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G = C1024C4order 400 = 24·52

4th semidirect product of C102 and C4 acting faithfully

metabelian, supersoluble, monomial

Aliases: C1024C4, C5⋊D5.8D4, (C2×C10)⋊2F5, C53(C22⋊F5), C22⋊(C52⋊C4), C10.27(C2×F5), C527(C22⋊C4), (C2×C5⋊D5)⋊7C4, (C2×C52⋊C4)⋊3C2, C2.7(C2×C52⋊C4), (C5×C10).40(C2×C4), (C22×C5⋊D5).5C2, (C2×C5⋊D5).26C22, SmallGroup(400,162)

Series: Derived Chief Lower central Upper central

C1C5×C10 — C1024C4
C1C5C52C5⋊D5C2×C5⋊D5C2×C52⋊C4 — C1024C4
C52C5×C10 — C1024C4
C1C2C22

Generators and relations for C1024C4
 G = < a,b,c | a10=b10=c4=1, ab=ba, cac-1=a7b5, cbc-1=b3 >

Subgroups: 908 in 100 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C2 [×4], C4 [×2], C22, C22 [×4], C5 [×2], C5 [×2], C2×C4 [×2], C23, D5 [×14], C10 [×2], C10 [×8], C22⋊C4, F5 [×4], D10 [×20], C2×C10 [×2], C2×C10 [×2], C52, C2×F5 [×4], C22×D5 [×4], C5⋊D5 [×2], C5⋊D5, C5×C10, C5×C10, C22⋊F5 [×2], C52⋊C4 [×2], C2×C5⋊D5 [×2], C2×C5⋊D5 [×2], C102, C2×C52⋊C4 [×2], C22×C5⋊D5, C1024C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5 [×2], C2×F5 [×2], C22⋊F5 [×2], C52⋊C4, C2×C52⋊C4, C1024C4

Permutation representations of C1024C4
On 20 points - transitive group 20T103
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 6 4 9 2 7 5 10 3 8)(11 12 13 14 15 16 17 18 19 20)
(1 12)(2 20 5 14)(3 18 4 16)(6 19 8 15)(7 17)(9 13 10 11)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,6,4,9,2,7,5,10,3,8)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,20,5,14)(3,18,4,16)(6,19,8,15)(7,17)(9,13,10,11)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,6,4,9,2,7,5,10,3,8)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,20,5,14)(3,18,4,16)(6,19,8,15)(7,17)(9,13,10,11) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,6,4,9,2,7,5,10,3,8),(11,12,13,14,15,16,17,18,19,20)], [(1,12),(2,20,5,14),(3,18,4,16),(6,19,8,15),(7,17),(9,13,10,11)])

G:=TransitiveGroup(20,103);

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A···5F10A···10R
order12222244445···510···10
size112252550505050504···44···4

34 irreducible representations

dim111112444444
type++++++++++
imageC1C2C2C4C4D4F5C2×F5C22⋊F5C52⋊C4C2×C52⋊C4C1024C4
kernelC1024C4C2×C52⋊C4C22×C5⋊D5C2×C5⋊D5C102C5⋊D5C2×C10C10C5C22C2C1
# reps121222224448

Matrix representation of C1024C4 in GL4(𝔽41) generated by

73400
74000
2922640
3830361
,
13400
73400
135640
639361
,
00341
113940
135400
06340
G:=sub<GL(4,GF(41))| [7,7,29,38,34,40,22,30,0,0,6,36,0,0,40,1],[1,7,1,6,34,34,35,39,0,0,6,36,0,0,40,1],[0,1,1,0,0,1,35,6,34,39,40,34,1,40,0,0] >;

C1024C4 in GAP, Magma, Sage, TeX

C_{10}^2\rtimes_4C_4
% in TeX

G:=Group("C10^2:4C4");
// GroupNames label

G:=SmallGroup(400,162);
// by ID

G=gap.SmallGroup(400,162);
# by ID

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

G:=Group<a,b,c|a^10=b^10=c^4=1,a*b=b*a,c*a*c^-1=a^7*b^5,c*b*c^-1=b^3>;
// generators/relations

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