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G = D28.39D4order 448 = 26·7

9th non-split extension by D28 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.39D4, M4(2)⋊6D14, Dic14.39D4, (C2×Q8)⋊4D14, C4○D4.7D14, (C7×D4).14D4, C4.106(D4×D7), (C7×Q8).14D4, C8.C221D7, C28.198(C2×D4), C74(D4.9D4), D284C411C2, (Q8×C14)⋊4C22, (C22×D7).6D4, C22.37(D4×D7), C14.65C22≀C2, D42Dic78C2, D48D14.2C2, C28.C235C2, C28.23D47C2, D4.11(C7⋊D4), (C2×C28).17C23, Q8.11(C7⋊D4), (C4×Dic7)⋊6C22, C4.Dic79C22, C28.46D411C2, C4○D28.25C22, C2.33(C23⋊D14), (C2×D28).130C22, (C7×M4(2))⋊16C22, C4.54(C2×C7⋊D4), (C2×C14).36(C2×D4), (C7×C8.C22)⋊5C2, (C2×C4).17(C22×D7), (C7×C4○D4).15C22, SmallGroup(448,736)

Series: Derived Chief Lower central Upper central

C1C2×C28 — D28.39D4
C1C7C14C2×C14C2×C28C2×D28D48D14 — D28.39D4
C7C14C2×C28 — D28.39D4
C1C2C2×C4C8.C22

Generators and relations for D28.39D4
 G = < a,b,c,d | a28=b2=c4=d2=1, bab=dad=a-1, cac-1=a13, cbc-1=a5b, dbd=a26b, dcd=a14c-1 >

Subgroups: 972 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4.4D4, C8.C22, C8.C22, 2+ 1+4, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×D7, D4.9D4, C4.Dic7, C4×Dic7, D14⋊C4, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, Q82D7, Q8×C14, C7×C4○D4, C28.46D4, D284C4, D42Dic7, C28.C23, C28.23D4, C7×C8.C22, D48D14, D28.39D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.9D4, D4×D7, C2×C7⋊D4, C23⋊D14, D28.39D4

Smallest permutation representation of D28.39D4
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 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 112)(12 111)(13 110)(14 109)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 98)(26 97)(27 96)(28 95)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 84)
(2 14)(3 27)(4 12)(5 25)(6 10)(7 23)(9 21)(11 19)(13 17)(16 28)(18 26)(20 24)(29 51)(30 36)(31 49)(32 34)(33 47)(35 45)(37 43)(38 56)(39 41)(40 54)(42 52)(44 50)(46 48)(53 55)(57 68 71 82)(58 81 72 67)(59 66 73 80)(60 79 74 65)(61 64 75 78)(62 77 76 63)(69 84 83 70)(85 98 99 112)(86 111 100 97)(87 96 101 110)(88 109 102 95)(89 94 103 108)(90 107 104 93)(91 92 105 106)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(57 104)(58 103)(59 102)(60 101)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 105)

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

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

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

49 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G7A7B7C8A8B14A14B14C14D14E14F14G14H14I28A···28F28G···28O56A···56F
order122222244444447778814141414141414141428···2828···2856···56
size112428282822482828282228562224448884···48···88···8

49 irreducible representations

dim11111111222222222224448
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D7D14D14D14C7⋊D4C7⋊D4D4.9D4D4×D7D4×D7D28.39D4
kernelD28.39D4C28.46D4D284C4D42Dic7C28.C23C28.23D4C7×C8.C22D48D14Dic14D28C7×D4C7×Q8C22×D7C8.C22M4(2)C2×Q8C4○D4D4Q8C7C4C22C1
# reps11111111111123333662333

Matrix representation of D28.39D4 in GL8(𝔽113)

5579000000
6834000000
0055790000
0068340000
000098000
0000621500
000000980
00009801515
,
54842190000
1610844710000
7194108650000
69429750000
000065579883
000054636262
0000986200
0000104547498
,
5834000000
4455000000
0058340000
0044550000
00001000
00002611200
000000150
0000460015
,
0058340000
0044550000
5834000000
4455000000
00001128700
00000100
00004249112111
0000711901

G:=sub<GL(8,GF(113))| [55,68,0,0,0,0,0,0,79,34,0,0,0,0,0,0,0,0,55,68,0,0,0,0,0,0,79,34,0,0,0,0,0,0,0,0,98,62,0,98,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,15,0,0,0,0,0,0,0,15],[5,16,71,69,0,0,0,0,48,108,94,42,0,0,0,0,42,44,108,97,0,0,0,0,19,71,65,5,0,0,0,0,0,0,0,0,65,54,98,104,0,0,0,0,57,63,62,54,0,0,0,0,98,62,0,74,0,0,0,0,83,62,0,98],[58,44,0,0,0,0,0,0,34,55,0,0,0,0,0,0,0,0,58,44,0,0,0,0,0,0,34,55,0,0,0,0,0,0,0,0,1,26,0,4,0,0,0,0,0,112,0,60,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15],[0,0,58,44,0,0,0,0,0,0,34,55,0,0,0,0,58,44,0,0,0,0,0,0,34,55,0,0,0,0,0,0,0,0,0,0,112,0,42,71,0,0,0,0,87,1,49,19,0,0,0,0,0,0,112,0,0,0,0,0,0,0,111,1] >;

D28.39D4 in GAP, Magma, Sage, TeX

D_{28}._{39}D_4
% in TeX

G:=Group("D28.39D4");
// GroupNames label

G:=SmallGroup(448,736);
// by ID

G=gap.SmallGroup(448,736);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,184,570,1684,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^5*b,d*b*d=a^26*b,d*c*d=a^14*c^-1>;
// generators/relations

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