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## G = C22×D24order 192 = 26·3

### Direct product of C22 and D24

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×D24
G = < a,b,c,d | a2=b2=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1240 in 338 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C23, C12, C12, D6, C2×C6, C2×C8, D8, C22×C4, C2×D4, C24, C24, D12, D12, C2×C12, C22×S3, C22×C6, C22×C8, C2×D8, C22×D4, D24, C2×C24, C2×D12, C2×D12, C22×C12, S3×C23, C22×D8, C2×D24, C22×C24, C22×D12, C22×D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C24, D12, C22×S3, C2×D8, C22×D4, D24, C2×D12, S3×C23, C22×D8, C2×D24, C22×D12, C22×D24

Smallest permutation representation of C22×D24
On 96 points
Generators in S96
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 70)(26 71)(27 72)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)
(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 73)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A 4B 4C 4D 6A ··· 6G 8A ··· 8H 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 2 ··· 2 3 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 12 ··· 12 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D4 D6 D6 D8 D12 D12 D24 kernel C22×D24 C2×D24 C22×C24 C22×D12 C22×C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 2 1 3 1 6 1 8 6 2 16

Matrix representation of C22×D24 in GL7(𝔽73)

 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 0 57 16 0 0 0 0 0 57 57 0 0 0 0 0 0 0 33 4 0 0 0 0 0 56 40 0 0 0 0 0 0 0 1 72 0 0 0 0 0 1 0
,
 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 53 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 1

G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,57,57,0,0,0,0,0,16,57,0,0,0,0,0,0,0,33,56,0,0,0,0,0,4,40,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0],[72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,53,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1] >;

C22×D24 in GAP, Magma, Sage, TeX

C_2^2\times D_{24}
% in TeX

G:=Group("C2^2xD24");
// GroupNames label

G:=SmallGroup(192,1299);
// by ID

G=gap.SmallGroup(192,1299);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,192,1684,102,6278]);
// Polycyclic

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

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