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## G = C2×D24⋊C2order 192 = 26·3

### Direct product of C2 and D24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D24⋊C2
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×Q8⋊3S3 — C2×D24⋊C2
 Lower central C3 — C6 — C12 — C2×D24⋊C2
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C2×D24⋊C2
G = < a,b,c,d | a2=b24=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b17, dcd=b4c >

Subgroups: 728 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×S3, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, S3×C8, D24, C2×C3⋊C8, Q82S3, C2×C24, C3×Q16, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q83S3, Q83S3, C6×Q8, C2×C4○D8, S3×C2×C8, C2×D24, D24⋊C2, C2×Q82S3, C6×Q16, C2×Q83S3, C2×D24⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C4○D8, C22×D4, S3×D4, S3×C23, C2×C4○D8, D24⋊C2, C2×S3×D4, C2×D24⋊C2

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 C4○D8 S3×D4 S3×D4 D24⋊C2 kernel C2×D24⋊C2 S3×C2×C8 C2×D24 D24⋊C2 C2×Q8⋊2S3 C6×Q16 C2×Q8⋊3S3 C2×Q16 C4×S3 C2×Dic3 C22×S3 C2×C8 Q16 C2×Q8 C6 C4 C22 C2 # reps 1 1 1 8 2 1 2 1 2 1 1 1 4 2 8 1 1 4

Matrix representation of C2×D24⋊C2 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
,
 0 16 0 0 0 0 41 41 0 0 0 0 0 0 32 25 0 0 0 0 35 0 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 0 16 0 0 0 0 32 0 0 0 0 0 0 0 32 25 0 0 0 0 35 41 0 0 0 0 0 0 72 1 0 0 0 0 0 1
,
 46 46 0 0 0 0 54 27 0 0 0 0 0 0 46 8 0 0 0 0 55 27 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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],[0,41,0,0,0,0,16,41,0,0,0,0,0,0,32,35,0,0,0,0,25,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[0,32,0,0,0,0,16,0,0,0,0,0,0,0,32,35,0,0,0,0,25,41,0,0,0,0,0,0,72,0,0,0,0,0,1,1],[46,54,0,0,0,0,46,27,0,0,0,0,0,0,46,55,0,0,0,0,8,27,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D24⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C2xD24:C2");
// GroupNames label

G:=SmallGroup(192,1324);
// by ID

G=gap.SmallGroup(192,1324);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,1123,185,136,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^24=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^17,d*c*d=b^4*c>;
// generators/relations

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