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G = C4×C28order 112 = 24·7

Abelian group of type [4,28]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C28, SmallGroup(112,19)

Series: Derived Chief Lower central Upper central

C1 — C4×C28
C1C2C22C2×C14C2×C28 — C4×C28
C1 — C4×C28
C1 — C4×C28

Generators and relations for C4×C28
 G = < a,b | a4=b28=1, ab=ba >


Smallest permutation representation of C4×C28
Regular action on 112 points
Generators in S112
(1 74 46 86)(2 75 47 87)(3 76 48 88)(4 77 49 89)(5 78 50 90)(6 79 51 91)(7 80 52 92)(8 81 53 93)(9 82 54 94)(10 83 55 95)(11 84 56 96)(12 57 29 97)(13 58 30 98)(14 59 31 99)(15 60 32 100)(16 61 33 101)(17 62 34 102)(18 63 35 103)(19 64 36 104)(20 65 37 105)(21 66 38 106)(22 67 39 107)(23 68 40 108)(24 69 41 109)(25 70 42 110)(26 71 43 111)(27 72 44 112)(28 73 45 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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)

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

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

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

C4×C28 is a maximal subgroup of
C42.D7  C28⋊C8  Dic14⋊C4  C282Q8  C28.6Q8  C42⋊D7  C284D4  C4.D28  C422D7  C42⋊(C7⋊C3)

112 conjugacy classes

class 1 2A2B2C4A···4L7A···7F14A···14R28A···28BT
order12224···47···714···1428···28
size11111···11···11···11···1

112 irreducible representations

dim111111
type++
imageC1C2C4C7C14C28
kernelC4×C28C2×C28C28C42C2×C4C4
# reps131261872

Matrix representation of C4×C28 in GL2(𝔽29) generated by

170
01
,
260
014
G:=sub<GL(2,GF(29))| [17,0,0,1],[26,0,0,14] >;

C4×C28 in GAP, Magma, Sage, TeX

C_4\times C_{28}
% in TeX

G:=Group("C4xC28");
// GroupNames label

G:=SmallGroup(112,19);
// by ID

G=gap.SmallGroup(112,19);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-2,140,286]);
// Polycyclic

G:=Group<a,b|a^4=b^28=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C28 in TeX

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