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## G = C7×D4⋊4D4order 448 = 26·7

### Direct product of C7 and D4⋊4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×D4⋊4D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — D4×C14 — C7×C8⋊C22 — C7×D4⋊4D4
 Lower central C1 — C2 — C2×C4 — C7×D4⋊4D4
 Upper central C1 — C14 — C2×C28 — C7×D4⋊4D4

Generators and relations for C7×D44D4
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 370 in 168 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C14, C14, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, C28, C28, C2×C14, C2×C14, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, C22×C14, D44D4, C4×C28, C7×M4(2), C7×D8, C7×SD16, D4×C14, D4×C14, C7×C4○D4, C7×C4○D4, C7×C4.D4, C7×C4≀C2, C7×C41D4, C7×C8⋊C22, C7×2+ 1+4, C7×D44D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, D44D4, D4×C14, C7×C22≀C2, C7×D44D4

Smallest permutation representation of C7×D44D4
On 56 points
Generators in S56
(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)
(1 43 35 38)(2 44 29 39)(3 45 30 40)(4 46 31 41)(5 47 32 42)(6 48 33 36)(7 49 34 37)(8 23 21 56)(9 24 15 50)(10 25 16 51)(11 26 17 52)(12 27 18 53)(13 28 19 54)(14 22 20 55)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 49)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 38)(16 39)(17 40)(18 41)(19 42)(20 36)(21 37)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 50)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 23 21 56)(9 24 15 50)(10 25 16 51)(11 26 17 52)(12 27 18 53)(13 28 19 54)(14 22 20 55)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)

G:=sub<Sym(56)| (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), (1,43,35,38)(2,44,29,39)(3,45,30,40)(4,46,31,41)(5,47,32,42)(6,48,33,36)(7,49,34,37)(8,23,21,56)(9,24,15,50)(10,25,16,51)(11,26,17,52)(12,27,18,53)(13,28,19,54)(14,22,20,55), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,49)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,38)(16,39)(17,40)(18,41)(19,42)(20,36)(21,37)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,50), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,23,21,56)(9,24,15,50)(10,25,16,51)(11,26,17,52)(12,27,18,53)(13,28,19,54)(14,22,20,55)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)>;

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), (1,43,35,38)(2,44,29,39)(3,45,30,40)(4,46,31,41)(5,47,32,42)(6,48,33,36)(7,49,34,37)(8,23,21,56)(9,24,15,50)(10,25,16,51)(11,26,17,52)(12,27,18,53)(13,28,19,54)(14,22,20,55), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,49)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,38)(16,39)(17,40)(18,41)(19,42)(20,36)(21,37)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,50), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,23,21,56)(9,24,15,50)(10,25,16,51)(11,26,17,52)(12,27,18,53)(13,28,19,54)(14,22,20,55)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43) );

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)], [(1,43,35,38),(2,44,29,39),(3,45,30,40),(4,46,31,41),(5,47,32,42),(6,48,33,36),(7,49,34,37),(8,23,21,56),(9,24,15,50),(10,25,16,51),(11,26,17,52),(12,27,18,53),(13,28,19,54),(14,22,20,55)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,22),(7,23),(8,49),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,38),(16,39),(17,40),(18,41),(19,42),(20,36),(21,37),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,50)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,23,21,56),(9,24,15,50),(10,25,16,51),(11,26,17,52),(12,27,18,53),(13,28,19,54),(14,22,20,55),(36,48),(37,49),(38,43),(39,44),(40,45),(41,46),(42,47)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43)]])

112 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 7A ··· 7F 8A 8B 14A ··· 14F 14G ··· 14L 14M ··· 14AJ 14AK ··· 14AP 28A ··· 28L 28M ··· 28AJ 56A ··· 56L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 7 ··· 7 8 8 14 ··· 14 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 4 4 4 4 8 2 2 4 4 4 4 1 ··· 1 8 8 1 ··· 1 2 ··· 2 4 ··· 4 8 ··· 8 2 ··· 2 4 ··· 4 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⋊4D4 C7×D4⋊4D4 kernel C7×D4⋊4D4 C7×C4.D4 C7×C4≀C2 C7×C4⋊1D4 C7×C8⋊C22 C7×2+ 1+4 D4⋊4D4 C4.D4 C4≀C2 C4⋊1D4 C8⋊C22 2+ 1+4 C7×D4 C7×Q8 C22×C14 D4 Q8 C23 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×D44D4 in GL4(𝔽113) generated by

 30 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 0 1 27 50 112 0 14 36 0 0 112 1 0 0 111 1
,
 86 0 0 88 99 0 1 93 1 1 0 77 2 0 0 27
,
 112 0 27 86 0 112 14 99 0 0 112 1 0 0 111 1
,
 0 1 0 50 1 0 0 63 0 0 112 1 0 0 0 1
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,0,0,1,0,0,0,27,14,112,111,50,36,1,1],[86,99,1,2,0,0,1,0,0,1,0,0,88,93,77,27],[112,0,0,0,0,112,0,0,27,14,112,111,86,99,1,1],[0,1,0,0,1,0,0,0,0,0,112,0,50,63,1,1] >;

C7×D44D4 in GAP, Magma, Sage, TeX

C_7\times D_4\rtimes_4D_4
% in TeX

G:=Group("C7xD4:4D4");
// GroupNames label

G:=SmallGroup(448,861);
// by ID

G=gap.SmallGroup(448,861);
# by ID

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

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

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