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## G = C24⋊Dic7order 448 = 26·7

### 1st semidirect product of C24 and Dic7 acting via Dic7/C7=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C14 — C24⋊Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — D4×C14 — C23⋊Dic7 — C24⋊Dic7
 Lower central C7 — C14 — C2×C14 — C22×C14 — C24⋊Dic7
 Upper central C1 — C2 — C22 — C2×D4 — C22≀C2

Generators and relations for C24⋊Dic7
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=e7, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 396 in 94 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C14, C14, C22⋊C4, C22⋊C4, M4(2), C2×D4, C2×D4, C24, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C4.D4, C22≀C2, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C2≀C4, C4.Dic7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, D4×C14, D4×C14, C23×C14, C28.D4, C23⋊Dic7, C7×C22≀C2, C24⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C23⋊C4, C2×Dic7, C7⋊D4, C2≀C4, C23.D7, C23⋊Dic7, C24⋊Dic7

Smallest permutation representation of C24⋊Dic7
On 56 points
Generators in S56
(8 39)(9 33)(10 41)(11 35)(12 29)(13 37)(14 31)(22 36)(23 30)(24 38)(25 32)(26 40)(27 34)(28 42)
(1 45)(2 53)(3 47)(4 55)(5 49)(6 43)(7 51)(8 39)(9 33)(10 41)(11 35)(12 29)(13 37)(14 31)(15 48)(16 56)(17 50)(18 44)(19 52)(20 46)(21 54)(22 36)(23 30)(24 38)(25 32)(26 40)(27 34)(28 42)
(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(1 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 15)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 44 36 51)(30 43 37 50)(31 56 38 49)(32 55 39 48)(33 54 40 47)(34 53 41 46)(35 52 42 45)

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

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

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

55 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 7A 7B 7C 8A 8B 14A ··· 14I 14J ··· 14AA 14AB 14AC 14AD 28A ··· 28I order 1 2 2 2 2 2 2 4 4 4 4 7 7 7 8 8 14 ··· 14 14 ··· 14 14 14 14 28 ··· 28 size 1 1 2 4 4 4 4 4 8 56 56 2 2 2 56 56 2 ··· 2 4 ··· 4 8 8 8 8 ··· 8

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + - + + image C1 C2 C2 C2 C4 C4 D4 D4 D7 Dic7 D14 Dic7 C7⋊D4 C7⋊D4 C23⋊C4 C2≀C4 C23⋊Dic7 C24⋊Dic7 kernel C24⋊Dic7 C28.D4 C23⋊Dic7 C7×C22≀C2 C7×C22⋊C4 C23×C14 C2×C28 C22×C14 C22≀C2 C22⋊C4 C2×D4 C24 C2×C4 C23 C14 C7 C2 C1 # reps 1 1 1 1 2 2 1 1 3 3 3 3 6 6 1 2 6 12

Matrix representation of C24⋊Dic7 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 91 0 0 0 0 0 112 0 0 0 0 0 55 0 1 0 0 0 55 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 108 0 112 0 0 0 108 0 0 112
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 7 0 0 0 0 0 50 97 0 0 0 0 0 0 1 0 0 0 0 0 72 112 0 0 0 0 0 0 1 0 0 0 108 0 0 112
,
 108 104 0 0 0 0 28 5 0 0 0 0 0 0 58 0 91 0 0 0 0 0 112 1 0 0 45 0 55 0 0 0 4 112 55 0

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,91,112,55,55,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,108,108,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[7,50,0,0,0,0,0,97,0,0,0,0,0,0,1,72,0,108,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[108,28,0,0,0,0,104,5,0,0,0,0,0,0,58,0,45,4,0,0,0,0,0,112,0,0,91,112,55,55,0,0,0,1,0,0] >;

C24⋊Dic7 in GAP, Magma, Sage, TeX

C_2^4\rtimes {\rm Dic}_7
% in TeX

G:=Group("C2^4:Dic7");
// GroupNames label

G:=SmallGroup(448,93);
// by ID

G=gap.SmallGroup(448,93);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,675,297,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=e^7,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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