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## G = C7×C4.4D4order 224 = 25·7

### Direct product of C7 and C4.4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C4.4D4
 Chief series C1 — C2 — C22 — C2×C14 — C22×C14 — C7×C22⋊C4 — C7×C4.4D4
 Lower central C1 — C22 — C7×C4.4D4
 Upper central C1 — C2×C14 — C7×C4.4D4

Generators and relations for C7×C4.4D4
G = < a,b,c,d | a7=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C4.4D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C4×C28, C7×C22⋊C4, D4×C14, Q8×C14, C7×C4.4D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4.4D4, C7×D4, C22×C14, D4×C14, C7×C4○D4, C7×C4.4D4

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

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

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

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

98 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 7A ··· 7F 14A ··· 14R 14S ··· 14AD 28A ··· 28AJ 28AK ··· 28AV order 1 2 2 2 2 2 4 ··· 4 4 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 2 ··· 2 4 4 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C7 C14 C14 C14 C14 D4 C4○D4 C7×D4 C7×C4○D4 kernel C7×C4.4D4 C4×C28 C7×C22⋊C4 D4×C14 Q8×C14 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C28 C14 C4 C2 # reps 1 1 4 1 1 6 6 24 6 6 2 4 12 24

Matrix representation of C7×C4.4D4 in GL4(𝔽29) generated by

 23 0 0 0 0 23 0 0 0 0 16 0 0 0 0 16
,
 17 0 0 0 5 12 0 0 0 0 17 24 0 0 0 12
,
 28 0 0 0 27 1 0 0 0 0 12 0 0 0 0 12
,
 12 17 0 0 0 17 0 0 0 0 1 27 0 0 1 28
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,16,0,0,0,0,16],[17,5,0,0,0,12,0,0,0,0,17,0,0,0,24,12],[28,27,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,17,17,0,0,0,0,1,1,0,0,27,28] >;

C7×C4.4D4 in GAP, Magma, Sage, TeX

C_7\times C_4._4D_4
% in TeX

G:=Group("C7xC4.4D4");
// GroupNames label

G:=SmallGroup(224,159);
// by ID

G=gap.SmallGroup(224,159);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,679,2090,266]);
// Polycyclic

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

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