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## G = D7×C4.10D4order 448 = 26·7

### Direct product of D7 and C4.10D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D7×C4.10D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C2×Q8×D7 — D7×C4.10D4
 Lower central C7 — C14 — C2×C14 — D7×C4.10D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D7×C4.10D4
G = < a,b,c,d,e | a7=b2=c4=1, d4=c2, e2=dcd-1=c-1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >

Subgroups: 684 in 146 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C23, D7, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4.10D4, C4.10D4, C2×M4(2), C22×Q8, C7⋊C8, C56, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C2×C4.10D4, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, Q8×D7, Q8×C14, C4.12D28, C28.10D4, C7×C4.10D4, D7×M4(2), C2×Q8×D7, D7×C4.10D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4.10D4, C2×C22⋊C4, C4×D7, C22×D7, C2×C4.10D4, C2×C4×D7, D4×D7, D7×C22⋊C4, D7×C4.10D4

Smallest permutation representation of D7×C4.10D4
On 112 points
Generators in S112
(1 76 58 96 99 68 86)(2 77 59 89 100 69 87)(3 78 60 90 101 70 88)(4 79 61 91 102 71 81)(5 80 62 92 103 72 82)(6 73 63 93 104 65 83)(7 74 64 94 97 66 84)(8 75 57 95 98 67 85)(9 35 46 105 54 29 20)(10 36 47 106 55 30 21)(11 37 48 107 56 31 22)(12 38 41 108 49 32 23)(13 39 42 109 50 25 24)(14 40 43 110 51 26 17)(15 33 44 111 52 27 18)(16 34 45 112 53 28 19)
(1 86)(2 87)(3 88)(4 81)(5 82)(6 83)(7 84)(8 85)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(41 108)(42 109)(43 110)(44 111)(45 112)(46 105)(47 106)(48 107)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 97)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 48)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(74 80 78 76)(81 83 85 87)(82 88 86 84)(89 91 93 95)(90 96 94 92)(97 103 101 99)(98 100 102 104)(105 107 109 111)(106 112 110 108)
(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)
(1 105 3 111 5 109 7 107)(2 110 8 112 6 106 4 108)(9 101 15 103 13 97 11 99)(10 102 12 100 14 98 16 104)(17 95 19 93 21 91 23 89)(18 92 24 94 22 96 20 90)(25 64 31 58 29 60 27 62)(26 57 28 63 30 61 32 59)(33 72 39 66 37 68 35 70)(34 65 36 71 38 69 40 67)(41 87 43 85 45 83 47 81)(42 84 48 86 46 88 44 82)(49 77 51 75 53 73 55 79)(50 74 56 76 54 78 52 80)

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

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

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

55 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 7 7 14 2 2 4 4 14 14 28 28 2 2 2 4 4 4 4 28 28 28 28 2 2 2 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 type + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 D7 D14 D14 C4×D7 C4.10D4 D4×D7 D7×C4.10D4 kernel D7×C4.10D4 C4.12D28 C28.10D4 C7×C4.10D4 D7×M4(2) C2×Q8×D7 C2×Dic14 C2×C4×D7 C4×D7 C4.10D4 M4(2) C2×Q8 C2×C4 D7 C4 C1 # reps 1 2 1 1 2 1 4 4 4 3 6 3 12 2 6 3

Matrix representation of D7×C4.10D4 in GL6(𝔽113)

 79 1 0 0 0 0 112 0 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
,
 25 88 0 0 0 0 34 88 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 1 0 0 0 0 0 15 98 1 91 0 0 0 50 72 112
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 15 98 1 91 0 0 0 1 0 0 0 0 72 0 0 15
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 93 20 99 67 0 0 77 36 49 98 0 0 74 88 0 0 0 0 0 17 106 97

G:=sub<GL(6,GF(113))| [79,112,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,34,0,0,0,0,88,88,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,15,0,0,0,112,0,98,50,0,0,0,0,1,72,0,0,0,0,91,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,15,0,72,0,0,0,98,1,0,0,0,1,1,0,0,0,0,0,91,0,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,93,77,74,0,0,0,20,36,88,17,0,0,99,49,0,106,0,0,67,98,0,97] >;

D7×C4.10D4 in GAP, Magma, Sage, TeX

D_7\times C_4._{10}D_4
% in TeX

G:=Group("D7xC4.10D4");
// GroupNames label

G:=SmallGroup(448,284);
// by ID

G=gap.SmallGroup(448,284);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,219,58,570,136,438,18822]);
// Polycyclic

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

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