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G = C2×C3⋊D24  order 288 = 25·32

Direct product of C2 and C3⋊D24

direct product, metabelian, supersoluble, monomial

Aliases: C2×C3⋊D24, C6⋊2D24, D12⋊17D6, C12.21D12, C62.43D4, C3⋊C8⋊23D6, (C3×C6)⋊3D8, C3⋊3(C2×D24), C32⋊6(C2×D8), C6⋊1(D4⋊S3), (C6×D12)⋊3C2, (C2×D12)⋊1S3, C6.65(C2×D12), (C2×C6).58D12, (C3×C12).62D4, (C2×C12).113D6, C4.5(C3⋊D12), (C3×D12)⋊21C22, C12.70(C3⋊D4), (C6×C12).73C22, (C3×C12).60C23, C12.80(C22×S3), C12⋊S3⋊16C22, C22.19(C3⋊D12), (C6×C3⋊C8)⋊9C2, (C2×C3⋊C8)⋊5S3, C4.50(C2×S32), C3⋊1(C2×D4⋊S3), (C2×C4).64S32, C6.1(C2×C3⋊D4), (C3×C3⋊C8)⋊27C22, (C3×C6).64(C2×D4), (C2×C12⋊S3)⋊9C2, C2.5(C2×C3⋊D12), (C2×C6).36(C3⋊D4), SmallGroup(288,472)

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, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×D8, C3×C12, S3×C6, C2×C3⋊S3, C62, D24, C2×C3⋊C8, D4⋊S3, C2×C24, C2×D12, C2×D12, C6×D4, C3×C3⋊C8, C3×D12, C3×D12, C12⋊S3, C12⋊S3, C6×C12, S3×C2×C6, C22×C3⋊S3, C2×D24, C2×D4⋊S3, C3⋊D24, C6×C3⋊C8, C6×D12, C2×C12⋊S3, C2×C3⋊D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C3⋊D4, C22×S3, C2×D8, S32, D24, D4⋊S3, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C2×D24, C2×D4⋊S3, C3⋊D24, C2×C3⋊D12, C2×C3⋊D24

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

(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 33 41)(26 42 34)(27 35 43)(28 44 36)(29 37 45)(30 46 38)(31 39 47)(32 48 40)

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A···6F6G6H6I6J6K6L6M8A8B8C8D12A12B12C12D12E···12J24A···24H
order12222222333446···6666666688881212121212···1224···24
size111112123636224222···244412121212666622224···46···6

48 irreducible representations

dim111112222222222222444444
type++++++++++++++++++++++
imageC1C2C2C2C2S3S3D4D4D6D6D6D8D12C3⋊D4D12C3⋊D4D24S32D4⋊S3C3⋊D12C2×S32C3⋊D12C3⋊D24
kernelC2×C3⋊D24C3⋊D24C6×C3⋊C8C6×D12C2×C12⋊S3C2×C3⋊C8C2×D12C3×C12C62C3⋊C8D12C2×C12C3×C6C12C12C2×C6C2×C6C6C2×C4C6C4C4C22C2
# reps141111111222422228121114

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

100000
010000
0072000
0007200
000010
000001
,
100000
010000
001000
000100
0000721
0000720
,
13500000
35280000
000100
0072100
000001
000010
,
7200000
710000
0072100
000100
000001
000010

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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