C2^4.285C2^3 - GroupNames

G = C₂⁴.₂₈₅C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₈₅C₂³, C₂³.₃₆₅C₂⁴, C₂².₁₇₁2₊¹⁺⁴, C₂².₁₂₆2_-¹⁺⁴, C₂.₁₇(D₄×Q₈), C₄⋊C₄.₃₃₁D₄, C₂²⋊C₄⋊₁₄Q₈, C₂³.₁₅(C₂×Q₈), C₂.₄₈(D₄⋊₅D₄), C₂³⋊Q₈.₄C₂, C₂.₁₆(D₄⋊₃Q₈), C₂³.Q₈.₄C₂, C₂².₈₂(C₂²×Q₈), (C₂²×C₄).₈₁₈C₂³, (C₂³×C₄).₃₅₈C₂², (C₂×C₄²).₅₀₈C₂², C₂².₂₄₅(C₂²×D₄), C₂³.₈Q₈.₁₅C₂, (C₂²×Q₈).₁₁₁C₂², C₂³.₆₅C₂³⋊₅₉C₂, C₂³.₇₈C₂³⋊₁₁C₂, C₂³.₆₇C₂³⋊₄₅C₂, C₂³.₆₃C₂³⋊₄₄C₂, C₂.₃₇(C₂².₁₉C₂⁴), C₂⁴.C₂².₁₄C₂, C₂.C₄².₁₂₂C₂², C₂.₁₈(C₂³.₃₇C₂³), C₂.₁₅(C₂².₅₀C₂⁴), C₂.₂₂(C₂².₃₆C₂⁴), (C₂×C₄×Q₈)⋊₁₉C₂, (C₂×C₄).₃₅(C₂×Q₈), (C₂×C₄).₃₄₀(C₂×D₄), (C₄×C₂²⋊C₄).₄₃C₂, (C₂×C₂²⋊Q₈).₂₈C₂, (C₂×C₄).₁₁₅(C₄○D₄), (C₂×C₄⋊C₄).₂₄₅C₂², C₂².₂₄₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₉₇C₂², SmallGroup(128,1197)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₈₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₂₈₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₈₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₈₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₈₅C₂³

Generators and relations for C₂⁴.₂₈₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=e²=a, g²=ba=ab, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 452 in 254 conjugacy classes, 108 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×Q₈, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂×C₄×Q₈, C₂×C₂²⋊Q₈, C₂⁴.₂₈₅C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂³.₃₇C₂³, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄×Q₈, D₄⋊₃Q₈, C₂².₅₀C₂⁴, C₂⁴.₂₈₅C₂³

Smallest permutation representation of C₂⁴.₂₈₅C₂³
►On 64 points

Generators in S₆₄

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

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

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

(2 42)(4 44)(5 62)(6 51)(7 64)(8 49)(10 58)(12 60)(14 26)(16 28)(17 63)(18 52)(19 61)(20 50)(21 40)(22 56)(23 38)(24 54)(30 46)(32 48)(33 39)(34 55)(35 37)(36 53)

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

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

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

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

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	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₂₈₅C₂³	C₄×C₂²⋊C₄	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³⋊Q₈	C₂³.₇₈C₂³	C₂³.Q₈	C₂×C₄×Q₈	C₂×C₂²⋊Q₈	C₂²⋊C₄	C₄⋊C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	1	3	2	1	1	1	2	1	1	1	4	4	12	1	1

Matrix representation of C₂⁴.₂₈₅C₂³ ►in GL₆(𝔽₅)

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

4	0	0	0	0	0
0	4	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	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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	2	0
0	0	0	0	0	3

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

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	1

2	0	0	0	0	0
0	2	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	0	1
0	0	0	0	4	0

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

C₂⁴.₂₈₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{285}C_2^3

% in TeX

G:=Group("C2^4.285C2^3");

// GroupNames label

G:=SmallGroup(128,1197);

// by ID

G=gap.SmallGroup(128,1197);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,675,192]);

// Polycyclic

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

// generators/relations

G = C24.285C23 order 128 = 27

125th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₈₅C₂³ order 128 = 2⁷

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