Copied to
clipboard

G = C335C4order 108 = 22·33

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

metabelian, supersoluble, monomial, A-group

Aliases: C335C4, C325Dic3, C3⋊(C3⋊Dic3), C6.3(C3⋊S3), (C3×C6).10S3, C2.(C33⋊C2), (C32×C6).3C2, SmallGroup(108,34)

Series: Derived Chief Lower central Upper central

C1C33 — C335C4
C1C3C32C33C32×C6 — C335C4
C33 — C335C4
C1C2

Generators and relations for C335C4
 G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 240 in 84 conjugacy classes, 57 normal (5 characteristic)
C1, C2, C3 [×13], C4, C6 [×13], C32 [×13], Dic3 [×13], C3×C6 [×13], C33, C3⋊Dic3 [×13], C32×C6, C335C4
Quotients: C1, C2, C4, S3 [×13], Dic3 [×13], C3⋊S3 [×13], C3⋊Dic3 [×13], C33⋊C2, C335C4

Character table of C335C4

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M
 size 11222222222222227272222222222222
ρ1111111111111111111111111111111    trivial
ρ2111111111111111-1-11111111111111    linear of order 2
ρ31-11111111111111-ii-1-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ41-11111111111111i-i-1-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ522-1-1-1-1-1-1-1-1-1222200-1-1-1-1-1-1-1-12222-1    orthogonal lifted from S3
ρ622-1-12-12-12-1-1-1-12-100-12-12-12-1-1-1-12-1-1    orthogonal lifted from S3
ρ7222-1-1-1-12-1-12-1-12-100-1-1-1-12-1-12-1-12-12    orthogonal lifted from S3
ρ822-1-12-1-12-12-12-1-1-100-12-1-12-12-12-1-1-1-1    orthogonal lifted from S3
ρ922-12-1-1-122-1-1-12-1-1002-1-1-122-1-1-12-1-1-1    orthogonal lifted from S3
ρ1022-12-12-1-1-12-1-1-12-1002-12-1-1-12-1-1-12-1-1    orthogonal lifted from S3
ρ11222-1-12-1-12-1-12-1-1-100-1-12-1-12-1-12-1-1-12    orthogonal lifted from S3
ρ1222-1-122-1-1-1-12-12-1-100-122-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ13222-1-1-12-1-12-1-12-1-100-1-1-12-1-12-1-12-1-12    orthogonal lifted from S3
ρ1422-12-1-12-1-1-122-1-1-1002-1-12-1-1-122-1-1-1-1    orthogonal lifted from S3
ρ1522222-1-1-1-1-1-1-1-1-120022-1-1-1-1-1-1-1-1-122    orthogonal lifted from S3
ρ1622-1-1-1222-1-1-1-1-1-1200-1-1222-1-1-1-1-1-12-1    orthogonal lifted from S3
ρ1722-1-1-1-1-1-1222-1-1-1200-1-1-1-1-1222-1-1-12-1    orthogonal lifted from S3
ρ182-2-1-12-12-12-1-1-1-12-1001-21-21-21111-211    symplectic lifted from Dic3, Schur index 2
ρ192-2-1-1-1222-1-1-1-1-1-120011-2-2-2111111-21    symplectic lifted from Dic3, Schur index 2
ρ202-2-12-1-1-122-1-1-12-1-100-2111-2-2111-2111    symplectic lifted from Dic3, Schur index 2
ρ212-22-1-1-1-12-1-12-1-12-1001111-211-211-21-2    symplectic lifted from Dic3, Schur index 2
ρ222-2-1-1-1-1-1-1-1-1-122220011111111-2-2-2-21    symplectic lifted from Dic3, Schur index 2
ρ232-2-12-1-12-1-1-122-1-1-100-211-2111-2-21111    symplectic lifted from Dic3, Schur index 2
ρ242-2-1-12-1-12-12-12-1-1-1001-211-21-21-21111    symplectic lifted from Dic3, Schur index 2
ρ252-22-1-12-1-12-1-12-1-1-10011-211-211-2111-2    symplectic lifted from Dic3, Schur index 2
ρ262-2-1-122-1-1-1-12-12-1-1001-2-21111-21-2111    symplectic lifted from Dic3, Schur index 2
ρ272-22-1-1-12-1-12-1-12-1-100111-211-211-211-2    symplectic lifted from Dic3, Schur index 2
ρ282-2-1-1-1-1-1-1222-1-1-120011111-2-2-2111-21    symplectic lifted from Dic3, Schur index 2
ρ292-2222-1-1-1-1-1-1-1-1-1200-2-2111111111-2-2    symplectic lifted from Dic3, Schur index 2
ρ302-2-12-12-1-1-12-1-1-12-100-21-2111-2111-211    symplectic lifted from Dic3, Schur index 2

Smallest permutation representation of C335C4
Regular action on 108 points
Generators in S108
(1 54 20)(2 17 55)(3 56 18)(4 19 53)(5 51 69)(6 70 52)(7 49 71)(8 72 50)(9 27 74)(10 75 28)(11 25 76)(12 73 26)(13 30 80)(14 77 31)(15 32 78)(16 79 29)(21 86 91)(22 92 87)(23 88 89)(24 90 85)(33 96 106)(34 107 93)(35 94 108)(36 105 95)(37 98 57)(38 58 99)(39 100 59)(40 60 97)(41 63 101)(42 102 64)(43 61 103)(44 104 62)(45 65 83)(46 84 66)(47 67 81)(48 82 68)
(1 49 60)(2 57 50)(3 51 58)(4 59 52)(5 38 18)(6 19 39)(7 40 20)(8 17 37)(9 43 85)(10 86 44)(11 41 87)(12 88 42)(13 81 34)(14 35 82)(15 83 36)(16 33 84)(21 62 28)(22 25 63)(23 64 26)(24 27 61)(29 106 46)(30 47 107)(31 108 48)(32 45 105)(53 100 70)(54 71 97)(55 98 72)(56 69 99)(65 95 78)(66 79 96)(67 93 80)(68 77 94)(73 89 102)(74 103 90)(75 91 104)(76 101 92)
(1 80 43)(2 44 77)(3 78 41)(4 42 79)(5 45 92)(6 89 46)(7 47 90)(8 91 48)(9 60 93)(10 94 57)(11 58 95)(12 96 59)(13 61 54)(14 55 62)(15 63 56)(16 53 64)(17 104 31)(18 32 101)(19 102 29)(20 30 103)(21 82 72)(22 69 83)(23 84 70)(24 71 81)(25 99 36)(26 33 100)(27 97 34)(28 35 98)(37 75 108)(38 105 76)(39 73 106)(40 107 74)(49 67 85)(50 86 68)(51 65 87)(52 88 66)
(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)

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

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

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

C335C4 is a maximal subgroup of
S3×C3⋊Dic3  Dic3×C3⋊S3  C336D4  C334Q8  C338Q8  C4×C33⋊C2  C3315D4  C33⋊C12  C334C12  C325Dic9  C348C4  C324CSU2(𝔽3)  C6210Dic3
C335C4 is a maximal quotient of
C337C8  C325Dic9  He36Dic3  C348C4  C6210Dic3

Matrix representation of C335C4 in GL7(𝔽13)

1000000
0010000
012120000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000100
000001212
0000010
,
1000000
0010000
012120000
0009000
0007300
0000001
000001212
,
5000000
0360000
03100000
0005500
0003800
00000107
00000103

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,9,7,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12],[5,0,0,0,0,0,0,0,3,3,0,0,0,0,0,6,10,0,0,0,0,0,0,0,5,3,0,0,0,0,0,5,8,0,0,0,0,0,0,0,10,10,0,0,0,0,0,7,3] >;

C335C4 in GAP, Magma, Sage, TeX

C_3^3\rtimes_5C_4
% in TeX

G:=Group("C3^3:5C4");
// GroupNames label

G:=SmallGroup(108,34);
// by ID

G=gap.SmallGroup(108,34);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,10,122,483,1804]);
// Polycyclic

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

Export

Character table of C335C4 in TeX

׿
×
𝔽