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## G = C22×C52⋊C4order 400 = 24·52

### Direct product of C22 and C52⋊C4

Aliases: C22×C52⋊C4, C1028C4, (C2×C10)⋊4F5, C103(C2×F5), C53(C22×F5), C5⋊D5.6C23, C527(C22×C4), (C2×C5⋊D5)⋊9C4, C5⋊D56(C2×C4), (C5×C10)⋊6(C2×C4), (C22×C5⋊D5).7C2, (C2×C5⋊D5).28C22, SmallGroup(400,217)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C22×C52⋊C4
 Chief series C1 — C5 — C52 — C5⋊D5 — C52⋊C4 — C2×C52⋊C4 — C22×C52⋊C4
 Lower central C52 — C22×C52⋊C4
 Upper central C1 — C22

Generators and relations for C22×C52⋊C4
G = < a,b,c,d,e | a2=b2=c5=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c2, ede-1=d3 >

Subgroups: 1052 in 140 conjugacy classes, 42 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, C10, C10, C22×C4, F5, D10, C2×C10, C2×C10, C52, C2×F5, C22×D5, C5⋊D5, C5⋊D5, C5×C10, C22×F5, C52⋊C4, C2×C5⋊D5, C102, C2×C52⋊C4, C22×C5⋊D5, C22×C52⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, F5, C2×F5, C22×F5, C52⋊C4, C2×C52⋊C4, C22×C52⋊C4

Smallest permutation representation of C22×C52⋊C4
On 40 points
Generators in S40
(1 18)(2 19)(3 20)(4 16)(5 17)(6 15)(7 11)(8 12)(9 13)(10 14)(21 38)(22 39)(23 40)(24 36)(25 37)(26 34)(27 35)(28 31)(29 32)(30 33)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(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)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 25 24 23 22)(26 30 29 28 27)(31 35 34 33 32)(36 40 39 38 37)
(1 34 8 36)(2 32 7 38)(3 35 6 40)(4 33 10 37)(5 31 9 39)(11 21 19 29)(12 24 18 26)(13 22 17 28)(14 25 16 30)(15 23 20 27)

G:=sub<Sym(40)| (1,18)(2,19)(3,20)(4,16)(5,17)(6,15)(7,11)(8,12)(9,13)(10,14)(21,38)(22,39)(23,40)(24,36)(25,37)(26,34)(27,35)(28,31)(29,32)(30,33), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)(36,40,39,38,37), (1,34,8,36)(2,32,7,38)(3,35,6,40)(4,33,10,37)(5,31,9,39)(11,21,19,29)(12,24,18,26)(13,22,17,28)(14,25,16,30)(15,23,20,27)>;

G:=Group( (1,18)(2,19)(3,20)(4,16)(5,17)(6,15)(7,11)(8,12)(9,13)(10,14)(21,38)(22,39)(23,40)(24,36)(25,37)(26,34)(27,35)(28,31)(29,32)(30,33), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)(36,40,39,38,37), (1,34,8,36)(2,32,7,38)(3,35,6,40)(4,33,10,37)(5,31,9,39)(11,21,19,29)(12,24,18,26)(13,22,17,28)(14,25,16,30)(15,23,20,27) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,16),(5,17),(6,15),(7,11),(8,12),(9,13),(10,14),(21,38),(22,39),(23,40),(24,36),(25,37),(26,34),(27,35),(28,31),(29,32),(30,33)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,25,24,23,22),(26,30,29,28,27),(31,35,34,33,32),(36,40,39,38,37)], [(1,34,8,36),(2,32,7,38),(3,35,6,40),(4,33,10,37),(5,31,9,39),(11,21,19,29),(12,24,18,26),(13,22,17,28),(14,25,16,30),(15,23,20,27)]])

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 5A ··· 5F 10A ··· 10R order 1 2 2 2 2 2 2 2 4 ··· 4 5 ··· 5 10 ··· 10 size 1 1 1 1 25 25 25 25 25 ··· 25 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 F5 C2×F5 C52⋊C4 C2×C52⋊C4 kernel C22×C52⋊C4 C2×C52⋊C4 C22×C5⋊D5 C2×C5⋊D5 C102 C2×C10 C10 C22 C2 # reps 1 6 1 6 2 2 6 4 12

Matrix representation of C22×C52⋊C4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 6 34 0 0 0 6 0 0 0 0 0 32 0 1 0 22 0 40 34
,
 1 0 0 0 0 0 6 34 0 0 0 6 0 0 0 0 22 0 34 40 0 0 32 1 0
,
 32 0 0 0 0 0 0 0 1 40 0 19 32 2 7 0 0 0 9 0 0 6 1 9 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,6,6,0,22,0,34,0,32,0,0,0,0,0,40,0,0,0,1,34],[1,0,0,0,0,0,6,6,22,0,0,34,0,0,32,0,0,0,34,1,0,0,0,40,0],[32,0,0,0,0,0,0,19,0,6,0,0,32,0,1,0,1,2,9,9,0,40,7,0,0] >;

C22×C52⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_5^2\rtimes C_4
% in TeX

G:=Group("C2^2xC5^2:C4");
// GroupNames label

G:=SmallGroup(400,217);
// by ID

G=gap.SmallGroup(400,217);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,1444,262,5765,1463]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^2,e*d*e^-1=d^3>;
// generators/relations

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