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## G = C2×C6.7S4order 288 = 25·32

### Direct product of C2 and C6.7S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C2×C6.7S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C6.7S4 — C2×C6.7S4
 Lower central C3×A4 — C2×C6.7S4
 Upper central C1 — C22

Generators and relations for C2×C6.7S4
G = < a,b,c,d,e,f | a2=b6=c2=d2=e3=1, f2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >

Subgroups: 708 in 158 conjugacy classes, 43 normal (16 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C32, Dic3, A4, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×C6, C2×Dic3, C2×A4, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C3⋊Dic3, C3×A4, C62, C6.D4, A4⋊C4, C22×Dic3, C22×A4, C23×C6, C2×C3⋊Dic3, C6×A4, C6×A4, C2×C6.D4, C2×A4⋊C4, C6.7S4, A4×C2×C6, C2×C6.7S4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C3⋊S3, C2×Dic3, S4, C3⋊Dic3, C2×C3⋊S3, A4⋊C4, C2×S4, C2×C3⋊Dic3, C3⋊S4, C2×A4⋊C4, C6.7S4, C2×C3⋊S4, C2×C6.7S4

Smallest permutation representation of C2×C6.7S4
On 72 points
Generators in S72
(1 53)(2 54)(3 49)(4 50)(5 51)(6 52)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 58)(20 59)(21 60)(22 55)(23 56)(24 57)(25 64)(26 65)(27 66)(28 61)(29 62)(30 63)(31 70)(32 71)(33 72)(34 67)(35 68)(36 69)
(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 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 10)(8 11)(9 12)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 52)(44 53)(45 54)(46 49)(47 50)(48 51)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(67 70)(68 71)(69 72)
(1 4)(2 5)(3 6)(7 72)(8 67)(9 68)(10 69)(11 70)(12 71)(13 16)(14 17)(15 18)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(55 64)(56 65)(57 66)(58 61)(59 62)(60 63)
(1 26 38)(2 27 39)(3 28 40)(4 29 41)(5 30 42)(6 25 37)(7 54 66)(8 49 61)(9 50 62)(10 51 63)(11 52 64)(12 53 65)(13 19 31)(14 20 32)(15 21 33)(16 22 34)(17 23 35)(18 24 36)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)
(1 17 4 14)(2 16 5 13)(3 15 6 18)(7 55 10 58)(8 60 11 57)(9 59 12 56)(19 39 22 42)(20 38 23 41)(21 37 24 40)(25 36 28 33)(26 35 29 32)(27 34 30 31)(43 51 46 54)(44 50 47 53)(45 49 48 52)(61 72 64 69)(62 71 65 68)(63 70 66 67)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A ··· 4H 6A 6B 6C 6D 6E 6F 6G 6H ··· 6P order 1 2 2 2 2 2 2 2 3 3 3 3 4 ··· 4 6 6 6 6 6 6 6 6 ··· 6 size 1 1 1 1 3 3 3 3 2 8 8 8 18 ··· 18 2 2 2 6 6 6 6 8 ··· 8

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 6 type + + + + + - + - + + + + - + image C1 C2 C2 C4 S3 S3 Dic3 D6 Dic3 D6 S4 A4⋊C4 C2×S4 C3⋊S4 C6.7S4 C2×C3⋊S4 kernel C2×C6.7S4 C6.7S4 A4×C2×C6 C6×A4 C22×A4 C23×C6 C2×A4 C2×A4 C22×C6 C22×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 4 3 1 6 3 2 1 2 4 2 1 2 1

Matrix representation of C2×C6.7S4 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 1 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 12 12 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 5 0 0 1 0 0
,
 5 5 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 5

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

C2×C6.7S4 in GAP, Magma, Sage, TeX

C_2\times C_6._7S_4
% in TeX

G:=Group("C2xC6.7S4");
// GroupNames label

G:=SmallGroup(288,916);
// by ID

G=gap.SmallGroup(288,916);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,451,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=c^2=d^2=e^3=1,f^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b^-1,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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