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## G = D7×C4⋊C4order 224 = 25·7

### Direct product of D7 and C4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×C4⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D7×C4⋊C4
 Lower central C7 — C14 — D7×C4⋊C4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D7×C4⋊C4
G = < a,b,c,d | a7=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 342 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, C23, D7, C14, C4⋊C4, C4⋊C4, C22×C4, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C4⋊C4, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, D7×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C4×D7, C22×D7, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 D4 Q8 D7 D14 C4×D7 D4×D7 Q8×D7 kernel D7×C4⋊C4 Dic7⋊C4 C4⋊Dic7 C7×C4⋊C4 C2×C4×D7 C4×D7 D14 D14 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 2 1 1 3 8 2 2 3 9 12 3 3

Matrix representation of D7×C4⋊C4 in GL4(𝔽29) generated by

 0 1 0 0 28 3 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 0 12 0 0 12 0
,
 12 0 0 0 0 12 0 0 0 0 0 1 0 0 28 0
G:=sub<GL(4,GF(29))| [0,28,0,0,1,3,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,0,12,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,0,28,0,0,1,0] >;

D7×C4⋊C4 in GAP, Magma, Sage, TeX

D_7\times C_4\rtimes C_4
% in TeX

G:=Group("D7xC4:C4");
// GroupNames label

G:=SmallGroup(224,86);
// by ID

G=gap.SmallGroup(224,86);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,188,50,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^4=d^4=1,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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