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## G = C24.9C23order 192 = 26·3

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

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
 Upper central C1 — C4 — C22×C4 — C2×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, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C24⋊C2, D24, Dic12, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, D8⋊C22, C4○D24, C8⋊D6, C8.D6, C6×M4(2), C2×C4○D12, C24.9C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C2×D12, 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 37)(26 48)(27 35)(28 46)(29 33)(30 44)(32 42)(34 40)(36 38)(39 47)(41 45)
(1 31 13 43)(2 32 14 44)(3 33 15 45)(4 34 16 46)(5 35 17 47)(6 36 18 48)(7 37 19 25)(8 38 20 26)(9 39 21 27)(10 40 22 28)(11 41 23 29)(12 42 24 30)
(1 43 13 31)(2 32 14 44)(3 45 15 33)(4 34 16 46)(5 47 17 35)(6 36 18 48)(7 25 19 37)(8 38 20 26)(9 27 21 39)(10 40 22 28)(11 29 23 41)(12 42 24 30)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 2 2 2 12 12 12 12 2 1 1 2 2 2 12 12 12 12 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D12 D12 D8⋊C22 C24.9C23 kernel C24.9C23 C4○D24 C8⋊D6 C8.D6 C6×M4(2) C2×C4○D12 C2×M4(2) C2×C12 C22×C6 C2×C8 M4(2) C22×C4 C2×C4 C23 C3 C1 # reps 1 4 4 4 1 2 1 3 1 2 4 1 6 2 2 4

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

 0 0 46 46 0 0 27 0 43 30 0 0 43 13 0 0
,
 0 72 0 0 72 0 0 0 0 0 59 66 0 0 7 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
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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