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G = C4○D4×C3⋊S3  order 288 = 25·32

Direct product of C4○D4 and C3⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C4○D4×C3⋊S3, C62.153C23, (C2×C12)⋊9D6, (C3×D4)⋊19D6, (C3×Q8)⋊20D6, (C6×C12)⋊15C22, C6.64(S3×C23), (C3×C6).63C24, C12⋊S3⋊28C22, C12.D6⋊11C2, C12.59D6⋊12C2, C12.26D6⋊11C2, (C3×C12).134C23, C12.115(C22×S3), (D4×C32)⋊26C22, C32⋊7D4⋊14C22, C3⋊Dic3.51C23, (Q8×C32)⋊23C22, C32⋊4Q8⋊26C22, D4⋊7(C2×C3⋊S3), C3⋊7(S3×C4○D4), Q8⋊7(C2×C3⋊S3), (D4×C3⋊S3)⋊11C2, (C3×C4○D4)⋊7S3, (Q8×C3⋊S3)⋊11C2, C32⋊20(C2×C4○D4), (C4×C3⋊S3)⋊17C22, (C32×C4○D4)⋊8C2, C4.25(C22×C3⋊S3), C2.12(C23×C3⋊S3), (C2×C3⋊S3).55C23, (C2×C6).17(C22×S3), C22.2(C22×C3⋊S3), (C2×C3⋊Dic3)⋊28C22, (C22×C3⋊S3).109C22, (C2×C4×C3⋊S3)⋊10C2, (C2×C4)⋊7(C2×C3⋊S3), SmallGroup(288,1013)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C4○D4×C3⋊S3
Chief series
C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C4○D4×C3⋊S3
Lower central
C32 — C3×C6 — C4○D4×C3⋊S3

Subgroups: 1700 in 492 conjugacy classes, 155 normal (16 characteristic)
C1, C2, C2 [×8], C3 [×4], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], S3 [×20], C6 [×4], C6 [×12], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], C32, Dic3 [×16], C12 [×16], D6 [×40], C2×C6 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], C3⋊S3 [×2], C3⋊S3 [×3], C3×C6, C3×C6 [×3], Dic6 [×12], C4×S3 [×40], D12 [×12], C2×Dic3 [×12], C3⋊D4 [×24], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], C22×S3 [×12], C2×C4○D4, C3⋊Dic3, C3⋊Dic3 [×3], C3×C12, C3×C12 [×3], C2×C3⋊S3, C2×C3⋊S3 [×3], C2×C3⋊S3 [×6], C62 [×3], S3×C2×C4 [×12], C4○D12 [×12], S3×D4 [×12], D4⋊2S3 [×12], S3×Q8 [×4], Q8⋊3S3 [×4], C3×C4○D4 [×4], C32⋊4Q8 [×3], C4×C3⋊S3, C4×C3⋊S3 [×9], C12⋊S3 [×3], C2×C3⋊Dic3 [×3], C32⋊7D4 [×6], C6×C12 [×3], D4×C32 [×3], Q8×C32, C22×C3⋊S3 [×3], S3×C4○D4 [×4], C2×C4×C3⋊S3 [×3], C12.59D6 [×3], D4×C3⋊S3 [×3], C12.D6 [×3], Q8×C3⋊S3, C12.26D6, C32×C4○D4, C4○D4×C3⋊S3

Quotients:
C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C4○D4 [×2], C24, C3⋊S3, C22×S3 [×28], C2×C4○D4, C2×C3⋊S3 [×7], S3×C23 [×4], C22×C3⋊S3 [×7], S3×C4○D4 [×4], C23×C3⋊S3, C4○D4×C3⋊S3

Generators and relations
 G = < a,b,c,d,e,f | a4=c2=d3=e3=f2=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=a2b, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

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

(9 11)(10 12)(17 19)(18 20)(25 27)(26 28)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
0012000
0001200
000080
000008
,
1200000
0120000
0012000
0001200
000084
000005
,
100000
010000
001000
000100
000010
0000912
,
100000
010000
000100
00121200
000010
000001
,
0120000
1120000
00121200
001000
000010
000001
,
010000
100000
00121200
000100
000010
000001

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,4,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,9,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C3D4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E···6P12A···12H12I···12T
order12222222223333444444444466666···612···1212···12
size11222991818182222112229918181822224···42···24···4

60 irreducible representations

dim11111111222224
type++++++++++++
imageC1C2C2C2C2C2C2C2S3D6D6D6C4○D4S3×C4○D4
kernelC4○D4×C3⋊S3C2×C4×C3⋊S3C12.59D6D4×C3⋊S3C12.D6Q8×C3⋊S3C12.26D6C32×C4○D4C3×C4○D4C2×C12C3×D4C3×Q8C3⋊S3C3
# reps1333311141212448

In GAP, Magma, Sage, TeX 

C_4\circ D_4\times C_3\rtimes S_3
 
% in TeX
 
G:=Group("C4oD4xC3:S3");
 
// GroupNames label
 
G:=SmallGroup(288,1013);
 
// by ID
 
G=gap.SmallGroup(288,1013);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,346,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=c^2=d^3=e^3=f^2=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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