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## G = C22.58(S3×D4)  order 192 = 26·3

### 9th central extension by C22 of S3×D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C22.58(S3×D4)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — S3×C22×C4 — C22.58(S3×D4)
 Lower central C3 — C2×C6 — C22.58(S3×D4)
 Upper central C1 — C23 — C2.C42

Generators and relations for C22.58(S3×D4)
G = < a,b,c,d,e,f | a2=b2=c3=d2=e4=1, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ede-1=bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, df=fd, fef-1=abe-1 >

Subgroups: 608 in 218 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2.C42, C2×C22⋊C4, C23×C4, D6⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C23.34D4, C6.C42, C6.C42, C3×C2.C42, C2×D6⋊C4, S3×C22×C4, C22.58(S3×D4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C2×C22⋊C4, C42⋊C2, C22.D4, S3×C2×C4, C4○D12, S3×D4, D42S3, Q83S3, C23.34D4, C422S3, S3×C22⋊C4, C23.9D6, C4⋊C47S3, D6.D4, C22.58(S3×D4)

Smallest permutation representation of C22.58(S3×D4)
On 96 points
Generators in S96
(1 59)(2 60)(3 57)(4 58)(5 84)(6 81)(7 82)(8 83)(9 85)(10 86)(11 87)(12 88)(13 69)(14 70)(15 71)(16 72)(17 23)(18 24)(19 21)(20 22)(25 42)(26 43)(27 44)(28 41)(29 52)(30 49)(31 50)(32 51)(33 56)(34 53)(35 54)(36 55)(37 68)(38 65)(39 66)(40 67)(45 61)(46 62)(47 63)(48 64)(73 77)(74 78)(75 79)(76 80)(89 95)(90 96)(91 93)(92 94)
(1 76)(2 73)(3 74)(4 75)(5 51)(6 52)(7 49)(8 50)(9 68)(10 65)(11 66)(12 67)(13 56)(14 53)(15 54)(16 55)(17 44)(18 41)(19 42)(20 43)(21 25)(22 26)(23 27)(24 28)(29 81)(30 82)(31 83)(32 84)(33 69)(34 70)(35 71)(36 72)(37 85)(38 86)(39 87)(40 88)(45 95)(46 96)(47 93)(48 94)(57 78)(58 79)(59 80)(60 77)(61 89)(62 90)(63 91)(64 92)
(1 93 44)(2 94 41)(3 95 42)(4 96 43)(5 65 55)(6 66 56)(7 67 53)(8 68 54)(9 15 50)(10 16 51)(11 13 52)(12 14 49)(17 76 47)(18 73 48)(19 74 45)(20 75 46)(21 78 61)(22 79 62)(23 80 63)(24 77 64)(25 57 89)(26 58 90)(27 59 91)(28 60 92)(29 87 69)(30 88 70)(31 85 71)(32 86 72)(33 81 39)(34 82 40)(35 83 37)(36 84 38)
(1 74)(2 4)(3 76)(5 12)(6 68)(7 10)(8 66)(9 52)(11 50)(13 15)(14 55)(16 53)(17 95)(18 46)(19 93)(20 48)(21 91)(22 64)(23 89)(24 62)(25 63)(26 92)(27 61)(28 90)(29 85)(30 38)(31 87)(32 40)(33 35)(34 72)(36 70)(37 81)(39 83)(41 96)(42 47)(43 94)(44 45)(49 65)(51 67)(54 56)(57 80)(58 60)(59 78)(69 71)(73 75)(77 79)(82 86)(84 88)
(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 14 76 53)(2 33 73 69)(3 16 74 55)(4 35 75 71)(5 95 51 45)(6 64 52 92)(7 93 49 47)(8 62 50 90)(9 26 68 22)(10 19 65 42)(11 28 66 24)(12 17 67 44)(13 60 56 77)(15 58 54 79)(18 87 41 39)(20 85 43 37)(21 38 25 86)(23 40 27 88)(29 94 81 48)(30 63 82 91)(31 96 83 46)(32 61 84 89)(34 59 70 80)(36 57 72 78)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 2 2 2 2 2 4 4 4 4 6 6 6 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + image C1 C2 C2 C2 C2 C4 S3 D4 D6 C4○D4 C4×S3 C4○D12 S3×D4 D4⋊2S3 Q8⋊3S3 kernel C22.58(S3×D4) C6.C42 C3×C2.C42 C2×D6⋊C4 S3×C22×C4 S3×C2×C4 C2.C42 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 C22 C22 # reps 1 3 1 2 1 8 1 4 3 8 4 8 2 1 1

Matrix representation of C22.58(S3×D4) in GL6(𝔽13)

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

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

C22.58(S3×D4) in GAP, Magma, Sage, TeX

C_2^2._{58}(S_3\times D_4)
% in TeX

G:=Group("C2^2.58(S3xD4)");
// GroupNames label

G:=SmallGroup(192,223);
// by ID

G=gap.SmallGroup(192,223);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,422,387,58,6278]);
// Polycyclic

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

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