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## G = C7×D8⋊2C4order 448 = 26·7

### Direct product of C7 and D8⋊2C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×D8⋊2C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C56 — C7×C4.Q8 — C7×D8⋊2C4
 Lower central C1 — C2 — C4 — C8 — C7×D8⋊2C4
 Upper central C1 — C14 — C2×C28 — C2×C56 — C7×D8⋊2C4

Generators and relations for C7×D82C4
G = < a,b,c,d | a7=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C14, C14, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C28, C28, C2×C14, C2×C14, C4.Q8, M5(2), C4○D8, C56, C2×C28, C2×C28, C7×D4, C7×Q8, D82C4, C112, C7×C4⋊C4, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C7×C4.Q8, C7×M5(2), C7×C4○D8, C7×D82C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, D8, SD16, C28, C2×C14, D4⋊C4, C2×C28, C7×D4, D82C4, C7×C22⋊C4, C7×D8, C7×SD16, C7×D4⋊C4, C7×D82C4

Smallest permutation representation of C7×D82C4
On 112 points
Generators in S112
(1 55 47 39 31 23 15)(2 56 48 40 32 24 16)(3 49 41 33 25 17 9)(4 50 42 34 26 18 10)(5 51 43 35 27 19 11)(6 52 44 36 28 20 12)(7 53 45 37 29 21 13)(8 54 46 38 30 22 14)(57 105 97 89 81 73 65)(58 106 98 90 82 74 66)(59 107 99 91 83 75 67)(60 108 100 92 84 76 68)(61 109 101 93 85 77 69)(62 110 102 94 86 78 70)(63 111 103 95 87 79 71)(64 112 104 96 88 80 72)
(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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 72)(16 71)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 80)(24 79)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 88)(32 87)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 96)(40 95)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 104)(48 103)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 112)(56 111)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)(57 58 61 62)(59 64 63 60)(65 66 69 70)(67 72 71 68)(73 74 77 78)(75 80 79 76)(81 82 85 86)(83 88 87 84)(89 90 93 94)(91 96 95 92)(97 98 101 102)(99 104 103 100)(105 106 109 110)(107 112 111 108)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A ··· 7F 8A 8B 8C 14A ··· 14F 14G ··· 14L 14M ··· 14R 16A 16B 16C 16D 28A ··· 28L 28M ··· 28AD 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 2 4 4 4 4 4 7 ··· 7 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 16 16 16 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 8 2 2 8 8 8 1 ··· 1 2 2 4 1 ··· 1 2 ··· 2 8 ··· 8 4 4 4 4 2 ··· 2 8 ··· 8 2 ··· 2 4 ··· 4 4 ··· 4

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 D4 D4 SD16 D8 C7×D4 C7×D4 C7×SD16 C7×D8 D8⋊2C4 C7×D8⋊2C4 kernel C7×D8⋊2C4 C7×C4.Q8 C7×M5(2) C7×C4○D8 C7×D8 C7×Q16 D8⋊2C4 C4.Q8 M5(2) C4○D8 D8 Q16 C56 C2×C28 C28 C2×C14 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 1 1 2 2 6 6 6 6 12 12 1 1 2 2 6 6 12 12 2 12

Matrix representation of C7×D82C4 in GL6(𝔽113)

 30 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 100 13 0 0 0 0 100 100 0 0 0 0 0 0 13 13 0 0 0 0 100 13
,
 0 112 0 0 0 0 112 0 0 0 0 0 0 0 0 0 13 13 0 0 0 0 100 13 0 0 100 13 0 0 0 0 100 100 0 0
,
 98 0 0 0 0 0 0 15 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 13 100 0 0 0 0 100 100

G:=sub<GL(6,GF(113))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,100,100,0,0,0,0,13,100,0,0,0,0,0,0,13,100,0,0,0,0,13,13],[0,112,0,0,0,0,112,0,0,0,0,0,0,0,0,0,100,100,0,0,0,0,13,100,0,0,13,100,0,0,0,0,13,13,0,0],[98,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,13,100,0,0,0,0,100,100] >;

C7×D82C4 in GAP, Magma, Sage, TeX

C_7\times D_8\rtimes_2C_4
% in TeX

G:=Group("C7xD8:2C4");
// GroupNames label

G:=SmallGroup(448,164);
// by ID

G=gap.SmallGroup(448,164);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,5106,136,4911,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations

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