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## G = C2×C23.11D6order 192 = 26·3

### Direct product of C2 and C23.11D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.11D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×D6⋊C4 — C2×C23.11D6
 Lower central C3 — C2×C6 — C2×C23.11D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.11D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 872 in 330 conjugacy classes, 119 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C2×C4.4D4, C23.11D6, C2×C4×Dic3, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, C22×Dic6, C22×C3⋊D4, C2×C23.11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C4.4D4, C22×D4, C2×C4○D4, C4○D12, S3×D4, D42S3, S3×C23, C2×C4.4D4, C23.11D6, C2×C4○D12, C2×S3×D4, C2×D42S3, C2×C23.11D6

Smallest permutation representation of C2×C23.11D6
On 96 points
Generators in S96
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 73)(13 93)(14 94)(15 95)(16 96)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(1 35)(2 65)(3 25)(4 67)(5 27)(6 69)(7 29)(8 71)(9 31)(10 61)(11 33)(12 63)(13 68)(14 28)(15 70)(16 30)(17 72)(18 32)(19 62)(20 34)(21 64)(22 36)(23 66)(24 26)(37 82)(38 86)(39 84)(40 88)(41 74)(42 90)(43 76)(44 92)(45 78)(46 94)(47 80)(48 96)(49 95)(50 81)(51 85)(52 83)(53 87)(54 73)(55 89)(56 75)(57 91)(58 77)(59 93)(60 79)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 49)(48 50)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 85)(83 86)(84 87)
(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 6 15 20)(2 19 16 5)(3 4 17 18)(7 12 21 14)(8 13 22 11)(9 10 23 24)(25 61 72 26)(27 71 62 36)(28 35 63 70)(29 69 64 34)(30 33 65 68)(31 67 66 32)(37 58 57 38)(39 56 59 48)(40 47 60 55)(41 54 49 46)(42 45 50 53)(43 52 51 44)(73 89 94 80)(74 79 95 88)(75 87 96 78)(76 77 85 86)(81 93 90 84)(82 83 91 92)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C4○D12 S3×D4 D4⋊2S3 kernel C2×C23.11D6 C23.11D6 C2×C4×Dic3 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 C22×Dic6 C22×C3⋊D4 C2×C22⋊C4 C2×Dic3 C22⋊C4 C22×C4 C24 C2×C6 C22 C22 C22 # reps 1 8 1 2 1 1 1 1 1 4 4 2 1 8 8 2 2

Matrix representation of C2×C23.11D6 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 8 1
,
 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 12 1 0 0 0 0 0 12 0 0 0 0 0 0 0 0 5 11 0 0 0 0 0 0 8
,
 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 8 2 0 0 0 0 0 1 5

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

C2×C23.11D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{11}D_6
% in TeX

G:=Group("C2xC2^3.11D6");
// GroupNames label

G:=SmallGroup(192,1050);
// by ID

G=gap.SmallGroup(192,1050);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,1571,297,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
// generators/relations

׿
×
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