C4^2.266C2^3 - GroupNames

G = C₄².₂₆₆C₂³ order 128 = 2⁷

127^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₂₆₆C₂³, (C₂×C₈)⋊₂₇D₄, C₄.₄₁(C₄×D₄), C₄⋊Q₈.₂₈C₄, C₈⋊₉D₄⋊₃₁C₂, C₈.₁₂₈(C₂×D₄), C₂².₃(C₄×D₄), C₄⋊D₄.₁₈C₄, C₄⋊₁D₄.₁₇C₄, C₄⋊C₈.₂₃₂C₂², C₄².₂₀₉(C₂×C₄), (C₂×C₈).₄₀₄C₂³, (C₂×C₄).₆₅₁C₂⁴, C₄.₄D₄.₁₅C₄, (C₄×D₄).₅₄C₂², C₄.₁₉₇(C₂²×D₄), C₄⋊M₄(2)⋊₃₄C₂, C₂³.₃₅(C₂²×C₄), C₈⋊C₄.₁₅₅C₂², C₂²⋊C₈.₁₄₀C₂², C₂.₁₅(Q₈○M₄(2)), (C₂×C₄²).₇₅₉C₂², C₂².₁₇₈(C₂³×C₄), (C₂²×C₄).₉₁₈C₂³, (C₂²×C₈).₄₃₄C₂², (C₂×M₄(2)).₃₄₈C₂², C₂².₂₆C₂⁴.₂₅C₂, C₂.₄₉(C₂×C₄×D₄), (C₂×C₈○D₄)⋊₂₂C₂, (C₂×C₈⋊C₄)⋊₃₄C₂, C₄⋊C₄.₁₁₅(C₂×C₄), C₄.₃₀₂(C₂×C₄○D₄), (C₂×C₄).₈₄₄(C₂×D₄), (C₂×D₄).₁₃₇(C₂×C₄), C₂²⋊C₄.₁₆(C₂×C₄), (C₂×Q₈).₁₅₅(C₂×C₄), (C₂²×C₈)⋊C₂⋊₂₉C₂, (C₂×C₄).₆₈₄(C₄○D₄), (C₂×C₄).₂₆₂(C₂²×C₄), (C₂²×C₄).₃₄₁(C₂×C₄), (C₂×C₄○D₄).₂₈₆C₂², SmallGroup(128,1664)

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

Derived series

C₁ — C₂² — C₄².₂₆₆C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₂×C₈○D₄ — C₄².₂₆₆C₂³

Lower central

C₁ — C₂² — C₄².₂₆₆C₂³

Upper central

C₁ — C₂×C₄ — C₄².₂₆₆C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₆₆C₂³

Generators and relations for C₄².₂₆₆C₂³
G = < a,b,c,d,e | a⁴=b⁴=e²=1, c²=b², d²=a²b, ab=ba, cac^-1=a^-1, dad^-1=ab², ae=ea, bc=cb, bd=db, be=eb, dcd^-1=a²c, ece=b²c, de=ed >

Subgroups: 364 in 242 conjugacy classes, 140 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₄×D₄, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₈○D₄, C₂×C₄○D₄, C₂×C₈⋊C₄, (C₂²×C₈)⋊C₂, C₄⋊M₄(2), C₈⋊₉D₄, C₂².₂₆C₂⁴, C₂×C₈○D₄, C₄².₂₆₆C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²×C₄, C₂×D₄, C₄○D₄, C₂⁴, C₄×D₄, C₂³×C₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄×D₄, Q₈○M₄(2), C₄².₂₆₆C₂³

Smallest permutation representation of C₄².₂₆₆C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	···	4N	8A	···	8H	8I	···	8T
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	1	1	2	2	4	4	4	4	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	1	2	2	4
type	+	+	+	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	C₄○D₄	Q₈○M₄(2)
kernel	C₄².₂₆₆C₂³	C₂×C₈⋊C₄	(C₂²×C₈)⋊C₂	C₄⋊M₄(2)	C₈⋊₉D₄	C₂².₂₆C₂⁴	C₂×C₈○D₄	C₄⋊D₄	C₄.₄D₄	C₄⋊₁D₄	C₄⋊Q₈	C₂×C₈	C₂×C₄	C₂
# reps	1	1	2	1	8	1	2	8	4	2	2	4	4	4

Matrix representation of C₄².₂₆₆C₂³ ►in GL₆(𝔽₁₇)

0	4	0	0	0	0
4	0	0	0	0	0
0	0	8	6	6	14
0	0	0	8	3	0
0	0	0	1	9	0
0	0	16	0	6	9

16	0	0	0	0	0
0	16	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	13	0
0	0	0	0	0	13

1	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	1	11
0	0	16	0	6	1
0	0	0	0	0	16
0	0	0	0	1	0

0	16	0	0	0	0
16	0	0	0	0	0
0	0	16	12	12	7
0	0	0	16	10	0
0	0	0	2	1	0
0	0	15	0	12	1

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	11
0	0	1	0	11	0
0	0	0	0	0	16
0	0	0	0	16	0

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,8,0,0,16,0,0,6,8,1,0,0,0,6,3,9,6,0,0,14,0,0,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,1,6,0,1,0,0,11,1,16,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,15,0,0,12,16,2,0,0,0,12,10,1,12,0,0,7,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,11,0,16,0,0,11,0,16,0] >;

C₄².₂₆₆C₂³ in GAP, Magma, Sage, TeX

C_4^2._{266}C_2^3

% in TeX

G:=Group("C4^2.266C2^3");

// GroupNames label

G:=SmallGroup(128,1664);

// by ID

G=gap.SmallGroup(128,1664);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2019,1018,124]);

// Polycyclic

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

// generators/relations

G = C42.266C23 order 128 = 27

127th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₆₆C₂³ order 128 = 2⁷

127^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²