 81 78)(22 79 82)(23 83 80)(24 77 84)(45 106 58)(46 59 107)(47 108 60)(48 57 105)(49 92 54)(50 55 89)(51 90 56)(52 53 91)(73 102 87)(74 88 103)(75 104 85)(76 86 101)
(1 40 101)(2 102 37)(3 38 103)(4 104 39)(5 86 42)(6 43 87)(7 88 44)(8 41 85)(9 15 57)(10 58 16)(11 13 59)(12 60 14)(17 56 23)(18 24 53)(19 54 21)(20 22 55)(25 90 80)(26 77 91)(27 92 78)(28 79 89)(29 46 36)(30 33 47)(31 48 34)(32 35 45)(49 81 62)(50 63 82)(51 83 64)(52 61 84)(65 74 72)(66 69 75)(67 76 70)(68 71 73)(93 108 98)(94 99 105)(95 106 100)(96 97 107)
(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)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)(97 98 99 100)(101 102 103 104)(105 106 107 108)
(2 4)(5 76)(6 75)(7 74)(8 73)(9 83)(10 82)(11 81)(12 84)(13 49)(14 52)(15 51)(16 50)(17 105)(18 108)(19 107)(20 106)(21 96)(22 95)(23 94)(24 93)(25 48)(26 47)(27 46)(28 45)(29 92)(30 91)(31 90)(32 89)(33 77)(34 80)(35 79)(36 78)(37 104)(38 103)(39 102)(40 101)(41 71)(42 70)(43 69)(44 72)(53 98)(54 97)(55 100)(56 99)(57 64)(58 63)(59 62)(60 61)(65 88)(66 87)(67 86)(68 85)

G:=sub<Sym(108)| (1,9,83)(2,84,10)(3,11,81)(4,82,12)(5,31,17)(6,18,32)(7,29,19)(8,20,30)(13,62,38)(14,39,63)(15,64,40)(16,37,61)(21,44,36)(22,33,41)(23,42,34)(24,35,43)(25,67,99)(26,100,68)(27,65,97)(28,98,66)(45,87,53)(46,54,88)(47,85,55)(48,56,86)(49,103,59)(50,60,104)(51,101,57)(52,58,102)(69,79,93)(70,94,80)(71,77,95)(72,96,78)(73,91,106)(74,107,92)(75,89,108)(76,105,90), (1,70,42)(2,43,71)(3,72,44)(4,41,69)(5,40,67)(6,68,37)(7,38,65)(8,66,39)(9,94,34)(10,35,95)(11,96,36)(12,33,93)(13,97,29)(14,30,98)(15,99,31)(16,32,100)(17,64,25)(18,26,61)(19,62,27)(20,28,63)(21,81,78)(22,79,82)(23,83,80)(24,77,84)(45,106,58)(46,59,107)(47,108,60)(48,57,105)(49,92,54)(50,55,89)(51,90,56)(52,53,91)(73,102,87)(74,88,103)(75,104,85)(76,86,101), (1,40,101)(2,102,37)(3,38,103)(4,104,39)(5,86,42)(6,43,87)(7,88,44)(8,41,85)(9,15,57)(10,58,16)(11,13,59)(12,60,14)(17,56,23)(18,24,53)(19,54,21)(20,22,55)(25,90,80)(26,77,91)(27,92,78)(28,79,89)(29,46,36)(30,33,47)(31,48,34)(32,35,45)(49,81,62)(50,63,82)(51,83,64)(52,61,84)(65,74,72)(66,69,75)(67,76,70)(68,71,73)(93,108,98)(94,99,105)(95,106,100)(96,97,107), (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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108), (2,4)(5,76)(6,75)(7,74)(8,73)(9,83)(10,82)(11,81)(12,84)(13,49)(14,52)(15,51)(16,50)(17,105)(18,108)(19,107)(20,106)(21,96)(22,95)(23,94)(24,93)(25,48)(26,47)(27,46)(28,45)(29,92)(30,91)(31,90)(32,89)(33,77)(34,80)(35,79)(36,78)(37,104)(38,103)(39,102)(40,101)(41,71)(42,70)(43,69)(44,72)(53,98)(54,97)(55,100)(56,99)(57,64)(58,63)(59,62)(60,61)(65,88)(66,87)(67,86)(68,85)>;

G:=Group( (1,9,83)(2,84,10)(3,11,81)(4,82,12)(5,31,17)(6,18,32)(7,29,19)(8,20,30)(13,62,38)(14,39,63)(15,64,40)(16,37,61)(21,44,36)(22,33,41)(23,42,34)(24,35,43)(25,67,99)(26,100,68)(27,65,97)(28,98,66)(45,87,53)(46,54,88)(47,85,55)(48,56,86)(49,103,59)(50,60,104)(51,101,57)(52,58,102)(69,79,93)(70,94,80)(71,77,95)(72,96,78)(73,91,106)(74,107,92)(75,89,108)(76,105,90), (1,70,42)(2,43,71)(3,72,44)(4,41,69)(5,40,67)(6,68,37)(7,38,65)(8,66,39)(9,94,34)(10,35,95)(11,96,36)(12,33,93)(13,97,29)(14,30,98)(15,99,31)(16,32,100)(17,64,25)(18,26,61)(19,62,27)(20,28,63)(21,81,78)(22,79,82)(23,83,80)(24,77,84)(45,106,58)(46,59,107)(47,108,60)(48,57,105)(49,92,54)(50,55,89)(51,90,56)(52,53,91)(73,102,87)(74,88,103)(75,104,85)(76,86,101), (1,40,101)(2,102,37)(3,38,103)(4,104,39)(5,86,42)(6,43,87)(7,88,44)(8,41,85)(9,15,57)(10,58,16)(11,13,59)(12,60,14)(17,56,23)(18,24,53)(19,54,21)(20,22,55)(25,90,80)(26,77,91)(27,92,78)(28,79,89)(29,46,36)(30,33,47)(31,48,34)(32,35,45)(49,81,62)(50,63,82)(51,83,64)(52,61,84)(65,74,72)(66,69,75)(67,76,70)(68,71,73)(93,108,98)(94,99,105)(95,106,100)(96,97,107), (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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108), (2,4)(5,76)(6,75)(7,74)(8,73)(9,83)(10,82)(11,81)(12,84)(13,49)(14,52)(15,51)(16,50)(17,105)(18,108)(19,107)(20,106)(21,96)(22,95)(23,94)(24,93)(25,48)(26,47)(27,46)(28,45)(29,92)(30,91)(31,90)(32,89)(33,77)(34,80)(35,79)(36,78)(37,104)(38,103)(39,102)(40,101)(41,71)(42,70)(43,69)(44,72)(53,98)(54,97)(55,100)(56,99)(57,64)(58,63)(59,62)(60,61)(65,88)(66,87)(67,86)(68,85) );

G=PermutationGroup([(1,9,83),(2,84,10),(3,11,81),(4,82,12),(5,31,17),(6,18,32),(7,29,19),(8,20,30),(13,62,38),(14,39,63),(15,64,40),(16,37,61),(21,44,36),(22,33,41),(23,42,34),(24,35,43),(25,67,99),(26,100,68),(27,65,97),(28,98,66),(45,87,53),(46,54,88),(47,85,55),(48,56,86),(49,103,59),(50,60,104),(51,101,57),(52,58,102),(69,79,93),(70,94,80),(71,77,95),(72,96,78),(73,91,106),(74,107,92),(75,89,108),(76,105,90)], [(1,70,42),(2,43,71),(3,72,44),(4,41,69),(5,40,67),(6,68,37),(7,38,65),(8,66,39),(9,94,34),(10,35,95),(11,96,36),(12,33,93),(13,97,29),(14,30,98),(15,99,31),(16,32,100),(17,64,25),(18,26,61),(19,62,27),(20,28,63),(21,81,78),(22,79,82),(23,83,80),(24,77,84),(45,106,58),(46,59,107),(47,108,60),(48,57,105),(49,92,54),(50,55,89),(51,90,56),(52,53,91),(73,102,87),(74,88,103),(75,104,85),(76,86,101)], [(1,40,101),(2,102,37),(3,38,103),(4,104,39),(5,86,42),(6,43,87),(7,88,44),(8,41,85),(9,15,57),(10,58,16),(11,13,59),(12,60,14),(17,56,23),(18,24,53),(19,54,21),(20,22,55),(25,90,80),(26,77,91),(27,92,78),(28,79,89),(29,46,36),(30,33,47),(31,48,34),(32,35,45),(49,81,62),(50,63,82),(51,83,64),(52,61,84),(65,74,72),(66,69,75),(67,76,70),(68,71,73),(93,108,98),(94,99,105),(95,106,100),(96,97,107)], [(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),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96),(97,98,99,100),(101,102,103,104),(105,106,107,108)], [(2,4),(5,76),(6,75),(7,74),(8,73),(9,83),(10,82),(11,81),(12,84),(13,49),(14,52),(15,51),(16,50),(17,105),(18,108),(19,107),(20,106),(21,96),(22,95),(23,94),(24,93),(25,48),(26,47),(27,46),(28,45),(29,92),(30,91),(31,90),(32,89),(33,77),(34,80),(35,79),(36,78),(37,104),(38,103),(39,102),(40,101),(41,71),(42,70),(43,69),(44,72),(53,98),(54,97),(55,100),(56,99),(57,64),(58,63),(59,62),(60,61),(65,88),(66,87),(67,86),(68,85)])

C3315D4 is a maximal subgroup of   C62.90D6  C62.93D6  S3×C327D4  C3⋊S3×C3⋊D4  C62.160D6  D4×C33⋊C2  C62.100D6
C3315D4 is a maximal quotient of   C62.146D6  C62.148D6  C3315D8  C3324SD16  C3327SD16  C3315Q16  C63.C2

57 conjugacy classes

class 1 2A2B2C3A···3M 4 6A···6AM
order12223···346···6
size112542···2542···2

57 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2S3D4D6C3⋊D4
kernelC3315D4C335C4C2×C33⋊C2C3×C62C62C33C3×C6C32
# reps11111311326

Matrix representation of C3315D4 in GL6(𝔽13)

010000
12120000
00121200
001000
000010
000001
,
010000
12120000
000100
00121200
000010
000001
,
100000
010000
001000
000100
0000121
0000120
,
1190000
1120000
000100
001000
0000411
000029
,
100000
12120000
000100
001000
000001
000010

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

C3315D4 in GAP, Magma, Sage, TeX

C_3^3\rtimes_{15}D_4
% in TeX

G:=Group("C3^3:15D4");
// GroupNames label

G:=SmallGroup(216,149);
// by ID

G=gap.SmallGroup(216,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,387,1444,5189]);
// 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=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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