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G = C33:5C4order 108 = 22·33

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

metabelian, supersoluble, monomial, A-group

Aliases: C33:5C4, C32:5Dic3, C3:(C3:Dic3), C6.3(C3:S3), (C3xC6).10S3, C2.(C33:C2), (C32xC6).3C2, SmallGroup(108,34)

Series: Derived Chief Lower central Upper central

C1C33 — C33:5C4
C1C3C32C33C32xC6 — C33:5C4
C33 — C33:5C4
C1C2

Generators and relations for C33:5C4
 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, C4, C6, C32, Dic3, C3xC6, C33, C3:Dic3, C32xC6, C33:5C4
Quotients: C1, C2, C4, S3, Dic3, C3:S3, C3:Dic3, C33:C2, C33:5C4

Character table of C33:5C4

 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 C33:5C4
Regular action on 108 points
Generators in S108
(1 27 74)(2 75 28)(3 25 76)(4 73 26)(5 101 41)(6 42 102)(7 103 43)(8 44 104)(9 81 47)(10 48 82)(11 83 45)(12 46 84)(13 61 49)(14 50 62)(15 63 51)(16 52 64)(17 55 68)(18 65 56)(19 53 66)(20 67 54)(21 35 70)(22 71 36)(23 33 72)(24 69 34)(29 99 38)(30 39 100)(31 97 40)(32 37 98)(57 78 90)(58 91 79)(59 80 92)(60 89 77)(85 105 95)(86 96 106)(87 107 93)(88 94 108)
(1 49 31)(2 32 50)(3 51 29)(4 30 52)(5 80 56)(6 53 77)(7 78 54)(8 55 79)(9 93 22)(10 23 94)(11 95 24)(12 21 96)(13 97 27)(14 28 98)(15 99 25)(16 26 100)(17 91 104)(18 101 92)(19 89 102)(20 103 90)(33 108 48)(34 45 105)(35 106 46)(36 47 107)(37 62 75)(38 76 63)(39 64 73)(40 74 61)(41 59 65)(42 66 60)(43 57 67)(44 68 58)(69 83 85)(70 86 84)(71 81 87)(72 88 82)
(1 22 43)(2 44 23)(3 24 41)(4 42 21)(5 25 69)(6 70 26)(7 27 71)(8 72 28)(9 57 49)(10 50 58)(11 59 51)(12 52 60)(13 81 78)(14 79 82)(15 83 80)(16 77 84)(17 108 37)(18 38 105)(19 106 39)(20 40 107)(29 95 65)(30 66 96)(31 93 67)(32 68 94)(33 75 104)(34 101 76)(35 73 102)(36 103 74)(45 92 63)(46 64 89)(47 90 61)(48 62 91)(53 86 100)(54 97 87)(55 88 98)(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)

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

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

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

C33:5C4 is a maximal subgroup of
S3xC3:Dic3  Dic3xC3:S3  C33:6D4  C33:4Q8  C33:8Q8  C4xC33:C2  C33:15D4  C33:C12  C33:4C12  C32:5Dic9  C34:8C4  C32:4CSU2(F3)  C62:10Dic3
C33:5C4 is a maximal quotient of
C33:7C8  C32:5Dic9  He3:6Dic3  C34:8C4  C62:10Dic3

Matrix representation of C33:5C4 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] >;

C33:5C4 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 C33:5C4 in TeX

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