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G = C2×C28.D4order 448 = 26·7

Direct product of C2 and C28.D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C28.D4, C24.2Dic7, (D4×C14).10C4, (C2×C28).190D4, C28.204(C2×D4), (C23×C14).4C4, (C2×D4).7Dic7, (C22×D4).3D7, (C2×D4).197D14, C142(C4.D4), C4.9(C23.D7), C28.32(C22⋊C4), (C2×C28).471C23, (C22×C4).148D14, C4.Dic722C22, C23.32(C2×Dic7), (D4×C14).239C22, C22.5(C22×Dic7), (C22×C28).196C22, C22.34(C23.D7), (D4×C2×C14).3C2, C73(C2×C4.D4), C4.90(C2×C7⋊D4), (C2×C28).117(C2×C4), C2.9(C2×C23.D7), C14.73(C2×C22⋊C4), (C2×C4.Dic7)⋊18C2, (C2×C4).24(C2×Dic7), (C2×C4).197(C7⋊D4), (C22×C14).15(C2×C4), (C2×C4).129(C22×D7), (C2×C14).194(C22×C4), (C2×C14).112(C22⋊C4), SmallGroup(448,750)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C2×C28.D4
C1C7C14C28C2×C28C4.Dic7C2×C4.Dic7 — C2×C28.D4
C7C14C2×C14 — C2×C28.D4
C1C22C22×C4C22×D4

Generators and relations for C2×C28.D4
 G = < a,b,c,d | a2=b28=1, c4=b14, d2=b21, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b13, dcd-1=b7c3 >

Subgroups: 532 in 186 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C28, C2×C14, C2×C14, C4.D4, C2×M4(2), C22×D4, C7⋊C8, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C4.D4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C28.D4, C2×C4.Dic7, D4×C2×C14, C2×C28.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C4.D4, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C2×C4.D4, C23.D7, C22×Dic7, C2×C7⋊D4, C28.D4, C2×C23.D7, C2×C28.D4

Smallest permutation representation of C2×C28.D4
On 112 points
Generators in S112
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(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 92 70 50 15 106 84 36)(2 91 71 49 16 105 57 35)(3 90 72 48 17 104 58 34)(4 89 73 47 18 103 59 33)(5 88 74 46 19 102 60 32)(6 87 75 45 20 101 61 31)(7 86 76 44 21 100 62 30)(8 85 77 43 22 99 63 29)(9 112 78 42 23 98 64 56)(10 111 79 41 24 97 65 55)(11 110 80 40 25 96 66 54)(12 109 81 39 26 95 67 53)(13 108 82 38 27 94 68 52)(14 107 83 37 28 93 69 51)
(1 99 22 92 15 85 8 106)(2 112 23 105 16 98 9 91)(3 97 24 90 17 111 10 104)(4 110 25 103 18 96 11 89)(5 95 26 88 19 109 12 102)(6 108 27 101 20 94 13 87)(7 93 28 86 21 107 14 100)(29 77 50 70 43 63 36 84)(30 62 51 83 44 76 37 69)(31 75 52 68 45 61 38 82)(32 60 53 81 46 74 39 67)(33 73 54 66 47 59 40 80)(34 58 55 79 48 72 41 65)(35 71 56 64 49 57 42 78)

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

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

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

82 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D7A7B7C8A···8H14A···14U14V···14AS28A···28L
order122222222244447778···814···1414···1428···28
size1111224444222222228···282···24···44···4

82 irreducible representations

dim111111222222244
type+++++++-+-+
imageC1C2C2C2C4C4D4D7D14Dic7D14Dic7C7⋊D4C4.D4C28.D4
kernelC2×C28.D4C28.D4C2×C4.Dic7D4×C2×C14D4×C14C23×C14C2×C28C22×D4C22×C4C2×D4C2×D4C24C2×C4C14C2
# reps14214443366624212

Matrix representation of C2×C28.D4 in GL6(𝔽113)

11200000
01120000
00112000
00011200
00001120
00000112
,
8500000
10040000
00831400
00493000
002483049
004083640
,
3100000
1121100000
001080
00001112
0001121120
002801120
,
3100000
1121100000
001008
00001121
00000112
002810112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[85,100,0,0,0,0,0,4,0,0,0,0,0,0,83,49,24,40,0,0,14,30,83,83,0,0,0,0,0,64,0,0,0,0,49,0],[3,112,0,0,0,0,10,110,0,0,0,0,0,0,1,0,0,28,0,0,0,0,112,0,0,0,8,1,112,112,0,0,0,112,0,0],[3,112,0,0,0,0,10,110,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,0,0,0,112,0,0,0,0,8,1,112,112] >;

C2×C28.D4 in GAP, Magma, Sage, TeX

C_2\times C_{28}.D_4
% in TeX

G:=Group("C2xC28.D4");
// GroupNames label

G:=SmallGroup(448,750);
// by ID

G=gap.SmallGroup(448,750);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=1,c^4=b^14,d^2=b^21,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^7*c^3>;
// generators/relations

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