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## G = C2×C23.8D6order 192 = 26·3

### Direct product of C2 and C23.8D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.8D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C2×C23.8D6
 Lower central C3 — C2×C6 — C2×C23.8D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.8D6
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=e5 >

Subgroups: 520 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C2×C422C2, C23.8D6, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C2×C23.8D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C422C2, C2×C4○D4, C4○D12, D42S3, S3×C23, C2×C422C2, C23.8D6, C2×C4○D12, C2×D42S3, C2×C23.8D6

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 C4○D12 D4⋊2S3 kernel C2×C23.8D6 C23.8D6 C2×C4×Dic3 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×C6.D4 C6×C22⋊C4 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C2×C6 C22 C22 # reps 1 8 1 2 1 2 1 1 4 2 1 12 8 4

Matrix representation of C2×C23.8D6 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 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 12 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 1 0 0 0 0 0 0 1
,
 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 12 0 0 0 0 0 0 12
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 11 0 0 0 0 0 1
,
 3 3 0 0 0 0 6 10 0 0 0 0 0 0 8 3 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,11,1],[3,6,0,0,0,0,3,10,0,0,0,0,0,0,8,0,0,0,0,0,3,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

C2×C23.8D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._8D_6
% in TeX

G:=Group("C2xC2^3.8D6");
// GroupNames label

G:=SmallGroup(192,1041);
// by ID

G=gap.SmallGroup(192,1041);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,1571,297,6278]);
// 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=e^5>;
// generators/relations

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