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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 [×3], C2 [×4], C2 [×4], C3, C4 [×8], C22 [×3], C22 [×4], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×26], C23, C23 [×10], Dic3 [×4], C12 [×4], D6 [×4], D6 [×12], C2×C6 [×3], C2×C6 [×4], C22⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×11], C24, C4×S3 [×8], C2×Dic3 [×2], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×8], C22×S3 [×6], C22×S3 [×4], C22×C6, C2.C42, C2.C42 [×3], C2×C22⋊C4 [×2], C23×C4, D6⋊C4 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23, C23.34D4, C6.C42, C6.C42 [×2], C3×C2.C42, C2×D6⋊C4 [×2], S3×C22×C4, C22.58(S3×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C4×S3 [×2], C22×S3, C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], S3×C2×C4, C4○D12 [×2], S3×D4 [×2], D42S3, Q83S3, C23.34D4, C422S3, S3×C22⋊C4, C23.9D6 [×2], C4⋊C47S3, D6.D4 [×2], C22.58(S3×D4)

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

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

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

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

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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