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G = C7×2+ 1+4order 224 = 25·7

Direct product of C7 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×2+ 1+4
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4 — D4×C14 — C7×2+ 1+4
 Lower central C1 — C2 — C7×2+ 1+4
 Upper central C1 — C14 — C7×2+ 1+4

Generators and relations for C7×2+ 1+4
G = < a,b,c,d,e | a7=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, D4, Q8, C23, C14, C14, C2×D4, C4○D4, C28, C2×C14, C2×C14, 2+ 1+4, C2×C28, C7×D4, C7×Q8, C22×C14, D4×C14, C7×C4○D4, C7×2+ 1+4
Quotients: C1, C2, C22, C7, C23, C14, C24, C2×C14, 2+ 1+4, C22×C14, C23×C14, C7×2+ 1+4

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

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

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

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

C7×2+ 1+4 is a maximal subgroup of   2+ 1+4⋊D7  2+ 1+4.D7  2+ 1+4.2D7  2+ 1+42D7  D28.32C23  D28.33C23  D14.C24
C7×2+ 1+4 is a maximal quotient of   C7×D42  C7×Q82

119 conjugacy classes

 class 1 2A 2B ··· 2J 4A ··· 4F 7A ··· 7F 14A ··· 14F 14G ··· 14BH 28A ··· 28AJ order 1 2 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

119 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C7 C14 C14 2+ 1+4 C7×2+ 1+4 kernel C7×2+ 1+4 D4×C14 C7×C4○D4 2+ 1+4 C2×D4 C4○D4 C7 C1 # reps 1 9 6 6 54 36 1 6

Matrix representation of C7×2+ 1+4 in GL4(𝔽29) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 26 1 2 0 28 0 1 28 1 0 1 0 27 0 1
,
 0 1 26 27 1 0 28 27 0 0 1 0 0 0 27 28
,
 0 26 28 1 0 28 0 1 1 1 0 27 0 27 0 1
,
 0 1 3 1 1 0 1 28 0 0 28 0 0 0 2 1
G:=sub<GL(4,GF(29))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,28,0,26,28,1,27,1,0,0,0,2,1,1,1],[0,1,0,0,1,0,0,0,26,28,1,27,27,27,0,28],[0,0,1,0,26,28,1,27,28,0,0,0,1,1,27,1],[0,1,0,0,1,0,0,0,3,1,28,2,1,28,0,1] >;

C7×2+ 1+4 in GAP, Magma, Sage, TeX

C_7\times 2_+^{1+4}
% in TeX

G:=Group("C7xES+(2,2)");
// GroupNames label

G:=SmallGroup(224,193);
// by ID

G=gap.SmallGroup(224,193);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-7,-2,1369,1052,2883]);
// Polycyclic

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

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