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## G = C2×D4×C7⋊C3order 336 = 24·3·7

### Direct product of C2, D4 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×D4×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C22×C7⋊C3 — C23×C7⋊C3 — C2×D4×C7⋊C3
 Lower central C7 — C14 — C2×D4×C7⋊C3
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×C7⋊C3
G = < a,b,c,d,e | a2=b4=c2=d7=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 350 in 108 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C7, C2×C4, D4, C23, C12, C2×C6, C14, C14, C14, C2×D4, C7⋊C3, C2×C12, C3×D4, C22×C6, C28, C2×C14, C2×C14, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C2×C7⋊C3, C6×D4, C2×C28, C7×D4, C22×C14, C4×C7⋊C3, C22×C7⋊C3, C22×C7⋊C3, C22×C7⋊C3, D4×C14, C2×C4×C7⋊C3, D4×C7⋊C3, C23×C7⋊C3, C2×D4×C7⋊C3
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C7⋊C3, C3×D4, C22×C6, C2×C7⋊C3, C6×D4, C22×C7⋊C3, D4×C7⋊C3, C23×C7⋊C3, C2×D4×C7⋊C3

Smallest permutation representation of C2×D4×C7⋊C3
On 56 points
Generators in S56
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
(1 50 8 43)(2 51 9 44)(3 52 10 45)(4 53 11 46)(5 54 12 47)(6 55 13 48)(7 56 14 49)(15 36 22 29)(16 37 23 30)(17 38 24 31)(18 39 25 32)(19 40 26 33)(20 41 27 34)(21 42 28 35)
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 7A 7B 12A 12B 12C 12D 14A ··· 14F 14G ··· 14N 28A 28B 28C 28D order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 7 7 12 12 12 12 14 ··· 14 14 ··· 14 28 28 28 28 size 1 1 1 1 2 2 2 2 7 7 2 2 7 ··· 7 14 ··· 14 3 3 14 14 14 14 3 ··· 3 6 ··· 6 6 6 6 6

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 6 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 D4×C7⋊C3 kernel C2×D4×C7⋊C3 C2×C4×C7⋊C3 D4×C7⋊C3 C23×C7⋊C3 D4×C14 C2×C28 C7×D4 C22×C14 C2×C7⋊C3 C14 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 4 2 2 2 8 4 2 4 2 2 8 4 4

Matrix representation of C2×D4×C7⋊C3 in GL7(𝔽337)

 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 336 2 0 0 0 0 0 336 1 0 0 0 0 0 0 0 0 336 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 336 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 125 0 0 0 0 0 1 124
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 124 212 0 0 0 0 0 336 213 1 0 0 0 0 336 1 0

G:=sub<GL(7,GF(337))| [336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[336,336,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,125,124],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,124,336,336,0,0,0,0,212,213,1,0,0,0,0,0,1,0] >;

C2×D4×C7⋊C3 in GAP, Magma, Sage, TeX

C_2\times D_4\times C_7\rtimes C_3
% in TeX

G:=Group("C2xD4xC7:C3");
// GroupNames label

G:=SmallGroup(336,165);
// by ID

G=gap.SmallGroup(336,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,260,455]);
// Polycyclic

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

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