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## G = C22⋊C4×C7⋊C3order 336 = 24·3·7

### Direct product of C22⋊C4 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C22⋊C4×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C14 — C22×C7⋊C3 — C23×C7⋊C3 — C22⋊C4×C7⋊C3
 Lower central C7 — C14 — C22⋊C4×C7⋊C3
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22⋊C4×C7⋊C3
G = < a,b,c,d,e | a2=b2=c4=d7=e3=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 230 in 68 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C7, C2×C4, C23, C12, C2×C6, C14, C14, C14, C22⋊C4, C7⋊C3, C2×C12, C22×C6, C28, C2×C14, C2×C14, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C2×C7⋊C3, C3×C22⋊C4, C2×C28, C22×C14, C4×C7⋊C3, C22×C7⋊C3, C22×C7⋊C3, C22×C7⋊C3, C7×C22⋊C4, C2×C4×C7⋊C3, C23×C7⋊C3, C22⋊C4×C7⋊C3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C7⋊C3, C2×C12, C3×D4, C2×C7⋊C3, C3×C22⋊C4, C4×C7⋊C3, C22×C7⋊C3, C2×C4×C7⋊C3, D4×C7⋊C3, C22⋊C4×C7⋊C3

Smallest permutation representation of C22⋊C4×C7⋊C3
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 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)
(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)| (1,15)(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)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42), (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( (1,15)(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)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42), (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([[(1,15),(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),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(1,43,15,29),(2,44,16,30),(3,45,17,31),(4,46,18,32),(5,47,19,33),(6,48,20,34),(7,49,21,35),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42)], [(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 3A 3B 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 7A 7B 12A ··· 12H 14A ··· 14F 14G 14H 14I 14J 28A ··· 28H order 1 2 2 2 2 2 3 3 4 4 4 4 6 ··· 6 6 6 6 6 7 7 12 ··· 12 14 ··· 14 14 14 14 14 28 ··· 28 size 1 1 1 1 2 2 7 7 2 2 2 2 7 ··· 7 14 14 14 14 3 3 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 C3 C4 C6 C6 C12 D4 C3×D4 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 D4×C7⋊C3 kernel C22⋊C4×C7⋊C3 C2×C4×C7⋊C3 C23×C7⋊C3 C7×C22⋊C4 C22×C7⋊C3 C2×C28 C22×C14 C2×C14 C2×C7⋊C3 C14 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 1 2 4 4 2 8 2 4 2 4 2 8 4

Matrix representation of C22⋊C4×C7⋊C3 in GL7(𝔽337)

 1 0 0 0 0 0 0 314 336 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
,
 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 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
,
 336 249 0 0 0 0 0 0 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 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 0 212 213 1 0 0 0 0 1 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 124 336 336 0 0 0 0 0 1 0

G:=sub<GL(7,GF(337))| [1,314,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[336,0,0,0,0,0,0,249,1,0,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,212,1,0,0,0,0,0,213,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,124,0,0,0,0,0,0,336,1,0,0,0,0,0,336,0] >;

C22⋊C4×C7⋊C3 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_7\rtimes C_3
% in TeX

G:=Group("C2^2:C4xC7:C3");
// GroupNames label

G:=SmallGroup(336,49);
// by ID

G=gap.SmallGroup(336,49);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,881]);
// Polycyclic

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