Copied to
clipboard

?

G = C23×D12  order 192 = 26·3

Direct product of C23 and D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23×D12, C12⋊2C24, D6⋊1C24, C6.3C25, C24.93D6, C3⋊1(D4×C23), C4⋊2(S3×C23), C6⋊1(C22×D4), (S3×C24)⋊4C2, (C23×C12)⋊9C2, (C23×C4)⋊11S3, (C22×C4)⋊48D6, (C22×C6)⋊16D4, C2.4(S3×C24), (C2×C12)⋊14C23, (C2×C6).325C24, (C22×S3)⋊7C23, (S3×C23)⋊22C22, (C22×C12)⋊61C22, C22.53(S3×C23), (C23×C6).115C22, (C22×C6).432C23, C23.356(C22×S3), (C2×C6)⋊12(C2×D4), (C2×C4)⋊11(C22×S3), SmallGroup(192,1512)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C23×D12
Chief series
C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C24 — C23×D12
Lower central
C3 — C6 — C23×D12

Subgroups: 3512 in 1362 conjugacy classes, 543 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×16], C3, C4 [×8], C22 [×35], C22 [×128], S3 [×16], C6, C6 [×14], C2×C4 [×28], D4 [×64], C23 [×15], C23 [×168], C12 [×8], D6 [×16], D6 [×112], C2×C6 [×35], C22×C4 [×14], C2×D4 [×112], C24, C24 [×44], D12 [×64], C2×C12 [×28], C22×S3 [×56], C22×S3 [×112], C22×C6 [×15], C23×C4, C22×D4 [×28], C25 [×2], C2×D12 [×112], C22×C12 [×14], S3×C23 [×28], S3×C23 [×16], C23×C6, D4×C23, C22×D12 [×28], C23×C12, S3×C24 [×2], C23×D12

Quotients:
C1, C2 [×31], C22 [×155], S3, D4 [×8], C23 [×155], D6 [×15], C2×D4 [×28], C24 [×31], D12 [×8], C22×S3 [×35], C22×D4 [×14], C25, C2×D12 [×28], S3×C23 [×15], D4×C23, C22×D12 [×14], S3×C24, C23×D12

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Smallest permutation representation
►On 96 points
Generators in S96
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 73)

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

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

(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)

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
0120000
0012000
0001200
000010
000001
,
1200000
010000
001000
000100
000010
000001
,
100000
010000
0012000
0001200
000010
000001
,
100000
010000
001100
0012000
0000103
0000107
,
100000
0120000
00121200
000100
00001212
000001

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

72 conjugacy classes

class 1 2A···2O2P···2AE 3 4A···4H6A···6O12A···12P
order12···22···234···46···612···12
size11···16···622···22···22···2

72 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2S3D4D6D6D12
kernelC23×D12C22×D12C23×C12S3×C24C23×C4C22×C6C22×C4C24C23
# reps128121814116

In GAP, Magma, Sage, TeX 

C_2^3\times D_{12}
 
% in TeX
 
G:=Group("C2^3xD12");
 
// GroupNames label
 
G:=SmallGroup(192,1512);
 
// by ID
 
G=gap.SmallGroup(192,1512);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1684,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀