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G = C22×C24order 96 = 25·3

Abelian group of type [2,2,24]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C24, SmallGroup(96,176)

Series: Derived Chief Lower central Upper central

C1 — C22×C24
C1C2C4C12C24C2×C24 — C22×C24
C1 — C22×C24
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···2G3A3B4A···4H6A···6N8A···8P12A···12P24A···24AF
order12···2334···46···68···812···1224···24
size11···1111···11···11···11···11···1

96 irreducible representations

dim111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C12C12C24
kernelC22×C24C2×C24C22×C12C22×C8C2×C12C22×C6C2×C8C22×C4C2×C6C2×C4C23C22
# reps1612621221612432

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

100
0720
001
,
100
0720
0072
,
6300
030
0072
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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