C4^2.301C2^3 - GroupNames

G = C₄².₃₀₁C₂³ order 128 = 2⁷

162^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₀₁C₂³, C₄.₁₇₅2₊¹⁺⁴, (C₈×D₄)⋊₄₉C₂, C₄⋊Q₈.₃₄C₄, C₈⋊₆D₄⋊₄₄C₂, C₈⋊₉D₄⋊₄₅C₂, C₄.₂₁(C₈○D₄), C₄⋊D₄.₂₉C₄, C₄⋊₁D₄.₂₁C₄, (C₄×C₈).₂₉C₂², C₄⋊C₈.₃₆₇C₂², C₄².₂₂₆(C₂×C₄), (C₂×C₄).₆₇₈C₂⁴, (C₂×C₈).₄₃₉C₂³, C₄.₄D₄.₂₂C₄, C₈⋊C₄.₉₈C₂², C₄².₆C₄⋊₅₃C₂, (C₄×D₄).₃₀₃C₂², C₂³.₄₅(C₂²×C₄), C₄².₁₂C₄⋊₅₅C₂, C₂.₃₂(Q₈○M₄(2)), C₂²⋊C₈.₁₄₆C₂², (C₂²×C₈).₄₅₁C₂², (C₂×C₄²).₇₈₅C₂², (C₂²×C₄).₉₄₅C₂³, C₂².₂₀₂(C₂³×C₄), C₂.₅₂(C₂².₁₁C₂⁴), (C₂×M₄(2)).₂₄₈C₂², C₂².₂₆C₂⁴.₂₉C₂, C₂.₃₁(C₂×C₈○D₄), C₄⋊C₄.₁₂₁(C₂×C₄), (C₂×D₄).₁₄₅(C₂×C₄), C₂²⋊C₄.₂₂(C₂×C₄), (C₂×C₄).₈₃(C₂²×C₄), (C₂×Q₈).₁₂₇(C₂×C₄), (C₂²×C₈)⋊C₂⋊₃₇C₂, (C₂²×C₄).₃₅₈(C₂×C₄), (C₂×C₄○D₄).₉₈C₂², SmallGroup(128,1713)

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

Derived series

C₁ — C₂² — C₄².₃₀₁C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂².₂₆C₂⁴ — C₄².₃₀₁C₂³

Lower central

C₁ — C₂² — C₄².₃₀₁C₂³

Upper central

C₁ — C₂×C₄ — C₄².₃₀₁C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₀₁C₂³

Generators and relations for C₄².₃₀₁C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=e²=1, c²=b, ab=ba, cac^-1=a^-1b², dad=a^-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a²b²c, ede=b²d >

Subgroups: 332 in 201 conjugacy classes, 126 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₄×D₄, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×C₄○D₄, (C₂²×C₈)⋊C₂, C₄².₁₂C₄, C₄².₆C₄, C₈×D₄, C₈⋊₉D₄, C₈⋊₆D₄, C₂².₂₆C₂⁴, C₄².₃₀₁C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, C₈○D₄, C₂³×C₄, 2₊¹⁺⁴, C₂².₁₁C₂⁴, C₂×C₈○D₄, Q₈○M₄(2), C₄².₃₀₁C₂³

Smallest permutation representation of C₄².₃₀₁C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2H	4A	4B	4C	4D	4E	4F	4G	4H	4I	···	4O	8A	···	8H	8I	···	8T
order	1	2	2	2	2	···	2	4	4	4	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	1	1	4	···	4	1	1	1	1	2	2	2	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+						+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₈○D₄	2₊¹⁺⁴	Q₈○M₄(2)
kernel	C₄².₃₀₁C₂³	(C₂²×C₈)⋊C₂	C₄².₁₂C₄	C₄².₆C₄	C₈×D₄	C₈⋊₉D₄	C₈⋊₆D₄	C₂².₂₆C₂⁴	C₄⋊D₄	C₄.₄D₄	C₄⋊₁D₄	C₄⋊Q₈	C₄	C₄	C₂
# reps	1	4	1	1	2	4	2	1	8	4	2	2	8	2	2

Matrix representation of C₄².₃₀₁C₂³ ►in GL₆(𝔽₁₇)

0	13	0	0	0	0
13	0	0	0	0	0
0	0	0	0	0	16
0	0	0	0	16	0
0	0	0	1	0	0
0	0	1	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	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	16	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	0	0	0	1

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

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	1	0

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₄².₃₀₁C₂³ in GAP, Magma, Sage, TeX

C_4^2._{301}C_2^3

% in TeX

G:=Group("C4^2.301C2^3");

// GroupNames label

G:=SmallGroup(128,1713);

// by ID

G=gap.SmallGroup(128,1713);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,521,124]);

// Polycyclic

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

// generators/relations

G = C42.301C23 order 128 = 27

162nd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₀₁C₂³ order 128 = 2⁷

162^nd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²