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## G = C4○D4×C3⋊S3order 288 = 25·32

### Direct product of C4○D4 and C3⋊S3

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
 Upper central C1 — C4 — C4○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, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C2×C4○D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, C324Q8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, C6×C12, D4×C32, Q8×C32, C22×C3⋊S3, S3×C4○D4, C2×C4×C3⋊S3, C12.59D6, D4×C3⋊S3, C12.D6, Q8×C3⋊S3, C12.26D6, C32×C4○D4, C4○D4×C3⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C3⋊S3, C22×S3, C2×C4○D4, C2×C3⋊S3, S3×C23, C22×C3⋊S3, S3×C4○D4, 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 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E ··· 6P 12A ··· 12H 12I ··· 12T order 1 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 2 2 2 9 9 18 18 18 2 2 2 2 1 1 2 2 2 9 9 18 18 18 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 S3×C4○D4 kernel C4○D4×C3⋊S3 C2×C4×C3⋊S3 C12.59D6 D4×C3⋊S3 C12.D6 Q8×C3⋊S3 C12.26D6 C32×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C3⋊S3 C3 # reps 1 3 3 3 3 1 1 1 4 12 12 4 4 8

Matrix representation of C4○D4×C3⋊S3 in GL6(𝔽13)

 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 4 0 0 0 0 0 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 0 0 0 0 0 9 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 12 0 0 0 0 1 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 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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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