C4^2.388C2^3 - GroupNames

G = C₄².₃₈₈C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₈₈C₂³, (C₄×D₈)⋊₄₁C₂, C₈⋊₂D₄⋊₆C₂, C₈⋊₇D₄⋊₃₆C₂, C₈⋊₄D₄⋊₁₉C₂, C₄⋊C₄.₂₄₈D₄, D₈⋊C₄⋊₁₄C₂, C₈.₁₁(C₄○D₄), C₂.₂₅(D₄○D₈), C₂²⋊C₄.₈₈D₄, C₈.₅Q₈⋊₁₈C₂, C₂³.₈₅(C₂×D₄), C₄⋊C₄.₁₁₅C₂³, (C₄×C₈).₂₂₄C₂², (C₂×C₈).₅₆₇C₂³, (C₂×C₄).₃₇₄C₂⁴, (C₂×D₈).₆₄C₂², (C₄×D₄).₉₅C₂², C₈○₂M₄(2)⋊₁₈C₂, C₄.Q₈.₂₆C₂², (C₂×D₄).₁₂₉C₂³, C₄⋊₁D₄.₆₅C₂², C₄⋊D₄.₃₆C₂², C₈⋊C₄.₁₃₁C₂², C₂.D₈.₁₈₅C₂², (C₂²×C₈).₃₀₃C₂², C₂².₆₃₄(C₂²×D₄), C₄².C₂.₂₀C₂², D₄⋊C₄.₂₀₇C₂², C₂².₃₄C₂⁴⋊₃C₂, (C₂²×C₄).₁₀₅₄C₂³, C₄².₂₉C₂²⋊₂₁C₂, C₄²⋊C₂.₃₃₁C₂², (C₂×M₄(2)).₂₈₄C₂², C₂.₇₁(C₂².₂₆C₂⁴), C₄.₅₉(C₂×C₄○D₄), (C₂×C₄).₁₄₆(C₂×D₄), SmallGroup(128,1908)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₈₈C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×C₈ — C₈○₂M₄(2) — C₄².₃₈₈C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₈₈C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₈₈C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₈₈C₂³

Generators and relations for C₄².₃₈₈C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=1, d²=b², e²=cbc=b^-1, ab=ba, ac=ca, dad^-1=ab², ae=ea, bd=db, be=eb, dcd^-1=a²c, ece^-1=b^-1c, de=ed >

Subgroups: 428 in 193 conjugacy classes, 88 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), D₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₄×C₈, C₈⋊C₄, D₄⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄×D₄, C₄⋊D₄, C₄⋊D₄, C₂².D₄, C₄².C₂, C₄⋊₁D₄, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₈○₂M₄(2), C₄×D₈, D₈⋊C₄, C₈⋊₇D₄, C₈⋊₂D₄, C₄².₂₉C₂², C₈⋊₄D₄, C₈.₅Q₈, C₂².₃₄C₂⁴, C₄².₃₈₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₂₆C₂⁴, D₄○D₈, C₄².₃₈₈C₂³

Smallest permutation representation of C₄².₃₈₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

(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)| (1,35,27,10)(2,36,28,11)(3,37,29,12)(4,38,30,13)(5,39,31,14)(6,40,32,15)(7,33,25,16)(8,34,26,9)(17,58,45,54)(18,59,46,55)(19,60,47,56)(20,61,48,49)(21,62,41,50)(22,63,42,51)(23,64,43,52)(24,57,44,53), (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), (1,42)(2,41)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(33,53)(34,52)(35,51)(36,50)(37,49)(38,56)(39,55)(40,54), (1,51,5,55)(2,52,6,56)(3,53,7,49)(4,54,8,50)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,38,21,34)(18,39,22,35)(19,40,23,36)(20,33,24,37)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (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( (1,35,27,10)(2,36,28,11)(3,37,29,12)(4,38,30,13)(5,39,31,14)(6,40,32,15)(7,33,25,16)(8,34,26,9)(17,58,45,54)(18,59,46,55)(19,60,47,56)(20,61,48,49)(21,62,41,50)(22,63,42,51)(23,64,43,52)(24,57,44,53), (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), (1,42)(2,41)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(33,53)(34,52)(35,51)(36,50)(37,49)(38,56)(39,55)(40,54), (1,51,5,55)(2,52,6,56)(3,53,7,49)(4,54,8,50)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,38,21,34)(18,39,22,35)(19,40,23,36)(20,33,24,37)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (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([[(1,35,27,10),(2,36,28,11),(3,37,29,12),(4,38,30,13),(5,39,31,14),(6,40,32,15),(7,33,25,16),(8,34,26,9),(17,58,45,54),(18,59,46,55),(19,60,47,56),(20,61,48,49),(21,62,41,50),(22,63,42,51),(23,64,43,52),(24,57,44,53)], [(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)], [(1,42),(2,41),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,64),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(33,53),(34,52),(35,51),(36,50),(37,49),(38,56),(39,55),(40,54)], [(1,51,5,55),(2,52,6,56),(3,53,7,49),(4,54,8,50),(9,45,13,41),(10,46,14,42),(11,47,15,43),(12,48,16,44),(17,38,21,34),(18,39,22,35),(19,40,23,36),(20,33,24,37),(25,61,29,57),(26,62,30,58),(27,63,31,59),(28,64,32,60)], [(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	2G	2H	4A	···	4F	4G	4H	4I	4J	4K	4L	4M	8A	8B	8C	8D	8E	···	8J
order	1	2	2	2	2	2	2	2	2	4	···	4	4	4	4	4	4	4	4	8	8	8	8	8	···	8
size	1	1	1	1	4	8	8	8	8	2	···	2	4	4	4	8	8	8	8	2	2	2	2	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○D₈

kernel

C₄².₃₈₈C₂³

C₈○₂M₄(2)

C₄×D₈

D₈⋊C₄

C₈⋊₇D₄

C₈⋊₂D₄

C₄².₂₉C₂²

C₈⋊₄D₄

C₈.₅Q₈

C₂².₃₄C₂⁴

C₂²⋊C₄

C₄⋊C₄

C₈

C₂

# reps

Matrix representation of C₄².₃₈₈C₂³ ►in GL₆(𝔽₁₇)

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

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

0	4	0	0	0	0
13	0	0	0	0	0
0	0	4	9	10	0
0	0	0	13	10	7
0	0	7	0	4	9
0	0	7	10	0	13

0	1	0	0	0	0
1	0	0	0	0	0
0	0	7	0	4	9
0	0	0	7	4	13
0	0	4	9	10	0
0	0	4	13	0	10

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	6	0	0
0	0	14	6	0	0
0	0	0	0	0	6
0	0	0	0	14	6

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

C₄².₃₈₈C₂³ in GAP, Magma, Sage, TeX

C_4^2._{388}C_2^3

% in TeX

G:=Group("C4^2.388C2^3");

// GroupNames label

G:=SmallGroup(128,1908);

// by ID

G=gap.SmallGroup(128,1908);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,520,1018,80,4037,124]);

// Polycyclic

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

// generators/relations

G = C42.388C23 order 128 = 27

249th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₈₈C₂³ order 128 = 2⁷

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