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G = C2×Dic7⋊C6  order 336 = 24·3·7

Direct product of C2 and Dic7⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C2×Dic7⋊C6, C23⋊2F7, C7⋊3(C6×D4), C14⋊2(C3×D4), C7⋊D4⋊4C6, D14⋊3(C2×C6), C7⋊C12⋊2C22, C22⋊3(C2×F7), (C22×C14)⋊2C6, Dic7⋊2(C2×C6), (C2×Dic7)⋊4C6, (C22×D7)⋊3C6, (C22×F7)⋊3C2, (C2×F7)⋊3C22, C2.10(C22×F7), C14.10(C22×C6), (C2×C7⋊D4)⋊C3, C7⋊C3⋊3(C2×D4), (C2×C7⋊C3)⋊2D4, (C2×C7⋊C12)⋊4C2, (C2×C14)⋊4(C2×C6), (C23×C7⋊C3)⋊1C2, (C2×C7⋊C3).10C23, (C22×C7⋊C3)⋊2C22, SmallGroup(336,130)

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 2A2B2C2D2E2F2G3A3B4A4B6A···6F6G···6N 7 12A12B12C12D14A···14G
order1222222233446···66···671212121214···14
size11112214147714147···714···146141414146···6

38 irreducible representations

dim111111111122666
type++++++++
imageC1C2C2C2C2C3C6C6C6C6D4C3×D4F7C2×F7Dic7⋊C6
kernelC2×Dic7⋊C6C2×C7⋊C12Dic7⋊C6C22×F7C23×C7⋊C3C2×C7⋊D4C2×Dic7C7⋊D4C22×D7C22×C14C2×C7⋊C3C14C23C22C2
# reps114112282224134

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

10000000
01000000
0033600000
0003360000
0000336000
0000033600
0000003360
0000000336
,
3360000000
0336000000
001125336000
000125336000
001124336000
00336212103360
00336212100336
00211882336212213
,
313249000000
6424000000
0000013360
0000010336
0021221310212213
0017915631433600
0018015631433600
0017915731433600
,
2080000000
101129000000
00124336212000
0033600000
00336336213000
0015818123100
00281179235124336336
0015818123010

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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