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## G = (C2×C24)⋊C4order 192 = 26·3

### 1st semidirect product of C2×C24 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C24)⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C23.26D6 — (C2×C24)⋊C4
 Lower central C3 — C6 — C2×C6 — (C2×C24)⋊C4
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for (C2×C24)⋊C4
G = < a,b,c | a2=b24=c4=1, ab=ba, cac-1=ab12, cbc-1=ab5 >

Subgroups: 200 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×C12, M4(2)⋊4C4, C2×C4.Dic3, C23.26D6, C6×M4(2), (C2×C24)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, M4(2)⋊4C4, C6.C42, (C2×C24)⋊C4

Smallest permutation representation of (C2×C24)⋊C4
On 48 points
Generators in S48
(1 47)(2 48)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)
(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 40 14 28)(3 11)(4 26 16 38)(5 21)(6 36 18 48)(8 46 20 34)(9 17)(10 32 22 44)(12 42 24 30)(15 23)(25 45)(27 31)(29 41)(33 37)(35 47)(39 43)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - - + - + image C1 C2 C2 C2 C4 C4 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 C4×S3 M4(2)⋊4C4 (C2×C24)⋊C4 kernel (C2×C24)⋊C4 C2×C4.Dic3 C23.26D6 C6×M4(2) C4.Dic3 C6.D4 C2×C24 C2×M4(2) C2×C12 C2×C12 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C3 C1 # reps 1 1 1 1 4 4 4 1 3 1 2 1 2 2 2 4 2 2 4

Matrix representation of (C2×C24)⋊C4 in GL4(𝔽73) generated by

 43 60 0 0 13 30 0 0 0 0 30 60 0 0 13 43
,
 36 69 46 44 4 32 2 46 36 55 41 4 55 36 69 37
,
 1 72 0 2 0 72 2 2 0 0 60 30 0 0 43 13
G:=sub<GL(4,GF(73))| [43,13,0,0,60,30,0,0,0,0,30,13,0,0,60,43],[36,4,36,55,69,32,55,36,46,2,41,69,44,46,4,37],[1,0,0,0,72,72,0,0,0,2,60,43,2,2,30,13] >;

(C2×C24)⋊C4 in GAP, Magma, Sage, TeX

(C_2\times C_{24})\rtimes C_4
% in TeX

G:=Group("(C2xC24):C4");
// GroupNames label

G:=SmallGroup(192,115);
// by ID

G=gap.SmallGroup(192,115);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,136,1684,6278]);
// Polycyclic

G:=Group<a,b,c|a^2=b^24=c^4=1,a*b=b*a,c*a*c^-1=a*b^12,c*b*c^-1=a*b^5>;
// generators/relations

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