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## G = D4×Dic7order 224 = 25·7

### Direct product of D4 and Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D4×Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — D4×Dic7
 Lower central C7 — C14 — D4×Dic7
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×Dic7
G = < a,b,c,d | a4=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 286 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C4×D4, C2×Dic7, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C22×C14, C4×Dic7, C4⋊Dic7, C23.D7, C22×Dic7, D4×C14, D4×Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C2×Dic7, C22×D7, D4×D7, D42D7, C22×Dic7, D4×Dic7

Smallest permutation representation of D4×Dic7
On 112 points
Generators in S112
(1 81 54 26)(2 82 55 27)(3 83 56 28)(4 84 43 15)(5 71 44 16)(6 72 45 17)(7 73 46 18)(8 74 47 19)(9 75 48 20)(10 76 49 21)(11 77 50 22)(12 78 51 23)(13 79 52 24)(14 80 53 25)(29 93 61 100)(30 94 62 101)(31 95 63 102)(32 96 64 103)(33 97 65 104)(34 98 66 105)(35 85 67 106)(36 86 68 107)(37 87 69 108)(38 88 70 109)(39 89 57 110)(40 90 58 111)(41 91 59 112)(42 92 60 99)
(1 26)(2 27)(3 28)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 99)(43 84)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)
(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 60 8 67)(2 59 9 66)(3 58 10 65)(4 57 11 64)(5 70 12 63)(6 69 13 62)(7 68 14 61)(15 89 22 96)(16 88 23 95)(17 87 24 94)(18 86 25 93)(19 85 26 92)(20 98 27 91)(21 97 28 90)(29 46 36 53)(30 45 37 52)(31 44 38 51)(32 43 39 50)(33 56 40 49)(34 55 41 48)(35 54 42 47)(71 109 78 102)(72 108 79 101)(73 107 80 100)(74 106 81 99)(75 105 82 112)(76 104 83 111)(77 103 84 110)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G ··· 4L 7A 7B 7C 14A ··· 14I 14J ··· 14U 28A ··· 28F order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 2 2 2 2 7 7 7 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 D4 D7 C4○D4 D14 Dic7 D14 D4×D7 D4⋊2D7 kernel D4×Dic7 C4×Dic7 C4⋊Dic7 C23.D7 C22×Dic7 D4×C14 C7×D4 Dic7 C2×D4 C14 C2×C4 D4 C23 C2 C2 # reps 1 1 1 2 2 1 8 2 3 2 3 12 6 3 3

Matrix representation of D4×Dic7 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 28 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 26 28 0 0 1 0 0 0 0 0 28 0 0 0 0 28
,
 0 12 0 0 12 0 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[26,1,0,0,28,0,0,0,0,0,28,0,0,0,0,28],[0,12,0,0,12,0,0,0,0,0,12,0,0,0,0,12] >;

D4×Dic7 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_7
% in TeX

G:=Group("D4xDic7");
// GroupNames label

G:=SmallGroup(224,129);
// by ID

G=gap.SmallGroup(224,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,188,6917]);
// Polycyclic

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

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