C4^2.698C2^3 - GroupNames

G = C₄².₆₉₈C₂³ order 128 = 2⁷

113^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂²=C₂

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₄².₆₉₈C₂³, C₄.₁₂₄2_-¹⁺⁴, C₄.₁₇₇2₊¹⁺⁴, C₈⋊₆D₄⋊₄₅C₂, C₈⋊₉D₄⋊₄₆C₂, C₈⋊₄Q₈⋊₄₄C₂, (C₄×D₄).₃₇C₄, (C₂×C₄)⋊₅M₄(2), (C₄×Q₈).₃₄C₄, C₄⋊C₈.₂₄₀C₂², (C₂×C₄).₆₈₆C₂⁴, (C₂×C₈).₄₄₅C₂³, (C₄×C₈).₃₄₃C₂², C₄².₂₃₄(C₂×C₄), C₄.₁₇(C₂×M₄(2)), C₄⋊M₄(2)⋊₃₆C₂, C₄².₆C₄⋊₅₄C₂, C₄²⋊C₂.₃₅C₄, (C₄×D₄).₃₀₄C₂², (C₄×Q₈).₂₈₅C₂², C₈⋊C₄.₁₀₄C₂², C₄².₁₂C₄⋊₅₆C₂, C₂².₅(C₂×M₄(2)), C₂²⋊C₈.₁₄₈C₂², C₂.₃₆(Q₈○M₄(2)), (C₂²×C₈).₄₅₂C₂², C₂³.₁₅₂(C₂²×C₄), C₂².₂₀₈(C₂³×C₄), (C₂×C₄²).₇₉₃C₂², C₂.₂₃(C₂²×M₄(2)), (C₂²×C₄).₁₂₈₇C₂³, (C₂×M₄(2)).₂₄₉C₂², C₂.₄₄(C₂³.₃₃C₂³), (C₂×C₄⋊C₈)⋊₅₀C₂, C₄⋊C₄.₂₃₃(C₂×C₄), (C₂×C₄○D₄).₂₉C₄, (C₄×C₄○D₄).₁₉C₂, (C₂×D₄).₂₃₇(C₂×C₄), C₂²⋊C₄.₇₉(C₂×C₄), (C₂×Q₈).₂₁₃(C₂×C₄), (C₂²×C₄).₃₆₄(C₂×C₄), (C₂×C₄).₂₈₀(C₂²×C₄), SmallGroup(128,1721)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂² — C₄².₆₉₈C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₂×C₄² — C₄×C₄○D₄ — C₄².₆₉₈C₂³

Lower central

C₁ — C₂² — C₄².₆₉₈C₂³

Upper central

C₁ — C₂×C₄ — C₄².₆₉₈C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₆₉₈C₂³

Generators and relations for C₄².₆₉₈C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=e²=1, c²=b, ab=ba, cac^-1=a^-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a²c, ece=b²c, ede=a²d >

Subgroups: 276 in 194 conjugacy classes, 134 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₄⋊C₈, C₂×C₄², C₂×C₄², C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₂²×C₈, C₂×M₄(2), C₂×C₄○D₄, C₂×C₄⋊C₈, C₄⋊M₄(2), C₄².₁₂C₄, C₄².₆C₄, C₈⋊₉D₄, C₈⋊₆D₄, C₈⋊₄Q₈, C₄×C₄○D₄, C₄².₆₉₈C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, M₄(2), C₂²×C₄, C₂⁴, C₂×M₄(2), C₂³×C₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₃C₂³, C₂²×M₄(2), Q₈○M₄(2), C₄².₆₉₈C₂³

Smallest permutation representation of C₄².₆₉₈C₂³
►On 64 points

Generators in S₆₄

(1 31 55 17)(2 18 56 32)(3 25 49 19)(4 20 50 26)(5 27 51 21)(6 22 52 28)(7 29 53 23)(8 24 54 30)(9 46 64 36)(10 37 57 47)(11 48 58 38)(12 39 59 41)(13 42 60 40)(14 33 61 43)(15 44 62 34)(16 35 63 45)

(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)

(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)

(1 16)(2 64)(3 10)(4 58)(5 12)(6 60)(7 14)(8 62)(9 56)(11 50)(13 52)(15 54)(17 45)(18 36)(19 47)(20 38)(21 41)(22 40)(23 43)(24 34)(25 37)(26 48)(27 39)(28 42)(29 33)(30 44)(31 35)(32 46)(49 57)(51 59)(53 61)(55 63)

(2 6)(4 8)(9 60)(10 57)(11 62)(12 59)(13 64)(14 61)(15 58)(16 63)(18 22)(20 24)(26 30)(28 32)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)(50 54)(52 56)

G:=sub<Sym(64)| (1,31,55,17)(2,18,56,32)(3,25,49,19)(4,20,50,26)(5,27,51,21)(6,22,52,28)(7,29,53,23)(8,24,54,30)(9,46,64,36)(10,37,57,47)(11,48,58,38)(12,39,59,41)(13,42,60,40)(14,33,61,43)(15,44,62,34)(16,35,63,45), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,16)(2,64)(3,10)(4,58)(5,12)(6,60)(7,14)(8,62)(9,56)(11,50)(13,52)(15,54)(17,45)(18,36)(19,47)(20,38)(21,41)(22,40)(23,43)(24,34)(25,37)(26,48)(27,39)(28,42)(29,33)(30,44)(31,35)(32,46)(49,57)(51,59)(53,61)(55,63), (2,6)(4,8)(9,60)(10,57)(11,62)(12,59)(13,64)(14,61)(15,58)(16,63)(18,22)(20,24)(26,30)(28,32)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46)(50,54)(52,56)>;

G:=Group( (1,31,55,17)(2,18,56,32)(3,25,49,19)(4,20,50,26)(5,27,51,21)(6,22,52,28)(7,29,53,23)(8,24,54,30)(9,46,64,36)(10,37,57,47)(11,48,58,38)(12,39,59,41)(13,42,60,40)(14,33,61,43)(15,44,62,34)(16,35,63,45), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,16)(2,64)(3,10)(4,58)(5,12)(6,60)(7,14)(8,62)(9,56)(11,50)(13,52)(15,54)(17,45)(18,36)(19,47)(20,38)(21,41)(22,40)(23,43)(24,34)(25,37)(26,48)(27,39)(28,42)(29,33)(30,44)(31,35)(32,46)(49,57)(51,59)(53,61)(55,63), (2,6)(4,8)(9,60)(10,57)(11,62)(12,59)(13,64)(14,61)(15,58)(16,63)(18,22)(20,24)(26,30)(28,32)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46)(50,54)(52,56) );

G=PermutationGroup([[(1,31,55,17),(2,18,56,32),(3,25,49,19),(4,20,50,26),(5,27,51,21),(6,22,52,28),(7,29,53,23),(8,24,54,30),(9,46,64,36),(10,37,57,47),(11,48,58,38),(12,39,59,41),(13,42,60,40),(14,33,61,43),(15,44,62,34),(16,35,63,45)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,16),(2,64),(3,10),(4,58),(5,12),(6,60),(7,14),(8,62),(9,56),(11,50),(13,52),(15,54),(17,45),(18,36),(19,47),(20,38),(21,41),(22,40),(23,43),(24,34),(25,37),(26,48),(27,39),(28,42),(29,33),(30,44),(31,35),(32,46),(49,57),(51,59),(53,61),(55,63)], [(2,6),(4,8),(9,60),(10,57),(11,62),(12,59),(13,64),(14,61),(15,58),(16,63),(18,22),(20,24),(26,30),(28,32),(33,43),(34,48),(35,45),(36,42),(37,47),(38,44),(39,41),(40,46),(50,54),(52,56)]])

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	···	4N	4O	···	4T	8A	···	8P
order	1	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	···	4	8	···	8
size	1	1	1	1	2	2	4	4	1	1	1	1	2	···	2	4	···	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	2	4	4	4
type	+	+	+	+	+	+	+	+	+						+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	M₄(2)	2₊¹⁺⁴	2_-¹⁺⁴	Q₈○M₄(2)
kernel	C₄².₆₉₈C₂³	C₂×C₄⋊C₈	C₄⋊M₄(2)	C₄².₁₂C₄	C₄².₆C₄	C₈⋊₉D₄	C₈⋊₆D₄	C₈⋊₄Q₈	C₄×C₄○D₄	C₄²⋊C₂	C₄×D₄	C₄×Q₈	C₂×C₄○D₄	C₂×C₄	C₄	C₄	C₂
# reps	1	1	2	1	2	4	2	2	1	6	6	2	2	8	1	1	2

Matrix representation of C₄².₆₉₈C₂³ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0

13	0	0	0	0	0
0	13	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

14	15	0	0	0	0
15	3	0	0	0	0
0	0	0	0	2	14
0	0	0	0	14	15
0	0	15	3	0	0
0	0	3	2	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

1	0	0	0	0	0
14	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	0	0	0	16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[14,15,0,0,0,0,15,3,0,0,0,0,0,0,0,0,15,3,0,0,0,0,3,2,0,0,2,14,0,0,0,0,14,15,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,14,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C₄².₆₉₈C₂³ in GAP, Magma, Sage, TeX

C_4^2._{698}C_2^3

% in TeX

G:=Group("C4^2.698C2^3");

// GroupNames label

G:=SmallGroup(128,1721);

// by ID

G=gap.SmallGroup(128,1721);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,80,124]);

// Polycyclic

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

// generators/relations

G = C42.698C23 order 128 = 27

113rd non-split extension by C42 of C23 acting via C23/C22=C2

G = C₄².₆₉₈C₂³ order 128 = 2⁷

113^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂²=C₂