C4oD4.4Q8 - GroupNames

G = C₄○D₄.₄Q₈ order 128 = 2⁷

2^nd non-split extension by C₄○D₄ of Q₈ acting via Q₈/C₄=C₂

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

Aliases: C₈○D₄⋊₂C₄, C₄○D₄.₄Q₈, D₄.₃(C₄⋊C₄), (C₂×C₈).₁₀₅D₄, Q₈.₃(C₄⋊C₄), (C₂×D₄).₂₀₂D₄, (C₂×Q₈).₁₆₀D₄, C₈.₄₈(C₂²⋊C₄), C₄.₇₂(C₂²⋊Q₈), C₄.₁₂₃(C₄⋊D₄), C₂.₁(D₄.₃D₄), M₄(2).₃₁(C₂×C₄), C₂².C₄²⋊₁₃C₂, C₂³.₂₆₁(C₄○D₄), (C₂²×C₈).₂₁₇C₂², (C₂²×C₄).₆₆₅C₂³, C₂³.₃₆D₄.₅C₂, C₂².₁₃(C₂²⋊Q₈), C₂².₁₁₂(C₄⋊D₄), C₂.₂₄(C₂³.₇Q₈), C₂².₁₅(C₄²⋊C₂), (C₂×M₄(2)).₁₅₈C₂², C₄.₇(C₂×C₄⋊C₄), (C₂×C₄.Q₈)⋊₂C₂, (C₂×C₄).₅(C₂×Q₈), (C₂×C₈).₆₁(C₂×C₄), (C₂×C₈○D₄).₄C₂, (C₂×C₈.C₄)⋊₅C₂, C₄○D₄.₂₈(C₂×C₄), (C₂×C₄).₂₃₂(C₂×D₄), C₄.₉₄(C₂×C₂²⋊C₄), (C₂×C₄).₄₇(C₄○D₄), (C₂×C₄⋊C₄).₃₉C₂², (C₂×C₄).₁₇₉(C₂²×C₄), (C₂×C₄○D₄).₂₅₉C₂², SmallGroup(128,547)

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

Derived series

C₁ — C₂×C₄ — C₄○D₄.₄Q₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×C₈○D₄ — C₄○D₄.₄Q₈

Lower central

C₁ — C₂ — C₂×C₄ — C₄○D₄.₄Q₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₄○D₄.₄Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₄○D₄.₄Q₈

Generators and relations for C₄○D₄.₄Q₈
G = < a,b,c,d,e | a⁴=c²=1, b²=d⁴=a², e²=a^-1bd², ab=ba, ac=ca, ad=da, eae^-1=a^-1, cbc=a²b, bd=db, be=eb, cd=dc, ece^-1=bc, ede^-1=d³ >

Subgroups: 244 in 132 conjugacy classes, 60 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₈.C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂×M₄(2), C₈○D₄, C₈○D₄, C₂×C₄○D₄, C₂².C₄², C₂³.₃₆D₄, C₂×C₄.Q₈, C₂×C₈.C₄, C₂×C₈○D₄, C₄○D₄.₄Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄⋊D₄, C₂²⋊Q₈, C₂³.₇Q₈, D₄.₃D₄, C₄○D₄.₄Q₈

Smallest permutation representation of C₄○D₄.₄Q₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	4
type	+	+	+	+	+	+		+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	D₄	D₄	D₄	Q₈	C₄○D₄	C₄○D₄	D₄.₃D₄
kernel	C₄○D₄.₄Q₈	C₂².C₄²	C₂³.₃₆D₄	C₂×C₄.Q₈	C₂×C₈.C₄	C₂×C₈○D₄	C₈○D₄	C₂×C₈	C₂×D₄	C₂×Q₈	C₄○D₄	C₂×C₄	C₂³	C₂
# reps	1	2	2	1	1	1	8	4	1	1	2	2	2	4

Matrix representation of C₄○D₄.₄Q₈ ►in GL₆(𝔽₁₇)

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	13	0
0	0	0	0	0	13

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	16	0	0
0	0	1	0	0	0
0	0	0	0	0	16
0	0	0	0	1	0

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	0	16	0
0	0	0	0	0	1

13	0	0	0	0	0
11	4	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	0	0	2	0
0	0	0	0	0	2

12	1	0	0	0	0
8	5	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	16	0	0	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[13,11,0,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[12,8,0,0,0,0,1,5,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₄○D₄.₄Q₈ in GAP, Magma, Sage, TeX

C_4\circ D_4._4Q_8

% in TeX

G:=Group("C4oD4.4Q8");

// GroupNames label

G:=SmallGroup(128,547);

// by ID

G=gap.SmallGroup(128,547);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,718,172,1027,124]);

// Polycyclic

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

// generators/relations

G = C4○D4.4Q8 order 128 = 27

2nd non-split extension by C4○D4 of Q8 acting via Q8/C4=C2

G = C₄○D₄.₄Q₈ order 128 = 2⁷

2^nd non-split extension by C₄○D₄ of Q₈ acting via Q₈/C₄=C₂