C2^3.17C4^2 - GroupNames

G = C₂³.₁₇C₄² order 128 = 2⁷

12^nd non-split extension by C₂³ of C₄² acting via C₄²/C₂²=C₂²

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

Aliases: C₂³.₁₇C₄², C₂²⋊C₈⋊₁₃C₄, C₈⋊₆(C₂²⋊C₄), (C₂×C₈).₃₈₂D₄, C₄.₁₆₀(C₄×D₄), (C₂×C₄)⋊₇M₄(2), C₂.₁(C₈⋊₆D₄), C₂.₂(C₈⋊₉D₄), (C₂×C₄).₂₄C₄², C₂⁴.₅₀(C₂×C₄), C₂².₈₅(C₄×D₄), (C₂×M₄(2))⋊₁₈C₄, C₂.₁₁(C₄×M₄(2)), C₂².₂₄(C₈○D₄), C₂².₅₂(C₂×C₄²), C₄.₇₂(C₄²⋊C₂), C₂.C₄².₁₄C₄, (C₂³×C₄).₂₂₅C₂², C₂³.₂₅₅(C₂²×C₄), (C₂×C₄²).₉₉₁C₂², (C₂²×C₈).₄₇₁C₂², C₂².₃₉(C₂×M₄(2)), C₂.₁₄(C₈○₂M₄(2)), (C₂²×C₄).₁₆₀₈C₂³, C₂².₇C₄²⋊₃₈C₂, (C₂²×M₄(2)).₁₈C₂, (C₂×C₄×C₈)⋊₃₇C₂, (C₂×C₈⋊C₄)⋊₂₀C₂, (C₂×C₈).₁₃₂(C₂×C₄), C₂.₁₁(C₄×C₂²⋊C₄), (C₂×C₄).₁₄₉₈(C₂×D₄), (C₄×C₂²⋊C₄).₁₁C₂, (C₂×C₂²⋊C₈).₄₃C₂, (C₂×C₂²⋊C₄).₂₄C₄, C₄.₁₀₈(C₂×C₂²⋊C₄), (C₂×C₄).₉₁₈(C₄○D₄), (C₂×C₄).₅₉₈(C₂²×C₄), (C₂²×C₄).₁₀₈(C₂×C₄), SmallGroup(128,485)

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

Derived series

C₁ — C₂² — C₂³.₁₇C₄²

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₂×C₄×C₈ — C₂³.₁₇C₄²

Lower central

C₁ — C₂² — C₂³.₁₇C₄²

Upper central

C₁ — C₂²×C₄ — C₂³.₁₇C₄²

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³.₁₇C₄²

Generators and relations for C₂³.₁₇C₄²
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e⁴=c, dad^-1=ab=ba, eae^-1=ac=ca, bc=cb, ede^-1=bd=db, be=eb, cd=dc, ce=ec >

Subgroups: 276 in 174 conjugacy classes, 88 normal (36 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₂².₇C₄², C₄×C₂²⋊C₄, C₂×C₄×C₈, C₂×C₈⋊C₄, C₂×C₂²⋊C₈, C₂²×M₄(2), C₂³.₁₇C₄²
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₄², C₂²⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, C₂×C₄², C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂×M₄(2), C₈○D₄, C₄×C₂²⋊C₄, C₄×M₄(2), C₈○₂M₄(2), C₈⋊₉D₄, C₈⋊₆D₄, C₂³.₁₇C₄²

Smallest permutation representation of C₂³.₁₇C₄²
►On 64 points

Generators in S₆₄

(2 6)(4 8)(10 14)(12 16)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(26 30)(28 32)(33 52)(34 49)(35 54)(36 51)(37 56)(38 53)(39 50)(40 55)(41 45)(43 47)

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

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

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

(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)

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4P	4Q	···	4V	8A	···	8P	8Q	···	8X
order	1	2	···	2	2	2	4	···	4	4	···	4	4	···	4	8	···	8	8	···	8
size	1	1	···	1	4	4	1	···	1	2	···	2	4	···	4	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	M₄(2)	C₄○D₄	C₈○D₄
kernel	C₂³.₁₇C₄²	C₂².₇C₄²	C₄×C₂²⋊C₄	C₂×C₄×C₈	C₂×C₈⋊C₄	C₂×C₂²⋊C₈	C₂²×M₄(2)	C₂.C₄²	C₂²⋊C₈	C₂×C₂²⋊C₄	C₂×M₄(2)	C₂×C₈	C₂×C₄	C₂×C₄	C₂²
# reps	1	2	1	1	1	1	1	4	8	4	8	4	8	4	8

Matrix representation of C₂³.₁₇C₄² ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
4	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	1	0
0	0	0	0	0	16

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

1	0	0	0	0	0
0	1	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

4	15	0	0	0	0
0	13	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	16	0
0	0	0	0	0	16

4	0	0	0	0	0
16	13	0	0	0	0
0	0	4	0	0	0
0	0	0	13	0	0
0	0	0	0	0	16
0	0	0	0	4	0

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

C₂³.₁₇C₄² in GAP, Magma, Sage, TeX

C_2^3._{17}C_4^2

% in TeX

G:=Group("C2^3.17C4^2");

// GroupNames label

G:=SmallGroup(128,485);

// by ID

G=gap.SmallGroup(128,485);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,100,172]);

// Polycyclic

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

// generators/relations

G = C23.17C42 order 128 = 27

12nd non-split extension by C23 of C42 acting via C42/C22=C22

G = C₂³.₁₇C₄² order 128 = 2⁷

12^nd non-split extension by C₂³ of C₄² acting via C₄²/C₂²=C₂²