C4^2.302C2^3 - GroupNames

G = C₄².₃₀₂C₂³ order 128 = 2⁷

163^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₀₂C₂³, C₄.₁₁₈2_-¹⁺⁴, C₈⋊₄Q₈⋊₃₉C₂, (C₄×Q₈).₃₁C₄, C₄⋊C₈.₂₃₇C₂², (C₄×C₈).₃₃₉C₂², C₄².₂₂₈(C₂×C₄), (C₂×C₈).₄₄₀C₂³, (C₂×C₄).₆₈₀C₂⁴, C₄.₁₆(C₂×M₄(2)), (C₂×C₄).₂₇M₄(2), (C₂²×Q₈).₃₃C₄, C₈⋊C₄.₉₉C₂², (C₄×Q₈).₂₈₂C₂², C₂²⋊C₈.₂₃₆C₂², C₂.₃₃(Q₈○M₄(2)), C₄².₆C₄.₃₂C₂, C₂³.₂₃₁(C₂²×C₄), C₂².₂₀₃(C₂³×C₄), (C₂×C₄²).₇₈₇C₂², C₂².₃₀(C₂×M₄(2)), C₂.₂₁(C₂²×M₄(2)), C₄².₁₂C₄.₄₅C₂, (C₂²×C₄).₁₂₈₄C₂³, C₂.₂₃(C₂³.₃₂C₂³), (C₂×C₄⋊C₄).₈₀C₄, (C₂×C₄×Q₈).₄₆C₂, C₄⋊C₄.₂₃₁(C₂×C₄), (C₂×Q₈).₂₁₁(C₂×C₄), (C₂×C₄).₂₇₈(C₂²×C₄), (C₂²×C₄).₃₅₉(C₂×C₄), SmallGroup(128,1715)

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

Derived series

C₁ — C₂² — C₄².₃₀₂C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₂×C₄×Q₈ — C₄².₃₀₂C₂³

Lower central

C₁ — C₂² — C₄².₃₀₂C₂³

Upper central

C₁ — C₂×C₄ — C₄².₃₀₂C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₀₂C₂³

Generators and relations for C₄².₃₀₂C₂³
G = < a,b,c,d,e | a⁴=b⁴=1, c²=b, d²=e²=a², ab=ba, cac^-1=a^-1b², dad^-1=a^-1, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=a²c, ece^-1=b²c, ede^-1=a²d >

Subgroups: 228 in 178 conjugacy classes, 134 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×Q₈, C₂²×Q₈, C₄².₁₂C₄, C₄².₆C₄, C₈⋊₄Q₈, C₂×C₄×Q₈, C₄².₃₀₂C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, M₄(2), C₂²×C₄, C₂⁴, C₂×M₄(2), C₂³×C₄, 2_-¹⁺⁴, C₂³.₃₂C₂³, C₂²×M₄(2), Q₈○M₄(2), C₄².₃₀₂C₂³

Smallest permutation representation of C₄².₃₀₂C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	···	4N	4O	···	4V	8A	···	8P
order	1	2	2	2	2	2	4	4	4	4	4	···	4	4	···	4	8	···	8
size	1	1	1	1	2	2	1	1	1	1	2	···	2	4	···	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+					-
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	M₄(2)	2_-¹⁺⁴	Q₈○M₄(2)
kernel	C₄².₃₀₂C₂³	C₄².₁₂C₄	C₄².₆C₄	C₈⋊₄Q₈	C₂×C₄×Q₈	C₂×C₄⋊C₄	C₄×Q₈	C₂²×Q₈	C₂×C₄	C₄	C₂
# reps	1	2	4	8	1	6	8	2	8	2	2

Matrix representation of C₄².₃₀₂C₂³ ►in GL₆(𝔽₁₇)

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	15	0	0
0	0	1	16	0	0
0	0	3	14	16	1
0	0	3	0	15	1

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

3	9	0	0	0	0
8	14	0	0	0	0
0	0	14	0	2	0
0	0	0	0	0	1
0	0	12	0	3	0
0	0	0	16	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	4	13	0	0
0	0	12	0	13	0
0	0	12	0	9	4

0	16	0	0	0	0
16	0	0	0	0	0
0	0	1	15	0	0
0	0	1	16	0	0
0	0	0	14	1	16
0	0	14	0	2	16

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

C₄².₃₀₂C₂³ in GAP, Magma, Sage, TeX

C_4^2._{302}C_2^3

% in TeX

G:=Group("C4^2.302C2^3");

// GroupNames label

G:=SmallGroup(128,1715);

// by ID

G=gap.SmallGroup(128,1715);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,219,100,675,124]);

// Polycyclic

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

// generators/relations

G = C42.302C23 order 128 = 27

163rd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₀₂C₂³ order 128 = 2⁷

163^rd non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²