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G = C7×D4⋊C4order 224 = 25·7

Direct product of C7 and D4⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C7×D4⋊C4, D41C28, C14.13D8, C28.60D4, C14.9SD16, C4⋊C41C14, (C2×C56)⋊4C2, (C2×C8)⋊2C14, (C7×D4)⋊4C4, C2.1(C7×D8), C4.1(C2×C28), C4.11(C7×D4), C28.28(C2×C4), (C2×D4).3C14, (D4×C14).9C2, (C2×C14).46D4, C2.1(C7×SD16), C22.8(C7×D4), C14.24(C22⋊C4), (C2×C28).114C22, (C7×C4⋊C4)⋊10C2, C2.6(C7×C22⋊C4), (C2×C4).17(C2×C14), SmallGroup(224,51)

Series: Derived Chief Lower central Upper central

C1C4 — C7×D4⋊C4
C1C2C22C2×C4C2×C28C7×C4⋊C4 — C7×D4⋊C4
C1C2C4 — C7×D4⋊C4
C1C2×C14C2×C28 — C7×D4⋊C4

Generators and relations for C7×D4⋊C4
 G = < a,b,c,d | a7=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

4C2
4C2
2C22
2C22
4C22
4C22
4C4
4C14
4C14
2C2×C4
2C23
2C8
2D4
2C2×C14
2C2×C14
4C2×C14
4C28
4C2×C14
2C56
2C22×C14
2C2×C28
2C7×D4

Smallest permutation representation of C7×D4⋊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 50 78 59)(2 51 79 60)(3 52 80 61)(4 53 81 62)(5 54 82 63)(6 55 83 57)(7 56 84 58)(8 35 110 37)(9 29 111 38)(10 30 112 39)(11 31 106 40)(12 32 107 41)(13 33 108 42)(14 34 109 36)(15 43 24 99)(16 44 25 100)(17 45 26 101)(18 46 27 102)(19 47 28 103)(20 48 22 104)(21 49 23 105)(64 91 73 92)(65 85 74 93)(66 86 75 94)(67 87 76 95)(68 88 77 96)(69 89 71 97)(70 90 72 98)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 85)(7 86)(8 101)(9 102)(10 103)(11 104)(12 105)(13 99)(14 100)(15 42)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 31)(23 32)(24 33)(25 34)(26 35)(27 29)(28 30)(43 108)(44 109)(45 110)(46 111)(47 112)(48 106)(49 107)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 74)(58 75)(59 76)(60 77)(61 71)(62 72)(63 73)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)
(1 22 76 40)(2 23 77 41)(3 24 71 42)(4 25 72 36)(5 26 73 37)(6 27 74 38)(7 28 75 39)(8 63 101 91)(9 57 102 85)(10 58 103 86)(11 59 104 87)(12 60 105 88)(13 61 99 89)(14 62 100 90)(15 69 33 80)(16 70 34 81)(17 64 35 82)(18 65 29 83)(19 66 30 84)(20 67 31 78)(21 68 32 79)(43 97 108 52)(44 98 109 53)(45 92 110 54)(46 93 111 55)(47 94 112 56)(48 95 106 50)(49 96 107 51)

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

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

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

C7×D4⋊C4 is a maximal subgroup of
Dic74D8  D4.D7⋊C4  Dic76SD16  Dic7.D8  Dic7.SD16  D4⋊Dic14  Dic142D4  D4.Dic14  C4⋊C4.D14  C28⋊Q8⋊C2  D4.2Dic14  Dic14.D4  (C8×Dic7)⋊C2  (D4×D7)⋊C4  D4⋊(C4×D7)  D42D7⋊C4  D4⋊D28  D14.D8  D14⋊D8  D4.6D28  D14.SD16  D14⋊SD16  C8⋊Dic7⋊C2  C7⋊C81D4  D43D28  C7⋊C8⋊D4  D4.D28  C561C4⋊C2  D4⋊D7⋊C4  D283D4  D28.D4  D8×C28  SD16×C28

98 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D7A···7F8A8B8C8D14A···14R14S···14AD28A···28L28M···28X56A···56X
order12222244447···7888814···1414···1428···2828···2856···56
size11114422441···122221···14···42···24···42···2

98 irreducible representations

dim111111111122222222
type+++++++
imageC1C2C2C2C4C7C14C14C14C28D4D4D8SD16C7×D4C7×D4C7×D8C7×SD16
kernelC7×D4⋊C4C7×C4⋊C4C2×C56D4×C14C7×D4D4⋊C4C4⋊C4C2×C8C2×D4D4C28C2×C14C14C14C4C22C2C2
# reps111146666241122661212

Matrix representation of C7×D4⋊C4 in GL4(𝔽113) generated by

1000
0100
00280
00028
,
112000
011200
00112111
0011
,
1000
011200
0012
000112
,
0100
112000
002626
0010087
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[112,0,0,0,0,112,0,0,0,0,112,1,0,0,111,1],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,2,112],[0,112,0,0,1,0,0,0,0,0,26,100,0,0,26,87] >;

C7×D4⋊C4 in GAP, Magma, Sage, TeX

C_7\times D_4\rtimes C_4
% in TeX

G:=Group("C7xD4:C4");
// GroupNames label

G:=SmallGroup(224,51);
// by ID

G=gap.SmallGroup(224,51);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,3363,1689,117]);
// Polycyclic

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

Export

Subgroup lattice of C7×D4⋊C4 in TeX

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