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G = D28.2C4order 224 = 25·7

The non-split extension by D28 of C4 acting through Inn(D28)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.2C4, C8.18D14, C56.23C22, C28.37C23, Dic14.2C4, (C2×C8)⋊7D7, (C8×D7)⋊6C2, C71(C8○D4), (C2×C56)⋊10C2, C8⋊D77C2, C4.10(C4×D7), C7⋊D4.2C4, C28.20(C2×C4), C4○D28.6C2, D14.1(C2×C4), (C2×C4).78D14, C7⋊C8.11C22, C22.2(C4×D7), C4.Dic711C2, Dic7.3(C2×C4), C4.37(C22×D7), C14.14(C22×C4), (C2×C28).98C22, (C4×D7).15C22, C2.15(C2×C4×D7), (C2×C14).16(C2×C4), SmallGroup(224,96)

Series: Derived Chief Lower central Upper central

C1C14 — D28.2C4
C1C7C14C28C4×D7C4○D28 — D28.2C4
C7C14 — D28.2C4
C1C8C2×C8

Generators and relations for D28.2C4
 G = < a,b,c | a28=b2=1, c4=a14, bab=a-1, ac=ca, bc=cb >

Subgroups: 206 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D7 [×2], C14, C14, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, Dic7 [×2], C28 [×2], D14 [×2], C2×C14, C8○D4, C7⋊C8 [×2], C56 [×2], Dic14, C4×D7 [×2], D28, C7⋊D4 [×2], C2×C28, C8×D7 [×2], C8⋊D7 [×2], C4.Dic7, C2×C56, C4○D28, D28.2C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D7, C22×C4, D14 [×3], C8○D4, C4×D7 [×2], C22×D7, C2×C4×D7, D28.2C4

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

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

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

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

D28.2C4 is a maximal subgroup of
D28.C8  Dic14.C8  D5611C4  D564C4  C8.20D28  C8.21D28  C8.24D28  D28.4C8  C16.12D14  C56.93D4  C56.50D4  C56.23D4  C56.44D4  C56.29D4  C28.70C24  D7×C8○D4  C56.49C23  D813D14  D28.29D4  D28.30D4  D815D14  D811D14  D8.10D14
D28.2C4 is a maximal quotient of
C8×Dic14  C5611Q8  C8×D28  C86D28  D14.C42  C42.243D14  C56⋊C4⋊C2  D14⋊C8⋊C2  D142M4(2)  Dic7⋊M4(2)  C42.27D14  D143M4(2)  C42.30D14  C42.31D14  C28.12C42  Dic7⋊C8⋊C2  C8×C7⋊D4  (C22×C8)⋊D7  C5632D4

68 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C8A8B8C8D8E8F8G8H8I8J14A···14I28A···28L56A···56X
order1222244444777888888888814···1428···2856···56
size11214141121414222111122141414142···22···22···2

68 irreducible representations

dim1111111112222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4D7D14D14C8○D4C4×D7C4×D7D28.2C4
kernelD28.2C4C8×D7C8⋊D7C4.Dic7C2×C56C4○D28Dic14D28C7⋊D4C2×C8C8C2×C4C7C4C22C1
# reps12211122436346624

Matrix representation of D28.2C4 in GL4(𝔽113) generated by

897900
10000
009828
00015
,
03400
10000
001120
0071
,
1000
0100
00950
00095
G:=sub<GL(4,GF(113))| [89,10,0,0,79,0,0,0,0,0,98,0,0,0,28,15],[0,10,0,0,34,0,0,0,0,0,112,7,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,95,0,0,0,0,95] >;

D28.2C4 in GAP, Magma, Sage, TeX

D_{28}._2C_4
% in TeX

G:=Group("D28.2C4");
// GroupNames label

G:=SmallGroup(224,96);
// by ID

G=gap.SmallGroup(224,96);
# by ID

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

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

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