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G = D2410C4order 192 = 26·3

10th semidirect product of D24 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2410C4, Dic1210C4, M4(2).26D6, C3⋊C8.35D4, C24⋊C24C4, C8.11(C4×S3), C24⋊C41C2, (C2×C8).69D6, C6.55(C4×D4), C33(C8.26D4), C8.C44S3, C24.29(C2×C4), C4○D24.4C2, C4.212(S3×D4), D12.C410C2, D12.10(C2×C4), C12.371(C2×D4), D12⋊C411C2, C12.54(C22×C4), (C2×C24).41C22, Dic6.10(C2×C4), (C2×C12).310C23, C4○D12.17C22, C2.15(Dic35D4), C22.1(Q83S3), (C4×Dic3).40C22, (C3×M4(2)).28C22, C4.46(S3×C2×C4), (C3×C8.C4)⋊4C2, (C2×C6).1(C4○D4), (C2×C3⋊C8).78C22, (C2×C4).413(C22×S3), SmallGroup(192,453)

Series: Derived Chief Lower central Upper central

C1C12 — D2410C4
C1C3C6C12C2×C12C4○D12C4○D24 — D2410C4
C3C6C12 — D2410C4
C1C4C2×C4C8.C4

Generators and relations for D2410C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a5, cbc-1=a10b >

Subgroups: 272 in 104 conjugacy classes, 45 normal (27 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], Dic3 [×3], C12 [×2], D6 [×2], C2×C6, C42, C2×C8, C2×C8 [×3], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], C3⋊C8 [×2], C24 [×2], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12, C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24, Dic12, C2×C3⋊C8, C4×Dic3, C2×C24, C3×M4(2) [×2], C4○D12 [×2], C8.26D4, C24⋊C4, D12⋊C4 [×2], C3×C8.C4, C4○D24, D12.C4 [×2], D2410C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C22×S3, C4×D4, S3×C2×C4, S3×D4, Q83S3, C8.26D4, Dic35D4, D2410C4

Smallest permutation representation of D2410C4
On 48 points
Generators in S48
(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)
(1 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)
(1 19 13 7)(2 24 14 12)(3 5 15 17)(4 10 16 22)(6 20 18 8)(9 11 21 23)(25 29)(26 34)(27 39)(28 44)(31 35)(32 40)(33 45)(37 41)(38 46)(43 47)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G6A6B8A···8F8G8H8I8J12A12B12C24A24B24C24D24E24F24G24H
order1222234444444668···888881212122424242424242424
size1121212211212121212244···4666622444448888

36 irreducible representations

dim1111111112222224444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D4D6D6C4○D4C4×S3S3×D4Q83S3C8.26D4D2410C4
kernelD2410C4C24⋊C4D12⋊C4C3×C8.C4C4○D24D12.C4C24⋊C2D24Dic12C8.C4C3⋊C8C2×C8M4(2)C2×C6C8C4C22C3C1
# reps1121124221212241124

Matrix representation of D2410C4 in GL4(𝔽5) generated by

0011
0024
0301
4024
,
1122
2440
0204
4320
,
1133
2012
0001
0024
G:=sub<GL(4,GF(5))| [0,0,0,4,0,0,3,0,1,2,0,2,1,4,1,4],[1,2,0,4,1,4,2,3,2,4,0,2,2,0,4,0],[1,2,0,0,1,0,0,0,3,1,0,2,3,2,1,4] >;

D2410C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes_{10}C_4
% in TeX

G:=Group("D24:10C4");
// GroupNames label

G:=SmallGroup(192,453);
// by ID

G=gap.SmallGroup(192,453);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,219,58,136,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b>;
// generators/relations

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