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## G = C4×C7⋊D4order 224 = 25·7

### Direct product of C4 and C7⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4×C7⋊D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C7⋊D4 — C4×C7⋊D4
 Lower central C7 — C14 — C4×C7⋊D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C4×C7⋊D4
G = < a,b,c,d | a4=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 334 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C4×D7, C2×C7⋊D4, C22×C28, C4×C7⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C7⋊D4, C22×D7, C2×C4×D7, C4○D28, C2×C7⋊D4, C4×C7⋊D4

Smallest permutation representation of C4×C7⋊D4
On 112 points
Generators in S112
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(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 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)
(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 57 8 64)(2 63 9 70)(3 62 10 69)(4 61 11 68)(5 60 12 67)(6 59 13 66)(7 58 14 65)(15 71 22 78)(16 77 23 84)(17 76 24 83)(18 75 25 82)(19 74 26 81)(20 73 27 80)(21 72 28 79)(29 85 36 92)(30 91 37 98)(31 90 38 97)(32 89 39 96)(33 88 40 95)(34 87 41 94)(35 86 42 93)(43 99 50 106)(44 105 51 112)(45 104 52 111)(46 103 53 110)(47 102 54 109)(48 101 55 108)(49 100 56 107)
(1 15)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 22)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(29 43)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 50)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(57 78)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 71)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(85 106)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 99)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D7 C4○D4 D14 D14 C7⋊D4 C4×D7 C4○D28 kernel C4×C7⋊D4 C4×Dic7 Dic7⋊C4 D14⋊C4 C23.D7 C2×C4×D7 C2×C7⋊D4 C22×C28 C7⋊D4 C28 C22×C4 C14 C2×C4 C23 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 8 2 3 2 6 3 12 12 12

Matrix representation of C4×C7⋊D4 in GL4(𝔽29) generated by

 17 0 0 0 0 17 0 0 0 0 12 0 0 0 0 12
,
 28 28 0 0 5 4 0 0 0 0 18 28 0 0 1 0
,
 0 8 0 0 11 0 0 0 0 0 11 2 0 0 26 18
,
 0 21 0 0 18 0 0 0 0 0 28 0 0 0 11 1
G:=sub<GL(4,GF(29))| [17,0,0,0,0,17,0,0,0,0,12,0,0,0,0,12],[28,5,0,0,28,4,0,0,0,0,18,1,0,0,28,0],[0,11,0,0,8,0,0,0,0,0,11,26,0,0,2,18],[0,18,0,0,21,0,0,0,0,0,28,11,0,0,0,1] >;

C4×C7⋊D4 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes D_4
% in TeX

G:=Group("C4xC7:D4");
// GroupNames label

G:=SmallGroup(224,123);
// by ID

G=gap.SmallGroup(224,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,50,6917]);
// Polycyclic

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

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