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G = C23.2F7  order 336 = 24·3·7

The non-split extension by C23 of F7 acting via F7/C7⋊C3=C2

metabelian, supersoluble, monomial

Aliases: C23.2F7, C23.D7⋊C3, (C2×C14)⋊3C12, C22⋊2(C7⋊C12), C14.9(C2×C12), (C2×Dic7)⋊2C6, C14.11(C3×D4), C22.7(C2×F7), (C22×C14).2C6, C2.3(Dic7⋊C6), (C2×C7⋊C12)⋊2C2, C2.5(C2×C7⋊C12), C7⋊2(C3×C22⋊C4), C7⋊C3⋊2(C22⋊C4), (C22×C7⋊C3)⋊1C4, (C2×C7⋊C3).11D4, (C2×C14).6(C2×C6), (C23×C7⋊C3).1C2, (C22×C7⋊C3).6C22, (C2×C7⋊C3).9(C2×C4), SmallGroup(336,22)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C23.2F7
Chief series
C1 — C7 — C14 — C2×C14 — C22×C7⋊C3 — C2×C7⋊C12 — C23.2F7
Lower central
C7 — C14 — C23.2F7
Upper central
C1 — C22 — C23

Generators and relations for C23.2F7
 G = < a,b,c,d,e | a2=b2=c2=d7=1, e6=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 272 in 68 conjugacy classes, 30 normal (14 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, Dic7, C2×C14, C2×C14, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C2×C7⋊C3, C3×C22⋊C4, C2×Dic7, C22×C14, C7⋊C12, C22×C7⋊C3, C22×C7⋊C3, C22×C7⋊C3, C23.D7, C2×C7⋊C12, C23×C7⋊C3, C23.2F7
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, F7, C3×C22⋊C4, C7⋊C12, C2×F7, C2×C7⋊C12, Dic7⋊C6, C23.2F7

Smallest permutation representation of C23.2F7
►On 56 points
Generators in S56
(1 6)(2 4)(3 8)(5 7)(9 15)(10 28)(11 17)(12 30)(13 19)(14 32)(16 22)(18 24)(20 26)(21 27)(23 29)(25 31)(33 52)(34 40)(35 54)(36 42)(37 56)(38 44)(39 46)(41 48)(43 50)(45 51)(47 53)(49 55)

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D6A···6F6G6H6I6J 7 12A···12H14A···14G
order1222223344446···66666712···1214···14
size11112277141414147···714141414614···146···6

38 irreducible representations

dim11111111226666
type+++++-+
imageC1C2C2C3C4C6C6C12D4C3×D4F7C7⋊C12C2×F7Dic7⋊C6
kernelC23.2F7C2×C7⋊C12C23×C7⋊C3C23.D7C22×C7⋊C3C2×Dic7C22×C14C2×C14C2×C7⋊C3C14C23C22C22C2
# reps12124428241214

Matrix representation of C23.2F7 ►in GL8(𝔽337)

10000000
119336000000
0033600000
0003360000
0000336000
00000100
00000010
00000001
,
10000000
01000000
0033600000
0003360000
0000336000
0000033600
0000003360
0000000336
,
3360000000
0336000000
0033600000
0003360000
0000336000
0000033600
0000003360
0000000336
,
10000000
01000000
001241251000
00100000
00010000
00000010
00000001
000001213212
,
209308000000
0128000000
00000100
00000001
00000212336336
0033600000
0000336000
0012511000

G:=sub<GL(8,GF(337))| [1,119,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,124,1,0,0,0,0,0,0,125,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,213,0,0,0,0,0,0,1,212],[209,0,0,0,0,0,0,0,308,128,0,0,0,0,0,0,0,0,0,0,0,336,0,125,0,0,0,0,0,0,0,1,0,0,0,0,0,0,336,1,0,0,1,0,212,0,0,0,0,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0] >;
 

C23.2F7 in GAP, Magma, Sage, TeX 

C_2^3._2F_7
 
% in TeX
 
G:=Group("C2^3.2F7");
 
// GroupNames label
 
G:=SmallGroup(336,22);
 
// by ID
 
G=gap.SmallGroup(336,22);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,10373,1745]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^7=1,e^6=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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^5>;
 
// generators/relations
 

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