C4^2.387C2^3 - GroupNames

G = C₄².₃₈₇C₂³ order 128 = 2⁷

248^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₃₈₇C₂³, (C₄×D₈)⋊₄₀C₂, C₈⋊D₄⋊₈C₂, C₄⋊C₄.₂₄₇D₄, (C₄×Q₁₆)⋊₄₀C₂, C₈⋊₂Q₈⋊₂₉C₂, C₈.₁₀(C₄○D₄), C₂.₂₄(D₄○D₈), C₈⋊₇D₄.₁₁C₂, C₂²⋊C₄.₈₇D₄, C₂.₂₄(Q₈○D₈), C₂³.₈₄(C₂×D₄), C₈.₁₈D₄⋊₃₆C₂, C₈.₁₂D₄⋊₁₉C₂, C₄⋊C₄.₁₁₄C₂³, (C₂×C₄).₃₇₃C₂⁴, (C₂×C₈).₅₆₆C₂³, (C₄×C₈).₂₂₃C₂², (C₄×D₄).₉₄C₂², C₄⋊Q₈.₁₁₆C₂², SD₁₆⋊C₄⋊₂₀C₂, (C₄×Q₈).₉₁C₂², C₈○₂M₄(2)⋊₁₇C₂, (C₂×D₈).₁₃₄C₂², (C₂×D₄).₁₂₈C₂³, C₄⋊D₄.₃₅C₂², (C₂×Q₈).₁₁₆C₂³, C₈⋊C₄.₁₃₀C₂², C₂.D₈.₁₈₄C₂², C₂²⋊Q₈.₃₅C₂², (C₂²×C₈).₃₀₂C₂², (C₂×Q₁₆).₁₃₀C₂², (C₂×SD₁₆).₂₃C₂², C₄.₄D₄.₃₅C₂², C₂².₆₃₃(C₂²×D₄), D₄⋊C₄.₂₀₆C₂², C₂².₃₆C₂⁴⋊₅C₂, (C₂²×C₄).₁₀₅₃C₂³, Q₈⋊C₄.₂₀₇C₂², C₄²⋊C₂.₃₃₀C₂², C₄².₂₈C₂²⋊₃₄C₂, (C₂×M₄(2)).₂₈₃C₂², C₂.₇₀(C₂².₂₆C₂⁴), C₄.₅₈(C₂×C₄○D₄), (C₂×C₄).₁₄₅(C₂×D₄), SmallGroup(128,1907)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₈₇C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×C₈ — C₈○₂M₄(2) — C₄².₃₈₇C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₈₇C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₈₇C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₈₇C₂³

Generators and relations for C₄².₃₈₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=d²=1, e²=cbc=b^-1, ab=ba, ac=ca, dad=ab², ae=ea, bd=db, be=eb, dcd=a²c, ece^-1=b^-1c, de=ed >

Subgroups: 348 in 181 conjugacy classes, 88 normal (44 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄×C₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₂.D₈, C₂.D₈, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₈○₂M₄(2), C₄×D₈, C₄×Q₁₆, SD₁₆⋊C₄, C₈⋊₇D₄, C₈.₁₈D₄, C₈⋊D₄, C₄².₂₈C₂², C₈.₁₂D₄, C₈⋊₂Q₈, C₂².₃₆C₂⁴, C₄².₃₈₇C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₂₆C₂⁴, D₄○D₈, Q₈○D₈, C₄².₃₈₇C₂³

Smallest permutation representation of C₄².₃₈₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○D₈

Q₈○D₈

kernel

C₄².₃₈₇C₂³

C₈○₂M₄(2)

C₄×D₈

C₄×Q₁₆

SD₁₆⋊C₄

C₈⋊₇D₄

C₈.₁₈D₄

C₈⋊D₄

C₄².₂₈C₂²

C₈.₁₂D₄

C₈⋊₂Q₈

C₂².₃₆C₂⁴

C₂²⋊C₄

C₄⋊C₄

C₈

C₂

# reps

Matrix representation of C₄².₃₈₇C₂³ ►in GL₆(𝔽₁₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	16	0	15	0
0	0	0	16	0	15
0	0	1	0	1	0
0	0	0	1	0	1

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
8	1	0	0	0	0
0	0	3	14	0	0
0	0	14	14	0	0
0	0	0	0	3	14
0	0	0	0	14	14

1	13	0	0	0	0
0	16	0	0	0	0
0	0	0	13	0	9
0	0	4	0	8	0
0	0	0	0	0	4
0	0	0	0	13	0

16	0	0	0	0	0
0	16	0	0	0	0
0	0	3	14	0	0
0	0	3	3	0	0
0	0	0	0	3	14
0	0	0	0	3	3

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

C₄².₃₈₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{387}C_2^3

% in TeX

G:=Group("C4^2.387C2^3");

// GroupNames label

G:=SmallGroup(128,1907);

// by ID

G=gap.SmallGroup(128,1907);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,520,1018,80,4037,124]);

// Polycyclic

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

// generators/relations

G = C42.387C23 order 128 = 27

248th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₈₇C₂³ order 128 = 2⁷

248^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²