Copied to
clipboard

G = C22×C28order 112 = 24·7

Abelian group of type [2,2,28]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C28, SmallGroup(112,37)

Series: Derived Chief Lower central Upper central

C1 — C22×C28
C1C2C14C28C2×C28 — C22×C28
C1 — C22×C28
C1 — C22×C28

Generators and relations for C22×C28
 G = < a,b,c | a2=b2=c28=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C7, C2×C4 [×6], C23, C14, C14 [×6], C22×C4, C28 [×4], C2×C14 [×7], C2×C28 [×6], C22×C14, C22×C28
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C7, C2×C4 [×6], C23, C14 [×7], C22×C4, C28 [×4], C2×C14 [×7], C2×C28 [×6], C22×C14, C22×C28

Smallest permutation representation of C22×C28
Regular action on 112 points
Generators in S112
(1 55)(2 56)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(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 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)

G:=sub<Sym(112)| (1,55)(2,56)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)>;

G:=Group( (1,55)(2,56)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112) );

G=PermutationGroup([(1,55),(2,56),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(1,93),(2,94),(3,95),(4,96),(5,97),(6,98),(7,99),(8,100),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)])

C22×C28 is a maximal subgroup of   C28.55D4  C14.C42  C28.48D4  C23.21D14  C23.23D14  C287D4

112 conjugacy classes

class 1 2A···2G4A···4H7A···7F14A···14AP28A···28AV
order12···24···47···714···1428···28
size11···11···11···11···11···1

112 irreducible representations

dim11111111
type+++
imageC1C2C2C4C7C14C14C28
kernelC22×C28C2×C28C22×C14C2×C14C22×C4C2×C4C23C22
# reps1618636648

Matrix representation of C22×C28 in GL3(𝔽29) generated by

100
0280
001
,
2800
0280
001
,
500
0200
002
G:=sub<GL(3,GF(29))| [1,0,0,0,28,0,0,0,1],[28,0,0,0,28,0,0,0,1],[5,0,0,0,20,0,0,0,2] >;

C22×C28 in GAP, Magma, Sage, TeX

C_2^2\times C_{28}
% in TeX

G:=Group("C2^2xC28");
// GroupNames label

G:=SmallGroup(112,37);
// by ID

G=gap.SmallGroup(112,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-2,280]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^28=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽