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G = D28.C4order 224 = 25·7

The non-split extension by D28 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.C4, C8.12D14, Dic14.C4, M4(2)⋊5D7, C28.39C23, C56.12C22, C7⋊D4.C4, (C8×D7)⋊8C2, C72(C8○D4), C4.5(C4×D7), C8⋊D76C2, C28.13(C2×C4), C4○D28.3C2, D14.2(C2×C4), (C2×C4).46D14, C7⋊C8.12C22, C22.1(C4×D7), (C7×M4(2))⋊4C2, Dic7.4(C2×C4), C4.39(C22×D7), C14.16(C22×C4), (C2×C28).26C22, (C4×D7).16C22, (C2×C7⋊C8)⋊3C2, C2.17(C2×C4×D7), (C2×C14).6(C2×C4), SmallGroup(224,102)

Series: Derived Chief Lower central Upper central

C1C14 — D28.C4
C1C7C14C28C4×D7C4○D28 — D28.C4
C7C14 — D28.C4
C1C4M4(2)

Generators and relations for D28.C4
 G = < a,b,c | a28=b2=1, c4=a14, bab=a-1, cac-1=a15, cbc-1=a14b >

Subgroups: 206 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, M4(2), M4(2), C4○D4, Dic7, C28, D14, C2×C14, C8○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C8×D7, C8⋊D7, C2×C7⋊C8, C7×M4(2), C4○D28, D28.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, D14, C8○D4, C4×D7, C22×D7, C2×C4×D7, D28.C4

Smallest permutation representation of D28.C4
On 112 points
Generators in S112
(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)
(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 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(50 56)(51 55)(52 54)(57 59)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(104 112)(105 111)(106 110)(107 109)
(1 98 83 43 15 112 69 29)(2 85 84 30 16 99 70 44)(3 100 57 45 17 86 71 31)(4 87 58 32 18 101 72 46)(5 102 59 47 19 88 73 33)(6 89 60 34 20 103 74 48)(7 104 61 49 21 90 75 35)(8 91 62 36 22 105 76 50)(9 106 63 51 23 92 77 37)(10 93 64 38 24 107 78 52)(11 108 65 53 25 94 79 39)(12 95 66 40 26 109 80 54)(13 110 67 55 27 96 81 41)(14 97 68 42 28 111 82 56)

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

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

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

D28.C4 is a maximal subgroup of
D28.2D4  D28.3D4  D28.6D4  D28.7D4  M4(2).22D14  C42.196D14  D5610C4  D567C4  C28.70C24  D7×C8○D4  C56.49C23  D85D14  D86D14  C56.C23  D28.44D4
D28.C4 is a maximal quotient of
C56⋊Q8  C89D28  D14.4C42  C42.185D14  C56⋊C4⋊C2  C7⋊D4⋊C8  D14⋊C8⋊C2  C7⋊C826D4  Dic14⋊C8  C28.M4(2)  D28⋊C8  C282M4(2)  C42.30D14  C42.31D14  C28.439(C2×D4)  C28.7C42  C56⋊D4  C5618D4  (C2×D28).14C4

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C8A8B8C8D8E8F8G8H8I8J14A14B14C14D14E14F28A···28F28G28H28I56A···56L
order1222244444777888888888814141414141428···2828282856···56
size112141411214142222222777714142224442···24444···4

50 irreducible representations

dim1111111112222224
type+++++++++
imageC1C2C2C2C2C2C4C4C4D7D14D14C8○D4C4×D7C4×D7D28.C4
kernelD28.C4C8×D7C8⋊D7C2×C7⋊C8C7×M4(2)C4○D28Dic14D28C7⋊D4M4(2)C8C2×C4C7C4C22C1
# reps1221112243634666

Matrix representation of D28.C4 in GL4(𝔽113) generated by

334200
7910400
00069
00180
,
010400
25000
0010
000112
,
15000
01500
000112
00150
G:=sub<GL(4,GF(113))| [33,79,0,0,42,104,0,0,0,0,0,18,0,0,69,0],[0,25,0,0,104,0,0,0,0,0,1,0,0,0,0,112],[15,0,0,0,0,15,0,0,0,0,0,15,0,0,112,0] >;

D28.C4 in GAP, Magma, Sage, TeX

D_{28}.C_4
% in TeX

G:=Group("D28.C4");
// GroupNames label

G:=SmallGroup(224,102);
// by ID

G=gap.SmallGroup(224,102);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,188,50,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^28=b^2=1,c^4=a^14,b*a*b=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^14*b>;
// generators/relations

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