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## G = C2×C23.26D6order 192 = 26·3

### Direct product of C2 and C23.26D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C23.26D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C2×C23.26D6
 Lower central C3 — C6 — C2×C23.26D6
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C2×C23.26D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 568 in 330 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, 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, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C22×C12, C22×C12, C23×C6, C2×C42⋊C2, C2×C4×Dic3, C2×C4⋊Dic3, C23.26D6, C2×C6.D4, C23×C12, C2×C23.26D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C24, C2×Dic3, C22×S3, C42⋊C2, C23×C4, C2×C4○D4, C4○D12, C22×Dic3, S3×C23, C2×C42⋊C2, C23.26D6, C2×C4○D12, C23×Dic3, C2×C23.26D6

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

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I 4J 4K 4L 4M ··· 4AB 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 1 ··· 1 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C4 S3 Dic3 D6 D6 C4○D4 C4○D12 kernel C2×C23.26D6 C2×C4×Dic3 C2×C4⋊Dic3 C23.26D6 C2×C6.D4 C23×C12 C22×C12 C23×C4 C22×C4 C22×C4 C24 C2×C6 C22 # reps 1 2 2 8 2 1 16 1 8 6 1 8 16

Matrix representation of C2×C23.26D6 in GL4(𝔽13) generated by

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

C2×C23.26D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{26}D_6
% in TeX

G:=Group("C2xC2^3.26D6");
// GroupNames label

G:=SmallGroup(192,1345);
// by ID

G=gap.SmallGroup(192,1345);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,1123,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,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,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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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