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G = C4○D4×C3⋊S3order 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⋊S328C22, C12.59D612C2, C12.D611C2, C12.26D611C2, (C3×C12).134C23, C12.115(C22×S3), (D4×C32)⋊26C22, C327D414C22, C3⋊Dic3.51C23, (Q8×C32)⋊23C22, C324Q826C22, D47(C2×C3⋊S3), C37(S3×C4○D4), Q87(C2×C3⋊S3), (D4×C3⋊S3)⋊11C2, (C3×C4○D4)⋊7S3, (Q8×C3⋊S3)⋊11C2, C3220(C2×C4○D4), (C4×C3⋊S3)⋊17C22, (C32×C4○D4)⋊8C2, C2.12(C23×C3⋊S3), C4.25(C22×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

C1C3×C6 — C4○D4×C3⋊S3
C1C3C32C3×C6C2×C3⋊S3C22×C3⋊S3C2×C4×C3⋊S3 — C4○D4×C3⋊S3
C32C3×C6 — C4○D4×C3⋊S3
C1C4C4○D4

Generators and relations for C4○D4×C3⋊S3
 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 >

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], D42S3 [×12], S3×Q8 [×4], Q83S3 [×4], C3×C4○D4 [×4], C324Q8 [×3], C4×C3⋊S3, C4×C3⋊S3 [×9], C12⋊S3 [×3], C2×C3⋊Dic3 [×3], C327D4 [×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

Smallest permutation representation of C4○D4×C3⋊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 18 3 20)(2 19 4 17)(5 68 7 66)(6 65 8 67)(9 42 11 44)(10 43 12 41)(13 62 15 64)(14 63 16 61)(21 47 23 45)(22 48 24 46)(25 58 27 60)(26 59 28 57)(29 55 31 53)(30 56 32 54)(33 50 35 52)(34 51 36 49)(37 70 39 72)(38 71 40 69)
(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 21 29)(2 22 30)(3 23 31)(4 24 32)(5 16 58)(6 13 59)(7 14 60)(8 15 57)(9 50 69)(10 51 70)(11 52 71)(12 49 72)(17 46 54)(18 47 55)(19 48 56)(20 45 53)(25 66 63)(26 67 64)(27 68 61)(28 65 62)(33 40 44)(34 37 41)(35 38 42)(36 39 43)
(1 33 16)(2 34 13)(3 35 14)(4 36 15)(5 29 44)(6 30 41)(7 31 42)(8 32 43)(9 68 55)(10 65 56)(11 66 53)(12 67 54)(17 49 64)(18 50 61)(19 51 62)(20 52 63)(21 40 58)(22 37 59)(23 38 60)(24 39 57)(25 45 71)(26 46 72)(27 47 69)(28 48 70)
(5 40)(6 37)(7 38)(8 39)(9 27)(10 28)(11 25)(12 26)(13 34)(14 35)(15 36)(16 33)(21 29)(22 30)(23 31)(24 32)(41 59)(42 60)(43 57)(44 58)(45 53)(46 54)(47 55)(48 56)(49 64)(50 61)(51 62)(52 63)(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), (1,18,3,20)(2,19,4,17)(5,68,7,66)(6,65,8,67)(9,42,11,44)(10,43,12,41)(13,62,15,64)(14,63,16,61)(21,47,23,45)(22,48,24,46)(25,58,27,60)(26,59,28,57)(29,55,31,53)(30,56,32,54)(33,50,35,52)(34,51,36,49)(37,70,39,72)(38,71,40,69), (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), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,16,58)(6,13,59)(7,14,60)(8,15,57)(9,50,69)(10,51,70)(11,52,71)(12,49,72)(17,46,54)(18,47,55)(19,48,56)(20,45,53)(25,66,63)(26,67,64)(27,68,61)(28,65,62)(33,40,44)(34,37,41)(35,38,42)(36,39,43), (1,33,16)(2,34,13)(3,35,14)(4,36,15)(5,29,44)(6,30,41)(7,31,42)(8,32,43)(9,68,55)(10,65,56)(11,66,53)(12,67,54)(17,49,64)(18,50,61)(19,51,62)(20,52,63)(21,40,58)(22,37,59)(23,38,60)(24,39,57)(25,45,71)(26,46,72)(27,47,69)(28,48,70), (5,40)(6,37)(7,38)(8,39)(9,27)(10,28)(11,25)(12,26)(13,34)(14,35)(15,36)(16,33)(21,29)(22,30)(23,31)(24,32)(41,59)(42,60)(43,57)(44,58)(45,53)(46,54)(47,55)(48,56)(49,64)(50,61)(51,62)(52,63)(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), (1,18,3,20)(2,19,4,17)(5,68,7,66)(6,65,8,67)(9,42,11,44)(10,43,12,41)(13,62,15,64)(14,63,16,61)(21,47,23,45)(22,48,24,46)(25,58,27,60)(26,59,28,57)(29,55,31,53)(30,56,32,54)(33,50,35,52)(34,51,36,49)(37,70,39,72)(38,71,40,69), (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), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,16,58)(6,13,59)(7,14,60)(8,15,57)(9,50,69)(10,51,70)(11,52,71)(12,49,72)(17,46,54)(18,47,55)(19,48,56)(20,45,53)(25,66,63)(26,67,64)(27,68,61)(28,65,62)(33,40,44)(34,37,41)(35,38,42)(36,39,43), (1,33,16)(2,34,13)(3,35,14)(4,36,15)(5,29,44)(6,30,41)(7,31,42)(8,32,43)(9,68,55)(10,65,56)(11,66,53)(12,67,54)(17,49,64)(18,50,61)(19,51,62)(20,52,63)(21,40,58)(22,37,59)(23,38,60)(24,39,57)(25,45,71)(26,46,72)(27,47,69)(28,48,70), (5,40)(6,37)(7,38)(8,39)(9,27)(10,28)(11,25)(12,26)(13,34)(14,35)(15,36)(16,33)(21,29)(22,30)(23,31)(24,32)(41,59)(42,60)(43,57)(44,58)(45,53)(46,54)(47,55)(48,56)(49,64)(50,61)(51,62)(52,63)(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)], [(1,18,3,20),(2,19,4,17),(5,68,7,66),(6,65,8,67),(9,42,11,44),(10,43,12,41),(13,62,15,64),(14,63,16,61),(21,47,23,45),(22,48,24,46),(25,58,27,60),(26,59,28,57),(29,55,31,53),(30,56,32,54),(33,50,35,52),(34,51,36,49),(37,70,39,72),(38,71,40,69)], [(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)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,16,58),(6,13,59),(7,14,60),(8,15,57),(9,50,69),(10,51,70),(11,52,71),(12,49,72),(17,46,54),(18,47,55),(19,48,56),(20,45,53),(25,66,63),(26,67,64),(27,68,61),(28,65,62),(33,40,44),(34,37,41),(35,38,42),(36,39,43)], [(1,33,16),(2,34,13),(3,35,14),(4,36,15),(5,29,44),(6,30,41),(7,31,42),(8,32,43),(9,68,55),(10,65,56),(11,66,53),(12,67,54),(17,49,64),(18,50,61),(19,51,62),(20,52,63),(21,40,58),(22,37,59),(23,38,60),(24,39,57),(25,45,71),(26,46,72),(27,47,69),(28,48,70)], [(5,40),(6,37),(7,38),(8,39),(9,27),(10,28),(11,25),(12,26),(13,34),(14,35),(15,36),(16,33),(21,29),(22,30),(23,31),(24,32),(41,59),(42,60),(43,57),(44,58),(45,53),(46,54),(47,55),(48,56),(49,64),(50,61),(51,62),(52,63),(65,70),(66,71),(67,72),(68,69)])

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

Matrix representation of C4○D4×C3⋊S3 in 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] >;

C4○D4×C3⋊S3 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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