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## G = D7×SL2(𝔽3)  order 336 = 24·3·7

### Direct product of D7 and SL2(𝔽3)

Aliases: D7×SL2(𝔽3), D14.2A4, Q8⋊(C3×D7), (Q8×D7)⋊1C3, (C7×Q8)⋊1C6, C2.3(A4×D7), C14.8(C2×A4), C73(C2×SL2(𝔽3)), (C7×SL2(𝔽3))⋊3C2, SmallGroup(336,132)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×Q8 — D7×SL2(𝔽3)
 Chief series C1 — C2 — C14 — C7×Q8 — C7×SL2(𝔽3) — D7×SL2(𝔽3)
 Lower central C7×Q8 — D7×SL2(𝔽3)
 Upper central C1 — C2

Generators and relations for D7×SL2(𝔽3)
G = < a,b,c,d,e | a7=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Smallest permutation representation of D7×SL2(𝔽3)
On 56 points
Generators in S56
(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)
(1 16)(2 15)(3 21)(4 20)(5 19)(6 18)(7 17)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 56)(22 32)(23 31)(24 30)(25 29)(26 35)(27 34)(28 33)(36 46)(37 45)(38 44)(39 43)(40 49)(41 48)(42 47)
(1 31 17 24)(2 32 18 25)(3 33 19 26)(4 34 20 27)(5 35 21 28)(6 29 15 22)(7 30 16 23)(8 41 55 48)(9 42 56 49)(10 36 50 43)(11 37 51 44)(12 38 52 45)(13 39 53 46)(14 40 54 47)
(1 12 17 52)(2 13 18 53)(3 14 19 54)(4 8 20 55)(5 9 21 56)(6 10 15 50)(7 11 16 51)(22 36 29 43)(23 37 30 44)(24 38 31 45)(25 39 32 46)(26 40 33 47)(27 41 34 48)(28 42 35 49)
(8 34 48)(9 35 49)(10 29 43)(11 30 44)(12 31 45)(13 32 46)(14 33 47)(22 36 50)(23 37 51)(24 38 52)(25 39 53)(26 40 54)(27 41 55)(28 42 56)

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

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), (1,16)(2,15)(3,21)(4,20)(5,19)(6,18)(7,17)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,56)(22,32)(23,31)(24,30)(25,29)(26,35)(27,34)(28,33)(36,46)(37,45)(38,44)(39,43)(40,49)(41,48)(42,47), (1,31,17,24)(2,32,18,25)(3,33,19,26)(4,34,20,27)(5,35,21,28)(6,29,15,22)(7,30,16,23)(8,41,55,48)(9,42,56,49)(10,36,50,43)(11,37,51,44)(12,38,52,45)(13,39,53,46)(14,40,54,47), (1,12,17,52)(2,13,18,53)(3,14,19,54)(4,8,20,55)(5,9,21,56)(6,10,15,50)(7,11,16,51)(22,36,29,43)(23,37,30,44)(24,38,31,45)(25,39,32,46)(26,40,33,47)(27,41,34,48)(28,42,35,49), (8,34,48)(9,35,49)(10,29,43)(11,30,44)(12,31,45)(13,32,46)(14,33,47)(22,36,50)(23,37,51)(24,38,52)(25,39,53)(26,40,54)(27,41,55)(28,42,56) );

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)], [(1,16),(2,15),(3,21),(4,20),(5,19),(6,18),(7,17),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,56),(22,32),(23,31),(24,30),(25,29),(26,35),(27,34),(28,33),(36,46),(37,45),(38,44),(39,43),(40,49),(41,48),(42,47)], [(1,31,17,24),(2,32,18,25),(3,33,19,26),(4,34,20,27),(5,35,21,28),(6,29,15,22),(7,30,16,23),(8,41,55,48),(9,42,56,49),(10,36,50,43),(11,37,51,44),(12,38,52,45),(13,39,53,46),(14,40,54,47)], [(1,12,17,52),(2,13,18,53),(3,14,19,54),(4,8,20,55),(5,9,21,56),(6,10,15,50),(7,11,16,51),(22,36,29,43),(23,37,30,44),(24,38,31,45),(25,39,32,46),(26,40,33,47),(27,41,34,48),(28,42,35,49)], [(8,34,48),(9,35,49),(10,29,43),(11,30,44),(12,31,45),(13,32,46),(14,33,47),(22,36,50),(23,37,51),(24,38,52),(25,39,53),(26,40,54),(27,41,55),(28,42,56)]])

35 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 7A 7B 7C 14A 14B 14C 21A ··· 21F 28A 28B 28C 42A ··· 42F order 1 2 2 2 3 3 4 4 6 6 6 6 6 6 7 7 7 14 14 14 21 ··· 21 28 28 28 42 ··· 42 size 1 1 7 7 4 4 6 42 4 4 28 28 28 28 2 2 2 2 2 2 8 ··· 8 12 12 12 8 ··· 8

35 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 4 4 6 type + + + - + + - + image C1 C2 C3 C6 D7 SL2(𝔽3) SL2(𝔽3) C3×D7 A4 C2×A4 D7×SL2(𝔽3) D7×SL2(𝔽3) A4×D7 kernel D7×SL2(𝔽3) C7×SL2(𝔽3) Q8×D7 C7×Q8 SL2(𝔽3) D7 D7 Q8 D14 C14 C1 C1 C2 # reps 1 1 2 2 3 2 4 6 1 1 3 6 3

Matrix representation of D7×SL2(𝔽3) in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 1 2 0 0 319 302
,
 336 0 0 0 0 336 0 0 0 0 1 2 0 0 0 336
,
 0 1 0 0 336 0 0 0 0 0 1 0 0 0 0 1
,
 208 128 0 0 128 129 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 208 128 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,1,319,0,0,2,302],[336,0,0,0,0,336,0,0,0,0,1,0,0,0,2,336],[0,336,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[208,128,0,0,128,129,0,0,0,0,1,0,0,0,0,1],[1,208,0,0,0,128,0,0,0,0,1,0,0,0,0,1] >;

D7×SL2(𝔽3) in GAP, Magma, Sage, TeX

D_7\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("D7xSL(2,3)");
// GroupNames label

G:=SmallGroup(336,132);
// by ID

G=gap.SmallGroup(336,132);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-7,-2,170,518,81,735,357,4324]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^4=e^3=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

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