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G = C24⋊S3order 144 = 24·32

5th semidirect product of C24 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C245S3, C12.56D6, C326M4(2), C83(C3⋊S3), (C3×C24)⋊8C2, C6.12(C4×S3), C32(C8⋊S3), C324C88C2, C3⋊Dic3.4C4, (C3×C12).47C22, C2.3(C4×C3⋊S3), (C4×C3⋊S3).5C2, (C2×C3⋊S3).4C4, C4.13(C2×C3⋊S3), (C3×C6).23(C2×C4), SmallGroup(144,86)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24⋊S3
C1C3C32C3×C6C3×C12C4×C3⋊S3 — C24⋊S3
C32C3×C6 — C24⋊S3
C1C4C8

Generators and relations for C24⋊S3
 G = < a,b,c | a24=b3=c2=1, ab=ba, cac=a5, cbc=b-1 >

Subgroups: 178 in 60 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, C8, C2×C4, C32, Dic3 [×4], C12 [×4], D6 [×4], M4(2), C3⋊S3, C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3 [×4], C324C8, C3×C24, C4×C3⋊S3, C24⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D6 [×4], M4(2), C3⋊S3, C4×S3 [×4], C2×C3⋊S3, C8⋊S3 [×4], C4×C3⋊S3, C24⋊S3

Smallest permutation representation of C24⋊S3
On 72 points
Generators in S72
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 57 28)(2 58 29)(3 59 30)(4 60 31)(5 61 32)(6 62 33)(7 63 34)(8 64 35)(9 65 36)(10 66 37)(11 67 38)(12 68 39)(13 69 40)(14 70 41)(15 71 42)(16 72 43)(17 49 44)(18 50 45)(19 51 46)(20 52 47)(21 53 48)(22 54 25)(23 55 26)(24 56 27)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 66)(26 71)(27 52)(28 57)(29 62)(30 67)(31 72)(32 53)(33 58)(34 63)(35 68)(36 49)(37 54)(38 59)(39 64)(40 69)(41 50)(42 55)(43 60)(44 65)(45 70)(46 51)(47 56)(48 61)

G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,57,28)(2,58,29)(3,59,30)(4,60,31)(5,61,32)(6,62,33)(7,63,34)(8,64,35)(9,65,36)(10,66,37)(11,67,38)(12,68,39)(13,69,40)(14,70,41)(15,71,42)(16,72,43)(17,49,44)(18,50,45)(19,51,46)(20,52,47)(21,53,48)(22,54,25)(23,55,26)(24,56,27), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,66)(26,71)(27,52)(28,57)(29,62)(30,67)(31,72)(32,53)(33,58)(34,63)(35,68)(36,49)(37,54)(38,59)(39,64)(40,69)(41,50)(42,55)(43,60)(44,65)(45,70)(46,51)(47,56)(48,61)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,57,28)(2,58,29)(3,59,30)(4,60,31)(5,61,32)(6,62,33)(7,63,34)(8,64,35)(9,65,36)(10,66,37)(11,67,38)(12,68,39)(13,69,40)(14,70,41)(15,71,42)(16,72,43)(17,49,44)(18,50,45)(19,51,46)(20,52,47)(21,53,48)(22,54,25)(23,55,26)(24,56,27), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,66)(26,71)(27,52)(28,57)(29,62)(30,67)(31,72)(32,53)(33,58)(34,63)(35,68)(36,49)(37,54)(38,59)(39,64)(40,69)(41,50)(42,55)(43,60)(44,65)(45,70)(46,51)(47,56)(48,61) );

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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,57,28),(2,58,29),(3,59,30),(4,60,31),(5,61,32),(6,62,33),(7,63,34),(8,64,35),(9,65,36),(10,66,37),(11,67,38),(12,68,39),(13,69,40),(14,70,41),(15,71,42),(16,72,43),(17,49,44),(18,50,45),(19,51,46),(20,52,47),(21,53,48),(22,54,25),(23,55,26),(24,56,27)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,66),(26,71),(27,52),(28,57),(29,62),(30,67),(31,72),(32,53),(33,58),(34,63),(35,68),(36,49),(37,54),(38,59),(39,64),(40,69),(41,50),(42,55),(43,60),(44,65),(45,70),(46,51),(47,56),(48,61)])

C24⋊S3 is a maximal subgroup of
S3×C8⋊S3  C246D6  C24.64D6  D12.4D6  C24.95D6  M4(2)×C3⋊S3  C24.47D6  C248D6  C247D6  C24.32D6  C24.35D6  He35M4(2)  C72⋊S3  C338M4(2)  C339M4(2)  C3315M4(2)
C24⋊S3 is a maximal quotient of
C12.30Dic6  C24⋊Dic3  C12.60D12  C72⋊S3  He36M4(2)  C338M4(2)  C339M4(2)  C3315M4(2)

42 conjugacy classes

class 1 2A2B3A3B3C3D4A4B4C6A6B6C6D8A8B8C8D12A···12H24A···24P
order12233334446666888812···1224···24
size11182222111822222218182···22···2

42 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4S3D6M4(2)C4×S3C8⋊S3
kernelC24⋊S3C324C8C3×C24C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C24C12C32C6C3
# reps111122442816

Matrix representation of C24⋊S3 in GL4(𝔽73) generated by

07200
17200
00578
006565
,
1000
0100
00072
00172
,
0100
1000
00721
0001
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,57,65,0,0,8,65],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[0,1,0,0,1,0,0,0,0,0,72,0,0,0,1,1] >;

C24⋊S3 in GAP, Magma, Sage, TeX

C_{24}\rtimes S_3
% in TeX

G:=Group("C24:S3");
// GroupNames label

G:=SmallGroup(144,86);
// by ID

G=gap.SmallGroup(144,86);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,50,964,3461]);
// Polycyclic

G:=Group<a,b,c|a^24=b^3=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

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