Copied to
clipboard

G = C2×C33⋊7D4  order 432 = 24·33

Direct product of C2 and C33⋊7D4

direct product, metabelian, supersoluble, monomial

Aliases: C2×C33⋊7D4, C62.114D6, (S3×C6)⋊19D6, (C3×C6)⋊10D12, (C32×C6)⋊7D4, C33⋊22(C2×D4), (S3×C62)⋊5C2, C3⋊Dic3⋊18D6, C6⋊2(C3⋊D12), C32⋊17(C2×D12), C6⋊1(C32⋊7D4), (C32×C6).59C23, (C3×C62).30C22, (S3×C2×C6)⋊7S3, C6.69(C2×S32), D6⋊6(C2×C3⋊S3), (C2×C6).43S32, C3⋊3(C2×C3⋊D12), (C3×C6)⋊7(C3⋊D4), (S3×C3×C6)⋊19C22, (C6×C3⋊Dic3)⋊8C2, C3⋊1(C2×C32⋊7D4), (C2×C3⋊Dic3)⋊11S3, C22.15(S3×C3⋊S3), C6.22(C22×C3⋊S3), C32⋊12(C2×C3⋊D4), (C22×S3)⋊3(C3⋊S3), (C3×C6).148(C22×S3), (C3×C3⋊Dic3)⋊14C22, (C2×C33⋊C2)⋊9C22, (C22×C33⋊C2)⋊1C2, C2.22(C2×S3×C3⋊S3), (C2×C6).24(C2×C3⋊S3), SmallGroup(432,681)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C2×C33⋊7D4
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊7D4 — C2×C33⋊7D4
Lower central
C33 — C32×C6 — C2×C33⋊7D4
Upper central
C1 — C22

Generators and relations for C2×C33⋊7D4
 G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=b-1, cd=dc, ece-1=fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 3000 in 452 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C2×C3⋊S3, C62, C62, C62, C2×D12, C2×C3⋊D4, S3×C32, C33⋊C2, C32×C6, C32×C6, C3⋊D12, C6×Dic3, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C3×C3⋊Dic3, S3×C3×C6, S3×C3×C6, C2×C33⋊C2, C2×C33⋊C2, C3×C62, C2×C3⋊D12, C2×C32⋊7D4, C33⋊7D4, C6×C3⋊Dic3, S3×C62, C22×C33⋊C2, C2×C33⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C3⋊D4, C22×S3, S32, C2×C3⋊S3, C2×D12, C2×C3⋊D4, C3⋊D12, C32⋊7D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C2×C3⋊D12, C2×C32⋊7D4, C33⋊7D4, C2×S3×C3⋊S3, C2×C33⋊7D4

Smallest permutation representation of C2×C33⋊7D4
►On 72 points
Generators in S72
(1 40)(2 37)(3 38)(4 39)(5 46)(6 47)(7 48)(8 45)(9 18)(10 19)(11 20)(12 17)(13 35)(14 36)(15 33)(16 34)(21 25)(22 26)(23 27)(24 28)(29 58)(30 59)(31 60)(32 57)(41 62)(42 63)(43 64)(44 61)(49 71)(50 72)(51 69)(52 70)(53 66)(54 67)(55 68)(56 65)

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6O6P···6AA6AB···6AQ12A12B12C12D
order122222223···33333446···66···66···612121212
size11116654542···2444418182···24···46···618181818

66 irreducible representations

dim1111122222222444
type+++++++++++++++
imageC1C2C2C2C2S3S3D4D6D6D6D12C3⋊D4S32C3⋊D12C2×S32
kernelC2×C33⋊7D4C33⋊7D4C6×C3⋊Dic3S3×C62C22×C33⋊C2C2×C3⋊Dic3S3×C2×C6C32×C6C3⋊Dic3S3×C6C62C3×C6C3×C6C2×C6C6C6
# reps14111142285416484

Matrix representation of C2×C33⋊7D4 ►in GL8(ℤ)

10000000
01000000
00-100000
000-10000
0000-1000
00000-100
00000010
00000001
,
01000000
-1-1000000
00100000
00010000
00001000
00000100
000000-11
000000-10
,
10000000
01000000
00100000
00010000
00001000
00000100
0000000-1
0000001-1
,
10000000
01000000
00100000
00010000
0000-1100
0000-1000
00000010
00000001
,
-10000000
11000000
00010000
00-100000
0000-1000
00000-100
00000001
00000010
,
10000000
-1-1000000
00100000
000-10000
00000-100
0000-1000
00000001
00000010

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C2×C33⋊7D4 in GAP, Magma, Sage, TeX 

C_2\times C_3^3\rtimes_7D_4
 
% in TeX
 
G:=Group("C2xC3^3:7D4");
 
// GroupNames label
 
G:=SmallGroup(432,681);
 
// by ID
 
G=gap.SmallGroup(432,681);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=f*b*f=b^-1,c*d=d*c,e*c*e^-1=f*c*f=c^-1,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁