C4^2:29D4

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

Aliases: C₄²⋊₂₉D₄, C₂³.₅₃₁C₂⁴, C₂⁴.₃₇₀C₂³, C₂².₃₀₈2₊⁽¹⁺⁴⁾, C₄²⋊₅C₄⋊₂₄C₂, C₂³⋊₂D₄⋊₂₆C₂, C₂³⋊Q₈⋊₂₉C₂, C₂³.₆₅(C₄○D₄), C₂³.₂₃D₄⋊₇₀C₂, C₂³.₁₁D₄⋊₆₀C₂, (C₂²×C₄).₁₄₁C₂³, (C₂×C₄²).₆₀₈C₂², (C₂³×C₄).₁₃₇C₂², C₂².₃₅₆(C₂²×D₄), (C₂²×D₄).₅₄₁C₂², (C₂²×Q₈).₁₅₆C₂², C₂.₄₀(C₂².₂₉C₂⁴), C₂.₈₂(C₂².₁₉C₂⁴), C₂.₄₀(C₂².₃₂C₂⁴), C₂.C₄².₂₅₆C₂², (C₂×C₄×D₄)⋊₅₃C₂, (C₂×C₄).₃₉₀(C₂×D₄), (C₂×C₄.₄D₄)⋊₂₁C₂, (C₂×C₄⋊C₄).₈₉₂C₂², C₂².₄₀₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₂₀C₂², SmallGroup(128,1363)

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

Derived series

C₁ — C₂³ — C₄²⋊₂₉D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³.₂₃D₄ — C₄²⋊₂₉D₄

Lower central

C₁ — C₂³ — C₄²⋊₂₉D₄

Upper central

C₁ — C₂³ — C₄²⋊₂₉D₄

Jennings

C₁ — C₂³ — C₄²⋊₂₉D₄

Subgroups: 724 in 314 conjugacy classes, 96 normal (18 characteristic)
C₁, C₂^[×3], C₂^[×4], C₂^[×6], C₄^[×14], C₂², C₂²^[×6], C₂²^[×34], C₂×C₄^[×6], C₂×C₄^[×38], D₄^[×28], Q₈^[×4], C₂³, C₂³^[×4], C₂³^[×26], C₄²^[×4], C₂²⋊C₄^[×20], C₄⋊C₄^[×2], C₂²×C₄^[×3], C₂²×C₄^[×8], C₂²×C₄^[×8], C₂×D₄^[×26], C₂×Q₈^[×4], C₂⁴^[×4], C₂.C₄²^[×10], C₂×C₄², C₂×C₂²⋊C₄^[×14], C₂×C₄⋊C₄, C₄×D₄^[×4], C₄.₄D₄^[×4], C₂³×C₄^[×2], C₂²×D₄^[×2], C₂²×D₄^[×4], C₂²×Q₈, C₄²⋊₅C₄, C₂³.₂₃D₄^[×4], C₂³⋊₂D₄^[×4], C₂³⋊Q₈^[×2], C₂³.₁₁D₄^[×2], C₂×C₄×D₄, C₂×C₄.₄D₄, C₄²⋊₂₉D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×4], C₂⁴, C₂²×D₄, C₂×C₄○D₄^[×2], 2₊⁽¹⁺⁴⁾^[×4], C₂².₁₉C₂⁴, C₂².₂₉C₂⁴^[×2], C₂².₃₂C₂⁴^[×4], C₄²⋊₂₉D₄

Generators and relations
G = < a,b,c,d | a⁴=b⁴=c⁴=d²=1, ab=ba, cac^-1=a^-1b², dad=a^-1, cbc^-1=a²b, bd=db, dcd=c^-1 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₈(𝔽₅)

0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	2	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0

2	0	0	0	0	0	0	0
0	2	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	1	2	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	1	2
0	0	0	0	0	0	0	4

0	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	4	4
0	0	0	0	1	0	0	0
0	0	0	0	4	4	0	0

0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	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

G:=sub<GL(8,GF(5))| [0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄	2₊⁽¹⁺⁴⁾
kernel	C₄²⋊₂₉D₄	C₄²⋊₅C₄	C₂³.₂₃D₄	C₂³⋊₂D₄	C₂³⋊Q₈	C₂³.₁₁D₄	C₂×C₄×D₄	C₂×C₄.₄D₄	C₄²	C₂³	C₂²
# reps	1	1	4	4	2	2	1	1	4	8	4

In GAP, Magma, Sage, TeX

C_4^2\rtimes_{29}D_4

% in TeX

G:=Group("C4^2:29D4");

// GroupNames label

G:=SmallGroup(128,1363);

// by ID

G=gap.SmallGroup(128,1363);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,185,136]);

// Polycyclic

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

// generators/relations

G = C₄²⋊₂₉D₄ order 128 = 2⁷

23^rd semidirect product of C₄² and D₄ acting via D₄/C₂=C₂²

0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	2	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0

2	0	0	0	0	0	0	0
0	2	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	1	2	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	1	2
0	0	0	0	0	0	0	4

0	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	4	4
0	0	0	0	1	0	0	0
0	0	0	0	4	4	0	0

0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	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	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	2	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0

2	0	0	0	0	0	0	0
0	2	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	1	2	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	1	2
0	0	0	0	0	0	0	4

0	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	4	4
0	0	0	0	1	0	0	0
0	0	0	0	4	4	0	0

0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	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

G = C42⋊29D4 order 128 = 27

23rd semidirect product of C42 and D4 acting via D4/C2=C22

G = C₄²⋊₂₉D₄ order 128 = 2⁷

23^rd semidirect product of C₄² and D₄ acting via D₄/C₂=C₂²

0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	2	0	0	0	0
0	0	0	4	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	0	0	4	0	0	0
0	0	0	0	0	4	0	0

2	0	0	0	0	0	0	0
0	2	0	0	0	0	0	0
0	0	3	0	0	0	0	0
0	0	0	3	0	0	0	0
0	0	0	0	1	2	0	0
0	0	0	0	0	4	0	0
0	0	0	0	0	0	1	2
0	0	0	0	0	0	0	4

0	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	4	4
0	0	0	0	1	0	0	0
0	0	0	0	4	4	0	0

0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	4	0	0	0	0	0
0	0	0	4	0	0	0	0
0	0	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