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## G = C7×D4.10D4order 448 = 26·7

### Direct product of C7 and D4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×D4.10D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — Q8×C14 — C7×C8.C22 — C7×D4.10D4
 Lower central C1 — C2 — C2×C4 — C7×D4.10D4
 Upper central C1 — C14 — C2×C28 — C7×D4.10D4

Generators and relations for C7×D4.10D4
G = < a,b,c,d,e | a7=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 242 in 142 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C14, C14, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C28, C28, C2×C14, C2×C14, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, D4.10D4, C4×C28, C7×C4⋊C4, C7×M4(2), C7×SD16, C7×Q16, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4.10D4, C7×C4≀C2, C7×C4⋊Q8, C7×C8.C22, C7×2- 1+4, C7×D4.10D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, D4.10D4, D4×C14, C7×C22≀C2, C7×D4.10D4

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

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C ··· 4H 4I 7A ··· 7F 8A 8B 14A ··· 14F 14G ··· 14L 14M ··· 14X 28A ··· 28L 28M ··· 28AV 28AW ··· 28BB 56A ··· 56L order 1 2 2 2 2 4 4 4 ··· 4 4 7 ··· 7 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 4 4 2 2 4 ··· 4 8 1 ··· 1 8 8 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 8 ··· 8 8 ··· 8

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 D4 C7×D4 C7×D4 C7×D4 D4.10D4 C7×D4.10D4 kernel C7×D4.10D4 C7×C4.10D4 C7×C4≀C2 C7×C4⋊Q8 C7×C8.C22 C7×2- 1+4 D4.10D4 C4.10D4 C4≀C2 C4⋊Q8 C8.C22 2- 1+4 C2×C28 C7×D4 C7×Q8 C2×C4 D4 Q8 C7 C1 # reps 1 1 2 1 2 1 6 6 12 6 12 6 2 2 2 12 12 12 2 12

Matrix representation of C7×D4.10D4 in GL4(𝔽113) generated by

 30 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 0 1 0 0 112 0 0 0 9 103 112 111 57 66 1 1
,
 112 89 14 63 101 51 64 78 74 88 0 0 31 82 82 63
,
 12 62 49 35 112 89 14 63 74 88 0 0 91 53 94 12
,
 88 39 0 0 39 25 0 0 1 24 99 50 25 64 39 14
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,9,57,1,0,103,66,0,0,112,1,0,0,111,1],[112,101,74,31,89,51,88,82,14,64,0,82,63,78,0,63],[12,112,74,91,62,89,88,53,49,14,0,94,35,63,0,12],[88,39,1,25,39,25,24,64,0,0,99,39,0,0,50,14] >;

C7×D4.10D4 in GAP, Magma, Sage, TeX

C_7\times D_4._{10}D_4
% in TeX

G:=Group("C7xD4.10D4");
// GroupNames label

G:=SmallGroup(448,864);
// by ID

G=gap.SmallGroup(448,864);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,1192,9804,4911,2468,172,7068]);
// Polycyclic

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

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