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## G = C22×C24order 96 = 25·3

### Abelian group of type [2,2,24]

Aliases: C22×C24, SmallGroup(96,176)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C24
 Chief series C1 — C2 — C4 — C12 — C24 — C2×C24 — C22×C24
 Lower central C1 — C22×C24
 Upper central C1 — C22×C24

Generators and relations for C22×C24
G = < a,b,c | a2=b2=c24=1, ab=ba, ac=ca, bc=cb >

Subgroups: 76, all normal (12 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C22 [×7], C6, C6 [×6], C8 [×4], C2×C4 [×6], C23, C12, C12 [×3], C2×C6 [×7], C2×C8 [×6], C22×C4, C24 [×4], C2×C12 [×6], C22×C6, C22×C8, C2×C24 [×6], C22×C12, C22×C24
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C8 [×4], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C2×C8 [×6], C22×C4, C24 [×4], C2×C12 [×6], C22×C6, C22×C8, C2×C24 [×6], C22×C12, C22×C24

Smallest permutation representation of C22×C24
Regular action on 96 points
Generators in S96
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 49)(24 50)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 73)(46 74)(47 75)(48 76)
(1 45)(2 46)(3 47)(4 48)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(49 95)(50 96)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)
(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)

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

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

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

C22×C24 is a maximal subgroup of
C24.98D4  (C2×C24)⋊5C4  C12.9C42  C12.10C42  C12.12C42  Dic3⋊C8⋊C2  C23.27D12  (C22×C8)⋊7S3  C2433D4  C23.28D12  C2430D4  C2429D4  C24.82D4

96 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 6A ··· 6N 8A ··· 8P 12A ··· 12P 24A ··· 24AF order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 kernel C22×C24 C2×C24 C22×C12 C22×C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 6 1 2 6 2 12 2 16 12 4 32

Matrix representation of C22×C24 in GL3(𝔽73) generated by

 1 0 0 0 72 0 0 0 1
,
 1 0 0 0 72 0 0 0 72
,
 63 0 0 0 3 0 0 0 72
G:=sub<GL(3,GF(73))| [1,0,0,0,72,0,0,0,1],[1,0,0,0,72,0,0,0,72],[63,0,0,0,3,0,0,0,72] >;

C22×C24 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC24");
// GroupNames label

G:=SmallGroup(96,176);
// by ID

G=gap.SmallGroup(96,176);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,88]);
// Polycyclic

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

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