C4^2.128D4 - GroupNames

G = C₄².₁₂₈D₄ order 128 = 2⁷

110^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄².₁₂₈D₄, M₄(2).₇Q₈, C₄.₃₄(C₄⋊Q₈), C₂².₁₅(C₄×Q₈), C₄.₈₅(C₂²⋊Q₈), C₄.₅₃(C₄.₄D₄), (C₄×M₄(2)).₂₈C₂, C₄⋊M₄(2).₃₆C₂, C₂³.₂₀₂(C₂²×C₄), (C₂×C₄²).₃₄₀C₂², (C₂²×C₄).₇₀₇C₂³, C₂².C₄².₁₃C₂, (C₂×M₄(2)).₂₁₉C₂², C₂.₂₀(C₂³.₆₇C₂³), C₂.₂₉(M₄(2).₈C₂²), (C₂×C₄⋊C₄).₂₆C₄, (C₂×C₄).₁₄(C₂×Q₈), (C₂×C₄).₇₁(C₄○D₄), (C₂×C₄).₁₃₆₉(C₂×D₄), (C₂²×C₄).₂₈(C₂×C₄), (C₂×C₄².C₂).₈C₂, (C₂×C₄⋊C₄).₁₀₀C₂², (C₂×C₄).₂₁₀(C₂²⋊C₄), C₂².₃₀₇(C₂×C₂²⋊C₄), SmallGroup(128,730)

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

Derived series

C₁ — C₂³ — C₄².₁₂₈D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×M₄(2) — C₄².₁₂₈D₄

Lower central

C₁ — C₂ — C₂³ — C₄².₁₂₈D₄

Upper central

C₁ — C₂² — C₂×C₄² — C₄².₁₂₈D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₄².₁₂₈D₄

Generators and relations for C₄².₁₂₈D₄
G = < a,b,c,d | a⁴=b⁴=1, c⁴=b², d²=a²b², ab=ba, ac=ca, dad^-1=a^-1b², cbc^-1=dbd^-1=b^-1, dcd^-1=a²b^-1c³ >

Subgroups: 196 in 114 conjugacy classes, 56 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₄², C₄², C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₄×C₈, C₈⋊C₄, C₄⋊C₈, C₂×C₄², C₂×C₄⋊C₄, C₄².C₂, C₂×M₄(2), C₂².C₄², C₄×M₄(2), C₄⋊M₄(2), C₂×C₄².C₂, C₄².₁₂₈D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₂²⋊C₄, C₄×Q₈, C₂²⋊Q₈, C₄.₄D₄, C₄⋊Q₈, C₂³.₆₇C₂³, M₄(2).₈C₂², C₄².₁₂₈D₄

Smallest permutation representation of C₄².₁₂₈D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	···	4H	4I	4J	4K	4L	4M	4N	8A	···	8H	8I	8J	8K	8L
order	1	2	2	2	2	2	4	···	4	4	4	4	4	4	4	8	···	8	8	8	8	8
size	1	1	1	1	2	2	2	···	2	4	4	8	8	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4
type	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₄	D₄	Q₈	C₄○D₄	M₄(2).₈C₂²
kernel	C₄².₁₂₈D₄	C₂².C₄²	C₄×M₄(2)	C₄⋊M₄(2)	C₂×C₄².C₂	C₂×C₄⋊C₄	C₄²	M₄(2)	C₂×C₄	C₂
# reps	1	4	1	1	1	8	4	4	4	4

Matrix representation of C₄².₁₂₈D₄ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	4	0	0	0	0
0	0	0	4	0	0
0	0	13	0	0	0
0	0	4	11	4	9
0	0	5	16	4	13

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	16	10	16	2
0	0	13	14	16	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	1	7	1	15
0	0	0	16	0	0
0	0	3	7	4	10

0	13	0	0	0	0
13	0	0	0	0	0
0	0	13	0	0	0
0	0	0	4	0	0
0	0	0	0	13	0
0	0	13	0	13	4

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,13,4,5,0,0,4,0,11,16,0,0,0,0,4,4,0,0,0,0,9,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,16,13,0,0,1,0,10,14,0,0,0,0,16,16,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,3,0,0,0,7,16,7,0,0,1,1,0,4,0,0,0,15,0,10],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,13,0,0,0,4,0,0,0,0,0,0,13,13,0,0,0,0,0,4] >;

C₄².₁₂₈D₄ in GAP, Magma, Sage, TeX

C_4^2._{128}D_4

% in TeX

G:=Group("C4^2.128D4");

// GroupNames label

G:=SmallGroup(128,730);

// by ID

G=gap.SmallGroup(128,730);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,436,2019,1018,2028,124]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^-1*c^3>;

// generators/relations

G = C42.128D4 order 128 = 27

110th non-split extension by C42 of D4 acting via D4/C2=C22

G = C₄².₁₂₈D₄ order 128 = 2⁷

110^th non-split extension by C₄² of D₄ acting via D₄/C₂=C₂²