Copied to
clipboard

## G = (C2×C42)⋊3S3order 192 = 26·3

### 2nd semidirect product of C2×C42 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C42)⋊3S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C2×C42)⋊3S3
 Lower central C3 — C2×C6 — (C2×C42)⋊3S3
 Upper central C1 — C23 — C2×C42

Generators and relations for (C2×C42)⋊3S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ebe=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 536 in 190 conjugacy classes, 71 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×10], C22 [×7], C22 [×10], S3 [×2], C6 [×7], C2×C4 [×6], C2×C4 [×16], C23, C23 [×8], Dic3 [×4], C12 [×6], D6 [×10], C2×C6 [×7], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×3], C24, C2×Dic3 [×2], C2×Dic3 [×8], C2×C12 [×6], C2×C12 [×6], C22×S3 [×2], C22×S3 [×6], C22×C6, C2.C42 [×2], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, Dic3⋊C4 [×2], D6⋊C4 [×4], D6⋊C4 [×4], C4×C12 [×2], C22×Dic3 [×3], C22×C12 [×3], S3×C23, C24.C22, C6.C42 [×2], C2×Dic3⋊C4, C2×D6⋊C4 [×3], C2×C4×C12, (C2×C42)⋊3S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22×C4, C2×D4 [×2], C4○D4 [×4], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, S3×C2×C4, C2×D12, C4○D12 [×4], C2×C3⋊D4, C24.C22, C422S3, C4×D12, C427S3, C423S3, C4×C3⋊D4, C23.28D6, C127D4, (C2×C42)⋊3S3

Smallest permutation representation of (C2×C42)⋊3S3
On 96 points
Generators in S96
(1 53)(2 54)(3 55)(4 56)(5 79)(6 80)(7 77)(8 78)(9 29)(10 30)(11 31)(12 32)(13 35)(14 36)(15 33)(16 34)(17 91)(18 92)(19 89)(20 90)(21 41)(22 42)(23 43)(24 44)(25 47)(26 48)(27 45)(28 46)(37 59)(38 60)(39 57)(40 58)(49 71)(50 72)(51 69)(52 70)(61 81)(62 82)(63 83)(64 84)(65 87)(66 88)(67 85)(68 86)(73 93)(74 94)(75 95)(76 96)
(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 47 35 87)(2 48 36 88)(3 45 33 85)(4 46 34 86)(5 69 57 17)(6 70 58 18)(7 71 59 19)(8 72 60 20)(9 73 61 21)(10 74 62 22)(11 75 63 23)(12 76 64 24)(13 65 53 25)(14 66 54 26)(15 67 55 27)(16 68 56 28)(29 93 81 41)(30 94 82 42)(31 95 83 43)(32 96 84 44)(37 89 77 49)(38 90 78 50)(39 91 79 51)(40 92 80 52)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 81 39)(14 82 40)(15 83 37)(16 84 38)(17 87 21)(18 88 22)(19 85 23)(20 86 24)(25 93 51)(26 94 52)(27 95 49)(28 96 50)(29 79 53)(30 80 54)(31 77 55)(32 78 56)(33 63 59)(34 64 60)(35 61 57)(36 62 58)(41 91 65)(42 92 66)(43 89 67)(44 90 68)(45 75 71)(46 76 72)(47 73 69)(48 74 70)
(2 54)(4 56)(5 9)(6 30)(7 11)(8 32)(10 80)(12 78)(14 36)(16 34)(17 23)(18 44)(19 21)(20 42)(22 90)(24 92)(25 27)(26 46)(28 48)(29 79)(31 77)(37 83)(38 64)(39 81)(40 62)(41 89)(43 91)(45 47)(49 93)(50 74)(51 95)(52 76)(57 61)(58 82)(59 63)(60 84)(65 67)(66 86)(68 88)(69 75)(70 96)(71 73)(72 94)(85 87)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A ··· 4L 4M ··· 4R 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D6 C4○D4 C4×S3 D12 C3⋊D4 C4○D12 kernel (C2×C42)⋊3S3 C6.C42 C2×Dic3⋊C4 C2×D6⋊C4 C2×C4×C12 D6⋊C4 C2×C42 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 3 1 8 1 4 3 8 4 4 4 16

Matrix representation of (C2×C42)⋊3S3 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 10 7 0 0 0 6 3 0 0 0 0 0 11 9 0 0 0 4 2
,
 5 0 0 0 0 0 3 6 0 0 0 7 10 0 0 0 0 0 8 0 0 0 0 0 8
,
 1 0 0 0 0 0 12 12 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 1 0
,
 12 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 12 12

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,10,6,0,0,0,7,3,0,0,0,0,0,11,4,0,0,0,9,2],[5,0,0,0,0,0,3,7,0,0,0,6,10,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,12,0],[12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,12] >;

(C2×C42)⋊3S3 in GAP, Magma, Sage, TeX

(C_2\times C_4^2)\rtimes_3S_3
% in TeX

G:=Group("(C2xC4^2):3S3");
// GroupNames label

G:=SmallGroup(192,499);
// by ID

G=gap.SmallGroup(192,499);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,758,58,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,e*b*e=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽