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G = (C2×C4).D28order 448 = 26·7

3rd non-split extension by C2×C4 of D28 acting via D28/C7=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C4).3D28, (C2×D28).3C4, (C2×C28).15D4, (C2×Q8).1D14, C4.10D45D7, (C4×Dic7).1C4, C28.10D41C2, C71(C42.C4), (Q8×C14).1C22, C14.13(C23⋊C4), C28.23D4.1C2, C22.14(D14⋊C4), C2.14(C23.1D14), (C2×C4).3(C4×D7), (C2×C28).3(C2×C4), (C2×C4).3(C7⋊D4), (C7×C4.10D4)⋊11C2, (C2×C14).7(C22⋊C4), SmallGroup(448,34)

Series: Derived Chief Lower central Upper central

C1C2×C28 — (C2×C4).D28
C1C7C14C2×C14C2×C28Q8×C14C28.23D4 — (C2×C4).D28
C7C14C2×C14C2×C28 — (C2×C4).D28
C1C2C22C2×Q8C4.10D4

Generators and relations for (C2×C4).D28
 G = < a,b,c,d | a2=b28=c2=1, d4=b14, ab=ba, ac=ca, dad-1=ab14, cbc=b-1, dbd-1=ab15, dcd-1=b7c >

Subgroups: 428 in 64 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C4.10D4, C4.10D4, C4.4D4, C7⋊C8, C56, D28, C2×Dic7, C2×C28, C7×Q8, C22×D7, C42.C4, C4.Dic7, C4×Dic7, D14⋊C4, C7×M4(2), C2×D28, Q8×C14, C28.10D4, C7×C4.10D4, C28.23D4, (C2×C4).D28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C42.C4, D14⋊C4, C23.1D14, (C2×C4).D28

Smallest permutation representation of (C2×C4).D28
On 112 points
Generators in S112
(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)
(1 102 38 66 15 88 52 80)(2 89 53 67 16 103 39 81)(3 104 40 68 17 90 54 82)(4 91 55 69 18 105 41 83)(5 106 42 70 19 92 56 84)(6 93 29 71 20 107 43 57)(7 108 44 72 21 94 30 58)(8 95 31 73 22 109 45 59)(9 110 46 74 23 96 32 60)(10 97 33 75 24 111 47 61)(11 112 48 76 25 98 34 62)(12 99 35 77 26 85 49 63)(13 86 50 78 27 100 36 64)(14 101 37 79 28 87 51 65)

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

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

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

46 conjugacy classes

class 1 2A2B2C4A4B4C4D4E7A7B7C8A8B8C8D14A14B14C14D14E14F28A···28F28G···28L56A···56L
order122244444777888814141414141428···2828···2856···56
size1125644428282228856562224444···48···88···8

46 irreducible representations

dim1111112222224448
type++++++++++
imageC1C2C2C2C4C4D4D7D14C4×D7D28C7⋊D4C23⋊C4C42.C4C23.1D14(C2×C4).D28
kernel(C2×C4).D28C28.10D4C7×C4.10D4C28.23D4C4×Dic7C2×D28C2×C28C4.10D4C2×Q8C2×C4C2×C4C2×C4C14C7C2C1
# reps1111222336661263

Matrix representation of (C2×C4).D28 in GL6(𝔽113)

100000
010000
001000
000100
00001120
002200112
,
112240000
89100000
0015000
00439800
0000150
0010403498
,
112240000
010000
001076200
0045600
006003787
00247076
,
100000
010000
000010
0010201021
00908700
008976110

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,22,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,89,0,0,0,0,24,10,0,0,0,0,0,0,15,43,0,104,0,0,0,98,0,0,0,0,0,0,15,34,0,0,0,0,0,98],[112,0,0,0,0,0,24,1,0,0,0,0,0,0,107,45,60,2,0,0,62,6,0,4,0,0,0,0,37,70,0,0,0,0,87,76],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,102,90,89,0,0,0,0,87,76,0,0,1,102,0,11,0,0,0,1,0,0] >;

(C2×C4).D28 in GAP, Magma, Sage, TeX

(C_2\times C_4).D_{28}
% in TeX

G:=Group("(C2xC4).D28");
// GroupNames label

G:=SmallGroup(448,34);
// by ID

G=gap.SmallGroup(448,34);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,184,1123,794,297,136,851,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^2=1,d^4=b^14,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^14,c*b*c=b^-1,d*b*d^-1=a*b^15,d*c*d^-1=b^7*c>;
// generators/relations

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