C2^4.355C2^3 - GroupNames

G = C₂⁴.₃₅₅C₂³ order 128 = 2⁷

195^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₅₅C₂³, C₂³.₅₀₇C₂⁴, C₂².₂₈₇2₊¹⁺⁴, (C₂²×C₄)⋊₅Q₈, C₂³.₆₄(C₂×Q₈), C₂³⋊Q₈.₁₄C₂, C₂.₆(C₂³⋊₂Q₈), (C₂³×C₄).₁₃₄C₂², (C₂×C₄²).₅₉₄C₂², (C₂²×C₄).₅₅₃C₂³, C₂³.₄Q₈.₁₃C₂, C₂².₁₂₈(C₂²×Q₈), C₂³.₃₄D₄.₂₁C₂, (C₂²×Q₈).₁₄₈C₂², C₂³.₆₇C₂³⋊₆₉C₂, C₂³.₆₃C₂³⋊₁₀₈C₂, C₂.₇₁(C₂².₄₅C₂⁴), C₂.C₄².₂₃₆C₂², C₂.₃₉(C₂³.₃₇C₂³), C₂.₃₂(C₂².₅₃C₂⁴), (C₂×C₄).₁₆₆(C₂×Q₈), (C₄×C₂²⋊C₄).₇₁C₂, (C₂×C₄).₁₆₅(C₄○D₄), (C₂×C₄⋊C₄).₃₄₆C₂², C₂².₃₈₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₁₆C₂², SmallGroup(128,1339)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₅₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₃₅₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₅₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₅₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₅₅C₂³

Generators and relations for C₂⁴.₃₅₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=c, g²=db=bd, eae^-1=ab=ba, ac=ca, faf^-1=ad=da, ag=ga, bc=cb, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 420 in 222 conjugacy classes, 100 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₂³.₃₄D₄, C₂³.₆₃C₂³, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₄Q₈, C₂⁴.₃₅₅C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₇C₂³, C₂³⋊₂Q₈, C₂².₄₅C₂⁴, C₂².₅₃C₂⁴, C₂⁴.₃₅₅C₂³

Smallest permutation representation of C₂⁴.₃₅₅C₂³
►On 64 points

Generators in S₆₄

(2 40)(4 38)(5 36)(6 62)(7 34)(8 64)(10 42)(12 44)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(22 50)(24 52)(26 54)(28 56)(29 45)(31 47)(46 58)(48 60)

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	C₄○D₄	2₊¹⁺⁴
kernel	C₂⁴.₃₅₅C₂³	C₄×C₂²⋊C₄	C₂³.₃₄D₄	C₂³.₆₃C₂³	C₂³.₆₇C₂³	C₂³⋊Q₈	C₂³.₄Q₈	C₂²×C₄	C₂×C₄	C₂²
# reps	1	2	1	4	4	2	2	4	16	2

Matrix representation of C₂⁴.₃₅₅C₂³ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	4

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	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

4	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	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	0	2	0	0
0	0	0	0	0	1
0	0	0	0	1	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	4	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

3	0	0	0	0	0
0	3	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	2	0
0	0	0	0	0	3

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3] >;

C₂⁴.₃₅₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{355}C_2^3

% in TeX

G:=Group("C2^4.355C2^3");

// GroupNames label

G:=SmallGroup(128,1339);

// by ID

G=gap.SmallGroup(128,1339);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,184,675,304]);

// Polycyclic

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

// generators/relations

G = C24.355C23 order 128 = 27

195th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₅₅C₂³ order 128 = 2⁷

195^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²