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## G = C2×C3⋊D24order 288 = 25·32

### Direct product of C2 and C3⋊D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C3⋊D24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C3⋊D24 — C2×C3⋊D24
 Lower central C32 — C3×C6 — C3×C12 — C2×C3⋊D24
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C3⋊D24
G = < a,b,c,d | a2=b3=c24=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 962 in 179 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×10], C6 [×2], C6 [×4], C6 [×5], C8 [×2], C2×C4, D4 [×6], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×20], C2×C6 [×2], C2×C6 [×5], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], D12 [×2], D12 [×11], C2×C12 [×2], C2×C12, C3×D4 [×3], C22×S3 [×5], C22×C6, C2×D8, C3×C12 [×2], S3×C6 [×4], C2×C3⋊S3 [×4], C62, D24 [×4], C2×C3⋊C8, D4⋊S3 [×4], C2×C24, C2×D12, C2×D12 [×3], C6×D4, C3×C3⋊C8 [×2], C3×D12 [×2], C3×D12, C12⋊S3 [×2], C12⋊S3, C6×C12, S3×C2×C6, C22×C3⋊S3, C2×D24, C2×D4⋊S3, C3⋊D24 [×4], C6×C3⋊C8, C6×D12, C2×C12⋊S3, C2×C3⋊D24
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], D8 [×2], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C2×D8, S32, D24 [×2], D4⋊S3 [×2], C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C2×D24, C2×D4⋊S3, C3⋊D24 [×2], C2×C3⋊D12, C2×C3⋊D24

Smallest permutation representation of C2×C3⋊D24
On 48 points
Generators in S48
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)
(1 17 9)(2 10 18)(3 19 11)(4 12 20)(5 21 13)(6 14 22)(7 23 15)(8 16 24)(25 41 33)(26 34 42)(27 43 35)(28 36 44)(29 45 37)(30 38 46)(31 47 39)(32 40 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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 ··· 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 12 12 36 36 2 2 4 2 2 2 ··· 2 4 4 4 12 12 12 12 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D8 D12 C3⋊D4 D12 C3⋊D4 D24 S32 D4⋊S3 C3⋊D12 C2×S32 C3⋊D12 C3⋊D24 kernel C2×C3⋊D24 C3⋊D24 C6×C3⋊C8 C6×D12 C2×C12⋊S3 C2×C3⋊C8 C2×D12 C3×C12 C62 C3⋊C8 D12 C2×C12 C3×C6 C12 C12 C2×C6 C2×C6 C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 1 1 1 4

Matrix representation of C2×C3⋊D24 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 13 50 0 0 0 0 35 28 0 0 0 0 0 0 0 1 0 0 0 0 72 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 72 0 0 0 0 0 7 1 0 0 0 0 0 0 72 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[13,35,0,0,0,0,50,28,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,7,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C3⋊D24 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{24}
% in TeX

G:=Group("C2xC3:D24");
// GroupNames label

G:=SmallGroup(288,472);
// by ID

G=gap.SmallGroup(288,472);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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