C4oD4.7Q8 - GroupNames

G = C₄○D₄.₇Q₈ order 128 = 2⁷

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

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

Aliases: C₈○D₄⋊₄C₄, D₄ ○(C₄.Q₈), Q₈ ○(C₄.Q₈), C₄○D₄.₇Q₈, D₄.₉(C₄⋊C₄), Q₈.₉(C₄⋊C₄), (C₂×D₄).₃₄₅D₄, C₄.₆(C₂²×Q₈), C₈.₁₄(C₂²×C₄), C₄.₅₀(C₂³×C₄), (C₂×Q₈).₂₆₄D₄, C₄⋊C₄.₃₅₁C₂³, M₄(2)⋊₁₆(C₂×C₄), (C₂×C₈).₅₈₉C₂³, (C₂×C₄).₁₈₈C₂⁴, C₂.₃(D₄○SD₁₆), C₂³.₄₃₃(C₂×D₄), M₄(2)⋊C₄⋊₈C₂, C₄.Q₈.₁₈₄C₂², C₂.D₈.₂₁₀C₂², C₂³.₂₅D₄⋊₂₁C₂, (C₂²×C₄).₉₀₇C₂³, (C₂²×C₈).₂₄₄C₂², C₂².₁₃₅(C₂²×D₄), C₄²⋊C₂.₇₈C₂², (C₂×M₄(2)).₂₅₈C₂², C₂³.₃₃C₂³.₄C₂, (C₂×C₈)⋊₉(C₂×C₄), C₄.₂₁(C₂×C₄⋊C₄), (C₂×C₄.Q₈)⋊₆C₂, (C₂×Q₈)○(C₄.Q₈), (C₂×C₈○D₄).₆C₂, C₂².₃(C₂×C₄⋊C₄), (C₂×C₄).₉₆(C₂×Q₈), C₄○D₄.₃₂(C₂×C₄), C₂.₂₇(C₂²×C₄⋊C₄), (C₂×C₄).₁₀₈₁(C₂×D₄), (C₂×C₄⋊C₄).₅₇₂C₂², (C₂×C₄).₂₅₁(C₂²×C₄), (C₂×C₄○D₄).₂₇₇C₂², SmallGroup(128,1644)

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

Derived series

C₁ — C₄ — C₄○D₄.₇Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×C₈○D₄ — C₄○D₄.₇Q₈

Lower central

C₁ — C₂ — C₄ — C₄○D₄.₇Q₈

Upper central

C₁ — C₂² — C₂×C₄○D₄ — C₄○D₄.₇Q₈

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄○D₄.₇Q₈

Generators and relations for C₄○D₄.₇Q₈
G = < a,b,c,d,e | a⁴=c²=1, b²=d⁴=a², e²=a^-1d², ab=ba, ac=ca, ad=da, eae^-1=a^-1, cbc=a²b, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 356 in 238 conjugacy classes, 172 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄.Q₈, C₄.Q₈, C₂.D₈, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²×C₈, C₂×M₄(2), C₈○D₄, C₂×C₄○D₄, C₂×C₄.Q₈, C₂³.₂₅D₄, M₄(2)⋊C₄, C₂³.₃₃C₂³, C₂×C₈○D₄, C₄○D₄.₇Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂²×C₄⋊C₄, D₄○SD₁₆, C₄○D₄.₇Q₈

Smallest permutation representation of C₄○D₄.₇Q₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	···	4H	4I	···	4X	8A	8B	8C	8D	8E	···	8J
order	1	2	2	2	2	···	2	4	···	4	4	···	4	8	8	8	8	8	···	8
size	1	1	1	1	2	···	2	2	···	2	4	···	4	2	2	2	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	4
type	+	+	+	+	+	+		+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	D₄	D₄	Q₈	D₄○SD₁₆
kernel	C₄○D₄.₇Q₈	C₂×C₄.Q₈	C₂³.₂₅D₄	M₄(2)⋊C₄	C₂³.₃₃C₂³	C₂×C₈○D₄	C₈○D₄	C₂×D₄	C₂×Q₈	C₄○D₄	C₂
# reps	1	3	3	6	2	1	16	3	1	4	4

Matrix representation of C₄○D₄.₇Q₈ ►in GL₆(𝔽₁₇)

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

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	1
0	0	0	0	16	0

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

10	2	0	0	0	0
9	7	0	0	0	0
0	0	12	5	0	0
0	0	12	12	0	0
0	0	0	0	12	5
0	0	0	0	12	12

14	3	0	0	0	0
8	3	0	0	0	0
0	0	0	0	14	3
0	0	0	0	3	3
0	0	14	3	0	0
0	0	3	3	0	0

G:=sub<GL(6,GF(17))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[10,9,0,0,0,0,2,7,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[14,8,0,0,0,0,3,3,0,0,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,14,3,0,0,0,0,3,3,0,0] >;

C₄○D₄.₇Q₈ in GAP, Magma, Sage, TeX

C_4\circ D_4._7Q_8

% in TeX

G:=Group("C4oD4.7Q8");

// GroupNames label

G:=SmallGroup(128,1644);

// by ID

G=gap.SmallGroup(128,1644);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,521,2804,172]);

// Polycyclic

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

// generators/relations

G = C4○D4.7Q8 order 128 = 27

5th non-split extension by C4○D4 of Q8 acting via Q8/C4=C2

G = C₄○D₄.₇Q₈ order 128 = 2⁷

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