C2^3.146C2^4 - GroupNames

G = C₂³.₁₄₆C₂⁴ order 128 = 2⁷

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

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

Aliases: C₂³.₁₄₆C₂⁴, C₄².₁₀₃C₂³, C₂².₁₁₂C₂⁵, C₄.₄₃2_-¹⁺⁴, C₂².₈2_-¹⁺⁴, (D₄×Q₈)⋊₂₅C₂, Q₈ ○₃(C₄⋊D₄), Q₈⋊₃Q₈⋊₂₃C₂, C₄⋊C₄.₃₀₄C₂³, (C₂×C₄).₁₀₂C₂⁴, Q₈.₄₃(C₄○D₄), C₄⋊Q₈.₂₂₃C₂², (C₂×D₄).₄₈₄C₂³, (C₄×D₄).₂₄₃C₂², C₂²⋊C₄.₃₅C₂³, (C₂×Q₈).₄₆₀C₂³, (C₄×Q₈).₂₃₀C₂², C₄⋊D₄.₂₄₃C₂², C₄²⋊₂C₂.₄C₂², (C₂×C₄²).₉₅₉C₂², C₂²⋊Q₈.₂₃₃C₂², C₂.₃₄(C₂×2_-¹⁺⁴), (C₂²×C₄).₁₂₁₂C₂³, C₄.₄D₄.₁₇₇C₂², C₄².C₂.₁₅₈C₂², (C₂²×Q₈).₃₆₅C₂², C₂².₄₆C₂⁴⋊₂₅C₂, C₄²⋊C₂.₂₃₈C₂², C₂².₅₀C₂⁴⋊₃₀C₂, C₂².₃₅C₂⁴⋊₁₅C₂, C₂³.₃₂C₂³⋊₁₈C₂, C₂³.₃₇C₂³⋊₄₇C₂, C₂².D₄.₃₄C₂², C₂³.₃₆C₂³.₃₂C₂, (C₂×C₄×Q₈)⋊₆₃C₂, C₄.₂₈₅(C₂×C₄○D₄), C₂.₆₈(C₂²×C₄○D₄), (C₂×C₄⋊C₄).₉₆₅C₂², SmallGroup(128,2255)

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

Derived series

C₁ — C₂² — C₂³.₁₄₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₂×C₄² — C₂×C₄×Q₈ — C₂³.₁₄₆C₂⁴

Lower central

C₁ — C₂² — C₂³.₁₄₆C₂⁴

Upper central

C₁ — C₂² — C₂³.₁₄₆C₂⁴

Jennings

C₁ — C₂² — C₂³.₁₄₆C₂⁴

Generators and relations for C₂³.₁₄₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=e²=f²=b, g²=cb=bc, ab=ba, dad^-1=ac=ca, ae=ea, af=fa, ag=ga, ede^-1=gdg^-1=bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >

Subgroups: 604 in 496 conjugacy classes, 392 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂²×Q₈, C₂²×Q₈, C₂×C₄×Q₈, C₂³.₃₂C₂³, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₂².₃₅C₂⁴, D₄×Q₈, D₄×Q₈, C₂².₄₆C₂⁴, C₂².₅₀C₂⁴, Q₈⋊₃Q₈, Q₈⋊₃Q₈, C₂³.₁₄₆C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2_-¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2_-¹⁺⁴, C₂³.₁₄₆C₂⁴

Smallest permutation representation of C₂³.₁₄₆C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4P	4Q	···	4AJ
order	1	2	2	2	2	2	2	2	4	···	4	4	···	4
size	1	1	1	1	2	2	4	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+	+	+		-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2_-¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₁₄₆C₂⁴	C₂×C₄×Q₈	C₂³.₃₂C₂³	C₂³.₃₆C₂³	C₂³.₃₇C₂³	C₂².₃₅C₂⁴	D₄×Q₈	C₂².₄₆C₂⁴	C₂².₅₀C₂⁴	Q₈⋊₃Q₈	Q₈	C₄	C₂²
# reps	1	1	2	3	3	6	3	6	3	4	8	2	2

Matrix representation of C₂³.₁₄₆C₂⁴ ►in GL₆(𝔽₅)

4	3	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	0
0	0	0	0	0	1

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	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

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

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

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

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

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

C₂³.₁₄₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{146}C_2^4

% in TeX

G:=Group("C2^3.146C2^4");

// GroupNames label

G:=SmallGroup(128,2255);

// by ID

G=gap.SmallGroup(128,2255);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,570,136,1684,242]);

// Polycyclic

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

// generators/relations

G = C23.146C24 order 128 = 27

46th non-split extension by C23 of C24 acting via C24/C23=C2

G = C₂³.₁₄₆C₂⁴ order 128 = 2⁷

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