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## G = C4○D4×C7⋊C3order 336 = 24·3·7

### Direct product of C4○D4 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4○D4×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C22×C7⋊C3 — C2×C4×C7⋊C3 — C4○D4×C7⋊C3
 Lower central C7 — C14 — C4○D4×C7⋊C3
 Upper central C1 — C4 — C4○D4

Generators and relations for C4○D4×C7⋊C3
G = < a,b,c,d,e | a4=c2=d7=e3=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 230 in 80 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C7, C2×C4, D4, Q8, C12, C2×C6, C14, C14, C4○D4, C7⋊C3, C2×C12, C3×D4, C3×Q8, C28, C28, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C3×C4○D4, C2×C28, C7×D4, C7×Q8, C4×C7⋊C3, C4×C7⋊C3, C22×C7⋊C3, C7×C4○D4, C2×C4×C7⋊C3, D4×C7⋊C3, Q8×C7⋊C3, C4○D4×C7⋊C3
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C7⋊C3, C22×C6, C2×C7⋊C3, C3×C4○D4, C22×C7⋊C3, C23×C7⋊C3, C4○D4×C7⋊C3

Smallest permutation representation of C4○D4×C7⋊C3
On 56 points
Generators in S56
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 43 22 50)(16 44 23 51)(17 45 24 52)(18 46 25 53)(19 47 26 54)(20 48 27 55)(21 49 28 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 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 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)| (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56), (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)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (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( (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56), (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)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (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([[(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,43,22,50),(16,44,23,51),(17,45,24,52),(18,46,25,53),(19,47,26,54),(20,48,27,55),(21,49,28,56)], [(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),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56)], [(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 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 7A 7B 12A 12B 12C 12D 12E ··· 12J 14A 14B 14C ··· 14H 28A 28B 28C 28D 28E ··· 28J order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 ··· 6 7 7 12 12 12 12 12 ··· 12 14 14 14 ··· 14 28 28 28 28 28 ··· 28 size 1 1 2 2 2 7 7 1 1 2 2 2 7 7 14 ··· 14 3 3 7 7 7 7 14 ··· 14 3 3 6 ··· 6 3 3 3 3 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 C4○D4 C3×C4○D4 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C4○D4×C7⋊C3 kernel C4○D4×C7⋊C3 C2×C4×C7⋊C3 D4×C7⋊C3 Q8×C7⋊C3 C7×C4○D4 C2×C28 C7×D4 C7×Q8 C7⋊C3 C7 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 3 1 2 6 6 2 2 4 2 6 6 2 4

Matrix representation of C4○D4×C7⋊C3 in GL5(𝔽337)

 189 0 0 0 0 0 189 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 189 0 0 0 0 0 148 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 0 1 0 0 0 1 0 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 336 124 1 0 0 0 124 1 0 0 336 125 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 125 1 213 0 0 1 0 0 0 0 1 1 212

G:=sub<GL(5,GF(337))| [189,0,0,0,0,0,189,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[189,0,0,0,0,0,148,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[0,1,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,336,0,0,124,124,125,0,0,1,1,1],[1,0,0,0,0,0,1,0,0,0,0,0,125,1,1,0,0,1,0,1,0,0,213,0,212] >;

C4○D4×C7⋊C3 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_7\rtimes C_3
% in TeX

G:=Group("C4oD4xC7:C3");
// GroupNames label

G:=SmallGroup(336,167);
// by ID

G=gap.SmallGroup(336,167);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,506,455]);
// Polycyclic

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