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G = C528M4(2)  order 400 = 24·52

4th semidirect product of C52 and M4(2) acting via M4(2)/C4=C4

metabelian, supersoluble, monomial

Aliases: C20.4F5, C528M4(2), (C5×C20).4C4, C4.(C52⋊C4), C52(C4.F5), C525C85C2, C10.22(C2×F5), C526C4.23C22, (C4×C5⋊D5).8C2, (C2×C5⋊D5).10C4, C2.4(C2×C52⋊C4), (C5×C10).35(C2×C4), SmallGroup(400,157)

Series: Derived Chief Lower central Upper central

C1C5×C10 — C528M4(2)
C1C5C52C5×C10C526C4C525C8 — C528M4(2)
C52C5×C10 — C528M4(2)
C1C2C4

Generators and relations for C528M4(2)
 G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, dad=a-1, cbc-1=b2, dbd=b-1, dcd=c5 >

Subgroups: 412 in 56 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5 [×2], C5 [×2], C8 [×2], C2×C4, D5 [×6], C10 [×2], C10 [×2], M4(2), Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], C52, C5⋊C8 [×4], C4×D5 [×4], C5⋊D5, C5×C10, C4.F5 [×2], C526C4, C5×C20, C2×C5⋊D5, C525C8 [×2], C4×C5⋊D5, C528M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, M4(2), F5 [×2], C2×F5 [×2], C4.F5 [×2], C52⋊C4, C2×C52⋊C4, C528M4(2)

Smallest permutation representation of C528M4(2)
On 40 points
Generators in S40
(1 22 15 26 39)(2 27 23 40 16)(3 33 28 9 24)(4 10 34 17 29)(5 18 11 30 35)(6 31 19 36 12)(7 37 32 13 20)(8 14 38 21 25)
(1 15 39 22 26)(2 40 27 16 23)(3 28 24 33 9)(4 17 10 29 34)(5 11 35 18 30)(6 36 31 12 19)(7 32 20 37 13)(8 21 14 25 38)
(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)
(2 6)(4 8)(9 28)(10 25)(11 30)(12 27)(13 32)(14 29)(15 26)(16 31)(17 38)(18 35)(19 40)(20 37)(21 34)(22 39)(23 36)(24 33)

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

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

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

34 conjugacy classes

class 1 2A2B4A4B4C5A···5F8A8B8C8D10A···10F20A···20L
order1224445···5888810···1020···20
size1150225254···4505050504···44···4

34 irreducible representations

dim111112444444
type+++++++
imageC1C2C2C4C4M4(2)F5C2×F5C4.F5C52⋊C4C2×C52⋊C4C528M4(2)
kernelC528M4(2)C525C8C4×C5⋊D5C5×C20C2×C5⋊D5C52C20C10C5C4C2C1
# reps121222224448

Matrix representation of C528M4(2) in GL4(𝔽41) generated by

0700
35600
173440
35610
,
40100
53500
614034
393577
,
713340
403567
2633400
1733400
,
6100
63500
3823400
312271
G:=sub<GL(4,GF(41))| [0,35,1,35,7,6,7,6,0,0,34,1,0,0,40,0],[40,5,6,39,1,35,1,35,0,0,40,7,0,0,34,7],[7,40,26,17,1,35,33,33,33,6,40,40,40,7,0,0],[6,6,38,31,1,35,23,22,0,0,40,7,0,0,0,1] >;

C528M4(2) in GAP, Magma, Sage, TeX

C_5^2\rtimes_8M_4(2)
% in TeX

G:=Group("C5^2:8M4(2)");
// GroupNames label

G:=SmallGroup(400,157);
// by ID

G=gap.SmallGroup(400,157);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,1444,496,5765,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^3,d*a*d=a^-1,c*b*c^-1=b^2,d*b*d=b^-1,d*c*d=c^5>;
// generators/relations

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