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G = C4○D28order 112 = 24·7

Central product of C4 and D28

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4D28, D285C2, C4Dic14, C4.16D14, Dic145C2, C14.4C23, C22.2D14, C28.16C22, D14.1C22, Dic7.2C22, (C2×C4)⋊3D7, (C2×C28)⋊4C2, (C4×D7)⋊4C2, C4(C7⋊D4), C71(C4○D4), C7⋊D43C2, C2.5(C22×D7), (C2×C14).11C22, SmallGroup(112,30)

Series: Derived Chief Lower central Upper central

C1C14 — C4○D28
C1C7C14D14C4×D7 — C4○D28
C7C14 — C4○D28
C1C4C2×C4

Generators and relations for C4○D28
 G = < a,b,c | a4=c2=1, b14=a2, ab=ba, ac=ca, cbc=a2b13 >

2C2
14C2
14C2
7C4
7C4
7C22
7C22
2C14
2D7
2D7
7C2×C4
7D4
7D4
7D4
7C2×C4
7Q8
7C4○D4

Smallest permutation representation of C4○D28
On 56 points
Generators in S56
(1 38 15 52)(2 39 16 53)(3 40 17 54)(4 41 18 55)(5 42 19 56)(6 43 20 29)(7 44 21 30)(8 45 22 31)(9 46 23 32)(10 47 24 33)(11 48 25 34)(12 49 26 35)(13 50 27 36)(14 51 28 37)
(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)
(1 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)

G:=sub<Sym(56)| (1,38,15,52)(2,39,16,53)(3,40,17,54)(4,41,18,55)(5,42,19,56)(6,43,20,29)(7,44,21,30)(8,45,22,31)(9,46,23,32)(10,47,24,33)(11,48,25,34)(12,49,26,35)(13,50,27,36)(14,51,28,37), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)>;

G:=Group( (1,38,15,52)(2,39,16,53)(3,40,17,54)(4,41,18,55)(5,42,19,56)(6,43,20,29)(7,44,21,30)(8,45,22,31)(9,46,23,32)(10,47,24,33)(11,48,25,34)(12,49,26,35)(13,50,27,36)(14,51,28,37), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56) );

G=PermutationGroup([(1,38,15,52),(2,39,16,53),(3,40,17,54),(4,41,18,55),(5,42,19,56),(6,43,20,29),(7,44,21,30),(8,45,22,31),(9,46,23,32),(10,47,24,33),(11,48,25,34),(12,49,26,35),(13,50,27,36),(14,51,28,37)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56)])

C4○D28 is a maximal subgroup of
Dic14⋊C4  D284C4  D28.2C4  D567C2  D28.C4  C8⋊D14  C8.D14  D4.D14  C28.C23  D4.8D14  D46D14  Q8.10D14  D7×C4○D4  D48D14  D4.10D14  D286C6  D285S3  D84⋊C2  D6.D14  Dic7.D6  D8411C2
C4○D28 is a maximal quotient of
C4×Dic14  C28.6Q8  C42⋊D7  C4×D28  C4.D28  C422D7  C23.D14  D14.D4  D14⋊D4  Dic7.D4  Dic7.Q8  D14.5D4  D14⋊Q8  C4⋊C4⋊D7  C28.48D4  C23.21D14  C4×C7⋊D4  C23.23D14  C287D4  D285S3  D84⋊C2  D6.D14  Dic7.D6  D8411C2

34 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C14A···14I28A···28L
order122224444477714···1428···28
size112141411214142222···22···2

34 irreducible representations

dim11111122222
type+++++++++
imageC1C2C2C2C2C2D7C4○D4D14D14C4○D28
kernelC4○D28Dic14C4×D7D28C7⋊D4C2×C28C2×C4C7C4C22C1
# reps112121326312

Matrix representation of C4○D28 in GL2(𝔽29) generated by

170
017
,
1726
182
,
10
528
G:=sub<GL(2,GF(29))| [17,0,0,17],[17,18,26,2],[1,5,0,28] >;

C4○D28 in GAP, Magma, Sage, TeX

C_4\circ D_{28}
% in TeX

G:=Group("C4oD28");
// GroupNames label

G:=SmallGroup(112,30);
// by ID

G=gap.SmallGroup(112,30);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,46,182,2404]);
// Polycyclic

G:=Group<a,b,c|a^4=c^2=1,b^14=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^13>;
// generators/relations

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Subgroup lattice of C4○D28 in TeX

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