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## G = C2×C23.21D6order 192 = 26·3

### Direct product of C2 and C23.21D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.21D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C2×C23.21D6
 Lower central C3 — C2×C6 — C2×C23.21D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.21D6
G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 872 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C2×C22.D4, C23.21D6, C2×C4⋊Dic3, C2×D6⋊C4, C6×C22⋊C4, C23×Dic3, C22×C3⋊D4, C2×C23.21D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C22×S3, C22.D4, C22×D4, C2×C4○D4, C2×D12, D42S3, S3×C23, C2×C22.D4, C23.21D6, C22×D12, C2×D42S3, C2×C23.21D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C 4D 4E ··· 4L 4M 4N 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 12 12 2 4 4 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 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 D12 D4⋊2S3 kernel C2×C23.21D6 C23.21D6 C2×C4⋊Dic3 C2×D6⋊C4 C6×C22⋊C4 C23×Dic3 C22×C3⋊D4 C2×C22⋊C4 C22×C6 C22⋊C4 C22×C4 C24 C2×C6 C23 C22 # reps 1 8 2 2 1 1 1 1 4 4 2 1 8 8 4

Matrix representation of C2×C23.21D6 in GL6(𝔽13)

 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 2 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1
,
 5 5 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 5 8 0 0 0 0 0 0 7 10 0 0 0 0 3 10
,
 5 5 0 0 0 0 3 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 7 10 0 0 0 0 3 6

G:=sub<GL(6,GF(13))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,0,0,7,3,0,0,0,0,10,10],[5,3,0,0,0,0,5,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,7,3,0,0,0,0,10,6] >;

C2×C23.21D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{21}D_6
% in TeX

G:=Group("C2xC2^3.21D6");
// GroupNames label

G:=SmallGroup(192,1051);
// by ID

G=gap.SmallGroup(192,1051);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,297,192,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

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