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## G = C2×Dic7⋊C6order 336 = 24·3·7

### Direct product of C2 and Dic7⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×Dic7⋊C6
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C22×F7 — C2×Dic7⋊C6
 Lower central C7 — C14 — C2×Dic7⋊C6
 Upper central C1 — C22 — C23

Generators and relations for C2×Dic7⋊C6
G = < a,b,c,d | a2=b14=d6=1, c2=b7, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b11, dcd-1=b7c >

Subgroups: 464 in 108 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C7, C2×C4, D4, C23, C23, C12, C2×C6, D7, C14, C14, C14, C2×D4, C7⋊C3, C2×C12, C3×D4, C22×C6, Dic7, D14, D14, C2×C14, C2×C14, C2×C14, F7, C2×C7⋊C3, C2×C7⋊C3, C2×C7⋊C3, C6×D4, C2×Dic7, C7⋊D4, C22×D7, C22×C14, C7⋊C12, C2×F7, C2×F7, C22×C7⋊C3, C22×C7⋊C3, C22×C7⋊C3, C2×C7⋊D4, C2×C7⋊C12, Dic7⋊C6, C22×F7, C23×C7⋊C3, C2×Dic7⋊C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, F7, C6×D4, C2×F7, Dic7⋊C6, C22×F7, C2×Dic7⋊C6

Smallest permutation representation of C2×Dic7⋊C6
On 56 points
Generators in S56
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 53)(30 54)(31 55)(32 56)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)
(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 45 8 52)(2 44 9 51)(3 43 10 50)(4 56 11 49)(5 55 12 48)(6 54 13 47)(7 53 14 46)(15 35 22 42)(16 34 23 41)(17 33 24 40)(18 32 25 39)(19 31 26 38)(20 30 27 37)(21 29 28 36)
(1 15)(2 24 12 16 10 26)(3 19 9 17 5 23)(4 28 6 18 14 20)(7 27 11 21 13 25)(8 22)(29 54 39 46 37 56)(30 49 36 47 32 53)(31 44 33 48 41 50)(34 43 38 51 40 55)(35 52)(42 45)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 7 12A 12B 12C 12D 14A ··· 14G order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 7 12 12 12 12 14 ··· 14 size 1 1 1 1 2 2 14 14 7 7 14 14 7 ··· 7 14 ··· 14 6 14 14 14 14 6 ··· 6

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 6 6 6 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 C3×D4 F7 C2×F7 Dic7⋊C6 kernel C2×Dic7⋊C6 C2×C7⋊C12 Dic7⋊C6 C22×F7 C23×C7⋊C3 C2×C7⋊D4 C2×Dic7 C7⋊D4 C22×D7 C22×C14 C2×C7⋊C3 C14 C23 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 2 4 1 3 4

Matrix representation of C2×Dic7⋊C6 in GL8(𝔽337)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336
,
 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 1 125 336 0 0 0 0 0 0 125 336 0 0 0 0 0 1 124 336 0 0 0 0 0 336 212 1 0 336 0 0 0 336 212 1 0 0 336 0 0 211 88 2 336 212 213
,
 313 249 0 0 0 0 0 0 64 24 0 0 0 0 0 0 0 0 0 0 0 1 336 0 0 0 0 0 0 1 0 336 0 0 212 213 1 0 212 213 0 0 179 156 314 336 0 0 0 0 180 156 314 336 0 0 0 0 179 157 314 336 0 0
,
 208 0 0 0 0 0 0 0 101 129 0 0 0 0 0 0 0 0 124 336 212 0 0 0 0 0 336 0 0 0 0 0 0 0 336 336 213 0 0 0 0 0 158 181 23 1 0 0 0 0 281 179 235 124 336 336 0 0 158 181 23 0 1 0

G:=sub<GL(8,GF(337))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,0,1,336,336,211,0,0,125,125,124,212,212,88,0,0,336,336,336,1,1,2,0,0,0,0,0,0,0,336,0,0,0,0,0,336,0,212,0,0,0,0,0,0,336,213],[313,64,0,0,0,0,0,0,249,24,0,0,0,0,0,0,0,0,0,0,212,179,180,179,0,0,0,0,213,156,156,157,0,0,0,0,1,314,314,314,0,0,1,1,0,336,336,336,0,0,336,0,212,0,0,0,0,0,0,336,213,0,0,0],[208,101,0,0,0,0,0,0,0,129,0,0,0,0,0,0,0,0,124,336,336,158,281,158,0,0,336,0,336,181,179,181,0,0,212,0,213,23,235,23,0,0,0,0,0,1,124,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0] >;

C2×Dic7⋊C6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_7\rtimes C_6
% in TeX

G:=Group("C2xDic7:C6");
// GroupNames label

G:=SmallGroup(336,130);
// by ID

G=gap.SmallGroup(336,130);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,506,10373,887]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=d^6=1,c^2=b^7,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^7*c>;
// generators/relations

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