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G = C24.9C23  order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.9C23, D24⋊10C22, M4(2)⋊19D6, C12.60C24, C23.26D12, Dic12⋊9C22, D12.23C23, Dic6.23C23, (C2×C8)⋊5D6, C4○D24⋊10C2, C8⋊D6⋊13C2, (C2×C24)⋊8C22, C4.73(C2×D12), C8.9(C22×S3), C12.239(C2×D4), C8.D6⋊13C2, (C2×C4).157D12, (C2×C12).205D4, (C6×M4(2))⋊5C2, (C2×M4(2))⋊5S3, C4.57(S3×C23), C6.27(C22×D4), C24⋊C2⋊10C22, C4○D12⋊17C22, (C2×D12)⋊53C22, C3⋊1(D8⋊C22), (C22×C6).120D4, C2.29(C22×D12), C22.22(C2×D12), (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

Derived series
C1 — C12 — C24.9C23
Chief series
C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×C4○D12 — C24.9C23
Lower central
C3 — C6 — C12 — C24.9C23

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

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

Smallest permutation representation
►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 41)(26 28)(27 39)(29 37)(30 48)(31 35)(32 46)(34 44)(36 42)(38 40)(43 47)

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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