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## G = C2×C2.D24order 192 = 26·3

### Direct product of C2 and C2.D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C2.D24
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C22×D12 — C2×C2.D24
 Lower central C3 — C6 — C12 — C2×C2.D24
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×C2.D24
G = < a,b,c,d | a2=b2=c24=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 728 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C24, C24, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C4⋊Dic3, C4⋊Dic3, C2×C24, C2×C24, C2×D12, C2×D12, C22×Dic3, C22×C12, S3×C23, C2×D4⋊C4, C2.D24, C2×C4⋊Dic3, C22×C24, C22×D12, C2×C2.D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C24⋊C2, D24, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×D4⋊C4, C2.D24, C2×C24⋊C2, C2×D24, C2×D6⋊C4, C2×C2.D24

Smallest permutation representation of C2×C2.D24
On 96 points
Generators in S96
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 49)(22 50)(23 51)(24 52)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 73)(70 74)(71 75)(72 76)
(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 64 33 92)(2 91 34 63)(3 62 35 90)(4 89 36 61)(5 60 37 88)(6 87 38 59)(7 58 39 86)(8 85 40 57)(9 56 41 84)(10 83 42 55)(11 54 43 82)(12 81 44 53)(13 52 45 80)(14 79 46 51)(15 50 47 78)(16 77 48 49)(17 72 25 76)(18 75 26 71)(19 70 27 74)(20 73 28 69)(21 68 29 96)(22 95 30 67)(23 66 31 94)(24 93 32 65)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 8A ··· 8H 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 12 12 12 12 2 2 2 2 2 12 12 12 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 D8 SD16 C4×S3 D12 C3⋊D4 D12 C24⋊C2 D24 kernel C2×C2.D24 C2.D24 C2×C4⋊Dic3 C22×C24 C22×D12 C2×D12 C22×C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 3 1 2 1 4 4 4 2 4 2 8 8

Matrix representation of C2×C2.D24 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 27 27 0 0 0 0 46 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 36 25 0 0 0 0 48 11
,
 46 46 0 0 0 0 0 27 0 0 0 0 0 0 1 1 0 0 0 0 0 72 0 0 0 0 0 0 36 25 0 0 0 0 62 37

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,46,0,0,0,0,27,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,36,48,0,0,0,0,25,11],[46,0,0,0,0,0,46,27,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,36,62,0,0,0,0,25,37] >;

C2×C2.D24 in GAP, Magma, Sage, TeX

C_2\times C_2.D_{24}
% in TeX

G:=Group("C2xC2.D24");
// GroupNames label

G:=SmallGroup(192,671);
// by ID

G=gap.SmallGroup(192,671);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,142,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^24=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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