C7.F8 - GroupNames

G = C₇.F₈ order 392 = 2³·7²

The central extension by C₇ of F₈

metabelian, soluble, monomial, A-group

Aliases: C₇.F₈, C₂³⋊C₄₉, (C₂²×C₁₄).C₇, SmallGroup(392,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₇.F₈

Chief series

C₁ — C₂³ — C₂²×C₁₄ — C₇.F₈

Lower central

C₂³ — C₇.F₈

Upper central

C₁ — C₇

Generators and relations for C₇.F₈
G = < a,b,c,d,e | a⁷=b²=c²=d²=1, e⁷=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=dc=cd, ece^-1=b, ede^-1=c >

C₁

₇C₂

C₇

₇C₂²

₇C₁₄

₈C₄₉

C₂³

₇C₂×C₁₄

C₂²×C₁₄

C₇.F₈

Smallest permutation representation of C₇.F₈
►On 98 points

Generators in S₉₈

(1 8 15 22 29 36 43)(2 9 16 23 30 37 44)(3 10 17 24 31 38 45)(4 11 18 25 32 39 46)(5 12 19 26 33 40 47)(6 13 20 27 34 41 48)(7 14 21 28 35 42 49)(50 57 64 71 78 85 92)(51 58 65 72 79 86 93)(52 59 66 73 80 87 94)(53 60 67 74 81 88 95)(54 61 68 75 82 89 96)(55 62 69 76 83 90 97)(56 63 70 77 84 91 98)

(2 67)(4 69)(5 70)(6 71)(9 74)(11 76)(12 77)(13 78)(16 81)(18 83)(19 84)(20 85)(23 88)(25 90)(26 91)(27 92)(30 95)(32 97)(33 98)(34 50)(37 53)(39 55)(40 56)(41 57)(44 60)(46 62)(47 63)(48 64)

(3 68)(5 70)(6 71)(7 72)(10 75)(12 77)(13 78)(14 79)(17 82)(19 84)(20 85)(21 86)(24 89)(26 91)(27 92)(28 93)(31 96)(33 98)(34 50)(35 51)(38 54)(40 56)(41 57)(42 58)(45 61)(47 63)(48 64)(49 65)

(1 66)(4 69)(6 71)(7 72)(8 73)(11 76)(13 78)(14 79)(15 80)(18 83)(20 85)(21 86)(22 87)(25 90)(27 92)(28 93)(29 94)(32 97)(34 50)(35 51)(36 52)(39 55)(41 57)(42 58)(43 59)(46 62)(48 64)(49 65)

(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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98)

G:=sub<Sym(98)| (1,8,15,22,29,36,43)(2,9,16,23,30,37,44)(3,10,17,24,31,38,45)(4,11,18,25,32,39,46)(5,12,19,26,33,40,47)(6,13,20,27,34,41,48)(7,14,21,28,35,42,49)(50,57,64,71,78,85,92)(51,58,65,72,79,86,93)(52,59,66,73,80,87,94)(53,60,67,74,81,88,95)(54,61,68,75,82,89,96)(55,62,69,76,83,90,97)(56,63,70,77,84,91,98), (2,67)(4,69)(5,70)(6,71)(9,74)(11,76)(12,77)(13,78)(16,81)(18,83)(19,84)(20,85)(23,88)(25,90)(26,91)(27,92)(30,95)(32,97)(33,98)(34,50)(37,53)(39,55)(40,56)(41,57)(44,60)(46,62)(47,63)(48,64), (3,68)(5,70)(6,71)(7,72)(10,75)(12,77)(13,78)(14,79)(17,82)(19,84)(20,85)(21,86)(24,89)(26,91)(27,92)(28,93)(31,96)(33,98)(34,50)(35,51)(38,54)(40,56)(41,57)(42,58)(45,61)(47,63)(48,64)(49,65), (1,66)(4,69)(6,71)(7,72)(8,73)(11,76)(13,78)(14,79)(15,80)(18,83)(20,85)(21,86)(22,87)(25,90)(27,92)(28,93)(29,94)(32,97)(34,50)(35,51)(36,52)(39,55)(41,57)(42,58)(43,59)(46,62)(48,64)(49,65), (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)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98)>;

G:=Group( (1,8,15,22,29,36,43)(2,9,16,23,30,37,44)(3,10,17,24,31,38,45)(4,11,18,25,32,39,46)(5,12,19,26,33,40,47)(6,13,20,27,34,41,48)(7,14,21,28,35,42,49)(50,57,64,71,78,85,92)(51,58,65,72,79,86,93)(52,59,66,73,80,87,94)(53,60,67,74,81,88,95)(54,61,68,75,82,89,96)(55,62,69,76,83,90,97)(56,63,70,77,84,91,98), (2,67)(4,69)(5,70)(6,71)(9,74)(11,76)(12,77)(13,78)(16,81)(18,83)(19,84)(20,85)(23,88)(25,90)(26,91)(27,92)(30,95)(32,97)(33,98)(34,50)(37,53)(39,55)(40,56)(41,57)(44,60)(46,62)(47,63)(48,64), (3,68)(5,70)(6,71)(7,72)(10,75)(12,77)(13,78)(14,79)(17,82)(19,84)(20,85)(21,86)(24,89)(26,91)(27,92)(28,93)(31,96)(33,98)(34,50)(35,51)(38,54)(40,56)(41,57)(42,58)(45,61)(47,63)(48,64)(49,65), (1,66)(4,69)(6,71)(7,72)(8,73)(11,76)(13,78)(14,79)(15,80)(18,83)(20,85)(21,86)(22,87)(25,90)(27,92)(28,93)(29,94)(32,97)(34,50)(35,51)(36,52)(39,55)(41,57)(42,58)(43,59)(46,62)(48,64)(49,65), (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)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98) );

G=PermutationGroup([[(1,8,15,22,29,36,43),(2,9,16,23,30,37,44),(3,10,17,24,31,38,45),(4,11,18,25,32,39,46),(5,12,19,26,33,40,47),(6,13,20,27,34,41,48),(7,14,21,28,35,42,49),(50,57,64,71,78,85,92),(51,58,65,72,79,86,93),(52,59,66,73,80,87,94),(53,60,67,74,81,88,95),(54,61,68,75,82,89,96),(55,62,69,76,83,90,97),(56,63,70,77,84,91,98)], [(2,67),(4,69),(5,70),(6,71),(9,74),(11,76),(12,77),(13,78),(16,81),(18,83),(19,84),(20,85),(23,88),(25,90),(26,91),(27,92),(30,95),(32,97),(33,98),(34,50),(37,53),(39,55),(40,56),(41,57),(44,60),(46,62),(47,63),(48,64)], [(3,68),(5,70),(6,71),(7,72),(10,75),(12,77),(13,78),(14,79),(17,82),(19,84),(20,85),(21,86),(24,89),(26,91),(27,92),(28,93),(31,96),(33,98),(34,50),(35,51),(38,54),(40,56),(41,57),(42,58),(45,61),(47,63),(48,64),(49,65)], [(1,66),(4,69),(6,71),(7,72),(8,73),(11,76),(13,78),(14,79),(15,80),(18,83),(20,85),(21,86),(22,87),(25,90),(27,92),(28,93),(29,94),(32,97),(34,50),(35,51),(36,52),(39,55),(41,57),(42,58),(43,59),(46,62),(48,64),(49,65)], [(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),(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98)]])

56 conjugacy classes

class	1	2	7A	···	7F	14A	···	14F	49A	···	49AP
order	1	2	7	···	7	14	···	14	49	···	49
size	1	7	1	···	1	7	···	7	8	···	8

56 irreducible representations

dim	1	1	1	7	7
type	+			+
image	C₁	C₇	C₄₉	F₈	C₇.F₈
kernel	C₇.F₈	C₂²×C₁₄	C₂³	C₇	C₁
# reps	1	6	42	1	6

Matrix representation of C₇.F₈ ►in GL₇(𝔽₁₉₇)

114	0	0	0	0	0	0
0	114	0	0	0	0	0
0	0	114	0	0	0	0
0	0	0	114	0	0	0
0	0	0	0	114	0	0
0	0	0	0	0	114	0
0	0	0	0	0	0	114

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	196	0	0	0	0
0	0	0	196	0	0	0
0	0	0	0	196	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	196

1	0	0	0	0	0	0
0	196	0	0	0	0	0
0	0	196	0	0	0	0
0	0	0	196	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	196	0
0	0	0	0	0	0	1

196	0	0	0	0	0	0
0	196	0	0	0	0	0
0	0	196	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	196	0	0
0	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	0	1
114	0	0	0	0	0	0

G:=sub<GL(7,GF(197))| [114,0,0,0,0,0,0,0,114,0,0,0,0,0,0,0,114,0,0,0,0,0,0,0,114,0,0,0,0,0,0,0,114,0,0,0,0,0,0,0,114,0,0,0,0,0,0,0,114],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,196],[1,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,1],[196,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,196,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,114,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C₇.F₈ in GAP, Magma, Sage, TeX

C_7.F_8

% in TeX

G:=Group("C7.F8");

// GroupNames label

G:=SmallGroup(392,11);

// by ID

G=gap.SmallGroup(392,11);

# by ID

G:=PCGroup([5,-7,-7,-2,2,2,35,1472,3923,6129]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₇.F₈ in TeX

114	0	0	0	0	0	0
0	114	0	0	0	0	0
0	0	114	0	0	0	0
0	0	0	114	0	0	0
0	0	0	0	114	0	0
0	0	0	0	0	114	0
0	0	0	0	0	0	114

114	0	0	0	0	0	0
0	114	0	0	0	0	0
0	0	114	0	0	0	0
0	0	0	114	0	0	0
0	0	0	0	114	0	0
0	0	0	0	0	114	0
0	0	0	0	0	0	114

G = C7.F8 order 392 = 23·72

The central extension by C7 of F8

G = C₇.F₈ order 392 = 2³·7²

The central extension by C₇ of F₈

114	0	0	0	0	0	0
0	114	0	0	0	0	0
0	0	114	0	0	0	0
0	0	0	114	0	0	0
0	0	0	0	114	0	0
0	0	0	0	0	114	0
0	0	0	0	0	0	114