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G = C3312D4order 216 = 23·33

3rd semidirect product of C33 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial

Aliases: C3312D4, C328D12, (C3×C12)⋊5S3, C121(C3⋊S3), C4⋊(C33⋊C2), (C3×C6).64D6, (C32×C12)⋊1C2, C31(C12⋊S3), (C32×C6).28C22, C6.16(C2×C3⋊S3), (C2×C33⋊C2)⋊3C2, C2.4(C2×C33⋊C2), SmallGroup(216,147)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C3312D4
C1C3C32C33C32×C6C2×C33⋊C2 — C3312D4
C33C32×C6 — C3312D4
C1C2C4

Generators and relations for C3312D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1372 in 224 conjugacy classes, 87 normal (7 characteristic)
C1, C2, C2 [×2], C3 [×13], C4, C22 [×2], S3 [×26], C6 [×13], D4, C32 [×13], C12 [×13], D6 [×26], C3⋊S3 [×26], C3×C6 [×13], D12 [×13], C33, C3×C12 [×13], C2×C3⋊S3 [×26], C33⋊C2 [×2], C32×C6, C12⋊S3 [×13], C32×C12, C2×C33⋊C2 [×2], C3312D4
Quotients: C1, C2 [×3], C22, S3 [×13], D4, D6 [×13], C3⋊S3 [×13], D12 [×13], C2×C3⋊S3 [×13], C33⋊C2, C12⋊S3 [×13], C2×C33⋊C2, C3312D4

Smallest permutation representation of C3312D4
On 108 points
Generators in S108
(1 27 74)(2 28 75)(3 25 76)(4 26 73)(5 101 41)(6 102 42)(7 103 43)(8 104 44)(9 81 47)(10 82 48)(11 83 45)(12 84 46)(13 61 49)(14 62 50)(15 63 51)(16 64 52)(17 68 55)(18 65 56)(19 66 53)(20 67 54)(21 70 35)(22 71 36)(23 72 33)(24 69 34)(29 99 38)(30 100 39)(31 97 40)(32 98 37)(57 78 90)(58 79 91)(59 80 92)(60 77 89)(85 105 95)(86 106 96)(87 107 93)(88 108 94)
(1 49 31)(2 50 32)(3 51 29)(4 52 30)(5 80 56)(6 77 53)(7 78 54)(8 79 55)(9 93 22)(10 94 23)(11 95 24)(12 96 21)(13 97 27)(14 98 28)(15 99 25)(16 100 26)(17 104 91)(18 101 92)(19 102 89)(20 103 90)(33 48 108)(34 45 105)(35 46 106)(36 47 107)(37 75 62)(38 76 63)(39 73 64)(40 74 61)(41 59 65)(42 60 66)(43 57 67)(44 58 68)(69 83 85)(70 84 86)(71 81 87)(72 82 88)
(1 22 43)(2 23 44)(3 24 41)(4 21 42)(5 25 69)(6 26 70)(7 27 71)(8 28 72)(9 57 49)(10 58 50)(11 59 51)(12 60 52)(13 81 78)(14 82 79)(15 83 80)(16 84 77)(17 37 108)(18 38 105)(19 39 106)(20 40 107)(29 95 65)(30 96 66)(31 93 67)(32 94 68)(33 104 75)(34 101 76)(35 102 73)(36 103 74)(45 92 63)(46 89 64)(47 90 61)(48 91 62)(53 100 86)(54 97 87)(55 98 88)(56 99 85)
(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)
(1 4)(2 3)(5 33)(6 36)(7 35)(8 34)(9 66)(10 65)(11 68)(12 67)(13 39)(14 38)(15 37)(16 40)(17 83)(18 82)(19 81)(20 84)(21 43)(22 42)(23 41)(24 44)(25 75)(26 74)(27 73)(28 76)(29 50)(30 49)(31 52)(32 51)(45 55)(46 54)(47 53)(48 56)(57 96)(58 95)(59 94)(60 93)(61 100)(62 99)(63 98)(64 97)(69 104)(70 103)(71 102)(72 101)(77 107)(78 106)(79 105)(80 108)(85 91)(86 90)(87 89)(88 92)

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

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

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

C3312D4 is a maximal subgroup of
C337D8  C338D8  C3315SD16  C3317SD16  C3321SD16  C3312D8  C3315D8  C3327SD16  C12.40S32  C12.58S32  S3×C12⋊S3  C3⋊S3×D12  C62.160D6  D4×C33⋊C2  (Q8×C33)⋊C2
C3312D4 is a maximal quotient of
C3321SD16  C3312D8  C3312Q16  C62.147D6  C62.148D6

57 conjugacy classes

class 1 2A2B2C3A···3M 4 6A···6M12A···12Z
order12223···346···612···12
size1154542···222···22···2

57 irreducible representations

dim1112222
type+++++++
imageC1C2C2S3D4D6D12
kernelC3312D4C32×C12C2×C33⋊C2C3×C12C33C3×C6C32
# reps1121311326

Matrix representation of C3312D4 in GL6(𝔽13)

12120000
100000
000100
00121200
0000121
0000120
,
010000
12120000
000100
00121200
0000121
0000120
,
100000
010000
001000
000100
0000121
0000120
,
1200000
0120000
0012000
0001200
0000106
000073
,
1200000
110000
001000
00121200
000073
0000106

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,7,10,0,0,0,0,3,6] >;

C3312D4 in GAP, Magma, Sage, TeX

C_3^3\rtimes_{12}D_4
% in TeX

G:=Group("C3^3:12D4");
// GroupNames label

G:=SmallGroup(216,147);
// by ID

G=gap.SmallGroup(216,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,387,1444,5189]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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