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G = C24.9C23order 192 = 26·3

2nd non-split extension by C24 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.9C23, D2410C22, M4(2)⋊19D6, C12.60C24, C23.26D12, Dic129C22, D12.23C23, Dic6.23C23, (C2×C8)⋊5D6, C4○D2410C2, C8⋊D613C2, (C2×C24)⋊8C22, C4.73(C2×D12), C8.9(C22×S3), (C2×C4).157D12, (C2×C12).205D4, C8.D613C2, C12.239(C2×D4), (C2×M4(2))⋊5S3, (C6×M4(2))⋊5C2, C4.57(S3×C23), C6.27(C22×D4), C24⋊C210C22, C4○D1217C22, (C2×D12)⋊53C22, C31(D8⋊C22), C22.22(C2×D12), C2.29(C22×D12), (C22×C6).120D4, (C22×C4).283D6, (C2×C12).798C23, (C2×Dic6)⋊64C22, (C3×M4(2))⋊21C22, (C22×C12).268C22, (C2×C6).64(C2×D4), (C2×C4○D12)⋊27C2, (C2×C4).225(C22×S3), SmallGroup(192,1307)

Series: Derived Chief Lower central Upper central

C1C12 — C24.9C23
C1C3C6C12D12C2×D12C2×C4○D12 — C24.9C23
C3C6C12 — C24.9C23
C1C4C22×C4C2×M4(2)

Generators and relations for C24.9C23
 G = < a,b,c,d | a24=b2=1, c2=d2=a12, bab=a11, ac=ca, dad-1=a13, bc=cb, bd=db, cd=dc >

Subgroups: 728 in 262 conjugacy classes, 107 normal (21 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], S3 [×4], C6, C6 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×8], C2×C6, C2×C6 [×2], C2×C6, C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], C24 [×4], Dic6 [×4], Dic6 [×2], C4×S3 [×8], D12 [×4], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C24⋊C2 [×8], D24 [×4], Dic12 [×4], C2×C24 [×2], C3×M4(2) [×4], C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12 [×2], C4○D12 [×8], C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, D8⋊C22, C4○D24 [×4], C8⋊D6 [×4], C8.D6 [×4], C6×M4(2), C2×C4○D12 [×2], C24.9C23
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, D8⋊C22, C22×D12, C24.9C23

Smallest permutation representation of C24.9C23
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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 27)(26 38)(28 36)(29 47)(30 34)(31 45)(33 43)(35 41)(37 39)(40 48)(42 46)
(1 44 13 32)(2 45 14 33)(3 46 15 34)(4 47 16 35)(5 48 17 36)(6 25 18 37)(7 26 19 38)(8 27 20 39)(9 28 21 40)(10 29 22 41)(11 30 23 42)(12 31 24 43)
(1 32 13 44)(2 45 14 33)(3 34 15 46)(4 47 16 35)(5 36 17 48)(6 25 18 37)(7 38 19 26)(8 27 20 39)(9 40 21 28)(10 29 22 41)(11 42 23 30)(12 31 24 43)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order122222222344444444466666888812121212121224···24
size1122212121212211222121212122224444442222444···4

42 irreducible representations

dim1111112222222244
type++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D12D12D8⋊C22C24.9C23
kernelC24.9C23C4○D24C8⋊D6C8.D6C6×M4(2)C2×C4○D12C2×M4(2)C2×C12C22×C6C2×C8M4(2)C22×C4C2×C4C23C3C1
# reps1444121312416224

Matrix representation of C24.9C23 in GL4(𝔽73) generated by

004646
00270
433000
431300
,
07200
72000
005966
00714
,
27000
02700
00270
00027
,
46000
04600
00270
00027
G:=sub<GL(4,GF(73))| [0,0,43,43,0,0,30,13,46,27,0,0,46,0,0,0],[0,72,0,0,72,0,0,0,0,0,59,7,0,0,66,14],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[46,0,0,0,0,46,0,0,0,0,27,0,0,0,0,27] >;

C24.9C23 in GAP, Magma, Sage, TeX

C_{24}._9C_2^3
% in TeX

G:=Group("C24.9C2^3");
// GroupNames label

G:=SmallGroup(192,1307);
// by ID

G=gap.SmallGroup(192,1307);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,80,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=1,c^2=d^2=a^12,b*a*b=a^11,a*c=c*a,d*a*d^-1=a^13,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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