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G = Q8×F8  order 448 = 26·7

Direct product of Q8 and F8

direct product, metabelian, soluble, monomial

Aliases: Q8×F8, C4.(C2×F8), (C4×F8).C2, (Q8×C23)⋊C7, C23⋊(C7×Q8), (C23×C4).C14, C2.3(C22×F8), C24.3(C2×C14), (C2×F8).3C22, SmallGroup(448,1364)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — Q8×F8
Chief series
C1 — C23 — C24 — C2×F8 — C4×F8 — Q8×F8
Lower central
C23 — C24 — Q8×F8

Subgroups: 479 in 83 conjugacy classes, 18 normal (9 characteristic)
C1, C2, C2 [×2], C4 [×3], C4 [×3], C22 [×5], C7, C2×C4 [×12], Q8, Q8 [×9], C23, C23 [×2], C14, C22×C4 [×6], C2×Q8 [×16], C24, C28 [×3], C23×C4 [×3], C22×Q8 [×4], C7×Q8, F8, Q8×C23, C2×F8, C4×F8 [×3], Q8×F8

Quotients:
C1, C2 [×3], C22, C7, Q8, C14 [×3], C2×C14, C7×Q8, F8, C2×F8 [×3], C22×F8, Q8×F8

Generators and relations
 G = < a,b,c,d,e,f | a4=c2=d2=e2=f7=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=ed=de, fdf-1=c, fef-1=d >

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

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

(1 15)(4 18)(6 20)(7 21)(8 55)(10 50)(11 51)(12 52)(22 34)(23 35)(24 29)(27 32)(36 48)(37 49)(38 43)(41 46)

(1 15)(2 16)(5 19)(7 21)(9 56)(11 51)(12 52)(13 53)(23 35)(24 29)(25 30)(28 33)(37 49)(38 43)(39 44)(42 47)

(1 15)(2 16)(3 17)(6 20)(10 50)(12 52)(13 53)(14 54)(22 34)(24 29)(25 30)(26 31)(36 48)(38 43)(39 44)(40 45)

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

Matrix representation ►G ⊆ GL9(𝔽29)

15210000000
21140000000
0028000000
0002800000
0000280000
0000028000
0000002800
0000000280
0000000028
,
0280000000
100000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
010000000
000001000
000000010
000000001
001000000
0028282828282828
000100000
000010000
,
100000000
010000000
000000010
000001000
0028282828282828
000100000
000000001
001000000
000000100
,
100000000
010000000
0028282828282828
000000001
000000010
000000100
000001000
000010000
000100000
,
100000000
010000000
0025000000
0000000025
0002500000
0000250000
004444444
0000025000
0000002500

G:=sub<GL(9,GF(29))| [15,21,0,0,0,0,0,0,0,21,14,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28],[0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,28,0,0,0,0,0,0,0,0,28,1,0,0,0,0,0,0,0,28,0,1,0,0,1,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,1,0,28,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,28,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,28,0,0,0,0,0,0,0,0,28,0,0,0,1,0,0,1,0,28,0,0,0,0,0,0,0,0,28,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0,0,0,0,1,0,0,0,28,0,0,0,1,0,0,0,0,28,0,0,1,0,0,0,0,0,28,0,1,0,0,0,0,0,0,28,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,25,0,0,0,4,0,0,0,0,0,0,25,0,4,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,0,4,0,25,0,0,0,0,0,0,4,0,0,0,0,0,25,0,0,4,0,0] >;
 

40 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F7A···7F14A···14F28A···28R
order12224444447···714···1428···28
size11772221414148···88···816···16

40 irreducible representations

dim1111142277
type++--++
imageC1C2C7C14Q8×F8Q8C7×Q8F8C2×F8
kernelQ8×F8C4×F8Q8×C23C23×C4C1F8C23Q8C4
# reps1361811613

In GAP, Magma, Sage, TeX 

Q_8\times F_8
 
% in TeX
 
G:=Group("Q8xF8");
 
// GroupNames label
 
G:=SmallGroup(448,1364);
 
// by ID
 
G=gap.SmallGroup(448,1364);
 
# by ID
 
G:=PCGroup([7,-2,-2,-7,-2,-2,2,2,196,421,204,998,2371,3450]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^2=f^7=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=e*d=d*e,f*d*f^-1=c,f*e*f^-1=d>;
 
// generators/relations
 

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