C4^2.292C2^3 - GroupNames

G = C₄².₂₉₂C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₂₉₂C₂³, C₈⋊₉D₄⋊₃₅C₂, C₈⋊₆D₄⋊₃₆C₂, C₈⋊₄Q₈⋊₃₅C₂, C₈.₆₂(C₄○D₄), C₄⋊D₄.₂₁C₄, C₂²⋊Q₈.₂₁C₄, C₄⋊C₈.₂₃₄C₂², (C₄×M₄(2))⋊₃₆C₂, (C₂×C₈).₄₂₈C₂³, C₄².₂₁₅(C₂×C₄), (C₄×C₈).₃₃₃C₂², (C₂×C₄).₆₆₄C₂⁴, C₄²⋊₂C₂.₁C₄, C₄.₄D₄.₁₇C₄, (C₄×D₄).₆₁C₂², C₄².C₂.₁₇C₄, (C₄×Q₈).₅₉C₂², C₈○₂M₄(2)⋊₃₄C₂, C₄².₆C₄⋊₄₇C₂, C₂³.₃₉(C₂²×C₄), C₂².D₄.₅C₄, C₈⋊C₄.₁₆₄C₂², C₂.₂₂(Q₈○M₄(2)), C₂²⋊C₈.₁₄₁C₂², (C₂²×C₈).₄₄₆C₂², (C₂×C₄²).₇₇₄C₂², C₂².₁₈₉(C₂³×C₄), (C₂²×C₄).₉₃₅C₂³, C₄².₇C₂²⋊₂₃C₂, C₄²⋊C₂.₃₀₇C₂², (C₂×M₄(2)).₃₆₆C₂², C₂³.₃₆C₂³.₁₃C₂, C₂.₄₆(C₄×C₄○D₄), C₄⋊C₄.₁₁₈(C₂×C₄), C₄.₃₁₅(C₂×C₄○D₄), (C₂×D₄).₁₄₁(C₂×C₄), C₂²⋊C₄.₁₇(C₂×C₄), (C₂×Q₈).₁₂₀(C₂×C₄), (C₂²×C₄).₃₄₈(C₂×C₄), (C₂×C₄).₂₇₂(C₂²×C₄), SmallGroup(128,1699)

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

Derived series

C₁ — C₂² — C₄².₂₉₂C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×C₈ — C₄×M₄(2) — C₄².₂₉₂C₂³

Lower central

C₁ — C₂² — C₄².₂₉₂C₂³

Upper central

C₁ — C₂×C₄ — C₄².₂₉₂C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₉₂C₂³

Generators and relations for C₄².₂₉₂C₂³
G = < a,b,c,d,e | a⁴=b⁴=e²=1, c²=d²=b, ab=ba, cac^-1=a^-1b², dad^-1=ab², ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a²c, ede=b²d >

Subgroups: 252 in 181 conjugacy classes, 128 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₂²×C₈, C₂×M₄(2), C₄×M₄(2), C₈○₂M₄(2), C₄².₆C₄, C₄².₇C₂², C₈⋊₉D₄, C₈⋊₆D₄, C₈⋊₄Q₈, C₂³.₃₆C₂³, C₄².₂₉₂C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₄○D₄, C₂⁴, C₂³×C₄, C₂×C₄○D₄, C₄×C₄○D₄, Q₈○M₄(2), C₄².₂₉₂C₂³

Smallest permutation representation of C₄².₂₉₂C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	···	4Q	8A	···	8H	8I	···	8T
order	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	1	1	4	4	4	1	1	1	1	2	2	2	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₄	C₄	C₄○D₄	Q₈○M₄(2)
kernel	C₄².₂₉₂C₂³	C₄×M₄(2)	C₈○₂M₄(2)	C₄².₆C₄	C₄².₇C₂²	C₈⋊₉D₄	C₈⋊₆D₄	C₈⋊₄Q₈	C₂³.₃₆C₂³	C₄⋊D₄	C₂²⋊Q₈	C₂².D₄	C₄.₄D₄	C₄².C₂	C₄²⋊₂C₂	C₈	C₂
# reps	1	1	2	1	2	4	2	2	1	2	2	4	2	2	4	8	4

Matrix representation of C₄².₂₉₂C₂³ ►in GL₆(𝔽₁₇)

15	9	0	0	0	0
7	2	0	0	0	0
0	0	4	8	3	1
0	0	0	12	0	1
0	0	12	5	13	0
0	0	0	10	0	5

1	0	0	0	0	0
0	1	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	13	0
0	0	0	0	0	13

16	0	0	0	0	0
9	1	0	0	0	0
0	0	0	3	16	10
0	0	0	3	0	9
0	0	4	4	0	14
0	0	0	8	0	14

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	3	1	9
0	0	0	3	0	9
0	0	13	4	0	0
0	0	0	8	0	14

9	2	0	0	0	0
11	8	0	0	0	0
0	0	8	4	3	1
0	0	12	0	6	1
0	0	5	12	4	0
0	0	10	0	8	5

G:=sub<GL(6,GF(17))| [15,7,0,0,0,0,9,2,0,0,0,0,0,0,4,0,12,0,0,0,8,12,5,10,0,0,3,0,13,0,0,0,1,1,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,3,3,4,8,0,0,16,0,0,0,0,0,10,9,14,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,3,3,4,8,0,0,1,0,0,0,0,0,9,9,0,14],[9,11,0,0,0,0,2,8,0,0,0,0,0,0,8,12,5,10,0,0,4,0,12,0,0,0,3,6,4,8,0,0,1,1,0,5] >;

C₄².₂₉₂C₂³ in GAP, Magma, Sage, TeX

C_4^2._{292}C_2^3

% in TeX

G:=Group("C4^2.292C2^3");

// GroupNames label

G:=SmallGroup(128,1699);

// by ID

G=gap.SmallGroup(128,1699);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,2019,80,124]);

// Polycyclic

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

// generators/relations

G = C42.292C23 order 128 = 27

153rd non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₉₂C₂³ order 128 = 2⁷

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