C4^2.307C2^3 - GroupNames

G = C₄².₃₀₇C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₀₇C₂³, C₄.₁₂₇2_-¹⁺⁴, C₄.₁₈₀2₊¹⁺⁴, (C₈×D₄)⋊₅₁C₂, C₈⋊₉D₄⋊₄₈C₂, C₈⋊₄Q₈⋊₄₆C₂, C₄⋊D₄.₃₀C₄, C₂²⋊Q₈.₃₀C₄, C₄⋊C₈.₃₇₂C₂², (C₂×C₄).₆₈₉C₂⁴, (C₄×C₈).₃₄₆C₂², (C₂×C₈).₄₄₇C₂³, C₄²⋊₂C₂.₄C₄, C₄².₂₃₇(C₂×C₄), C₄.₄D₄.₂₃C₄, C₂².₅(C₈○D₄), C₄².C₂.₂₃C₄, (C₄×Q₈).₆₂C₂², C₄².₆C₄⋊₅₆C₂, (C₄×D₄).₃₀₆C₂², C₂³.₄₆(C₂²×C₄), C₈⋊C₄.₁₀₆C₂², C₂.₃₇(Q₈○M₄(2)), C₂²⋊C₈.₁₄₉C₂², (C₂×C₄²).₇₉₆C₂², (C₂²×C₈).₄₅₄C₂², C₂².₂₁₁(C₂³×C₄), C₂².D₄.₁₁C₄, (C₂²×C₄).₁₂₈₉C₂³, C₄²⋊C₂.₉₁C₂², C₄².₇C₂²⋊₂₉C₂, C₄².₆C₂²⋊₃₂C₂, (C₂×M₄(2)).₂₅₂C₂², C₂³.₃₆C₂³.₁₆C₂, C₂.₄₇(C₂³.₃₃C₂³), (C₂×C₄⋊C₈)⋊₅₂C₂, C₂.₃₇(C₂×C₈○D₄), C₄⋊C₄.₁₂₄(C₂×C₄), (C₂×D₄).₁₈₅(C₂×C₄), C₂²⋊C₄.₄₅(C₂×C₄), (C₂×C₄).₈₇(C₂²×C₄), (C₂×Q₈).₁₂₉(C₂×C₄), (C₂²×C₄).₃₆₇(C₂×C₄), SmallGroup(128,1724)

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

Derived series

C₁ — C₂² — C₄².₃₀₇C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₂×C₄² — C₂³.₃₆C₂³ — C₄².₃₀₇C₂³

Lower central

C₁ — C₂² — C₄².₃₀₇C₂³

Upper central

C₁ — C₂×C₄ — C₄².₃₀₇C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₀₇C₂³

Generators and relations for C₄².₃₀₇C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=e²=1, c²=b, ab=ba, ac=ca, dad=a^-1b², eae=ab², bc=cb, bd=db, be=eb, cd=dc, ece=a²b²c, ede=a²b²d >

Subgroups: 252 in 178 conjugacy classes, 126 normal (40 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², 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, C₈⋊C₄, C₂²⋊C₈, C₄⋊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, C₂²×C₈, C₂×M₄(2), C₂×C₄⋊C₈, C₄².₆C₂², C₄².₆C₄, C₄².₇C₂², C₈×D₄, C₈⋊₉D₄, C₈⋊₄Q₈, C₂³.₃₆C₂³, C₄².₃₀₇C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, C₈○D₄, C₂³×C₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₃C₂³, C₂×C₈○D₄, Q₈○M₄(2), C₄².₃₀₇C₂³

Smallest permutation representation of C₄².₃₀₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	···	4P	8A	···	8H	8I	···	8T
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	1	1	2	2	4	4	1	1	1	1	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	4	4
type	+	+	+	+	+	+	+	+	+								+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₄	C₄	C₈○D₄	2₊¹⁺⁴	2_-¹⁺⁴	Q₈○M₄(2)
kernel	C₄².₃₀₇C₂³	C₂×C₄⋊C₈	C₄².₆C₂²	C₄².₆C₄	C₄².₇C₂²	C₈×D₄	C₈⋊₉D₄	C₈⋊₄Q₈	C₂³.₃₆C₂³	C₄⋊D₄	C₂²⋊Q₈	C₂².D₄	C₄.₄D₄	C₄².C₂	C₄²⋊₂C₂	C₂²	C₄	C₄	C₂
# reps	1	1	2	1	2	2	4	2	1	2	2	4	2	2	4	8	1	1	2

Matrix representation of C₄².₃₀₇C₂³ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	4	0	0	0	0
0	0	4	1	0	0
0	0	0	13	0	0
0	0	9	8	1	13
0	0	7	9	9	16

13	0	0	0	0	0
0	13	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

9	0	0	0	0	0
0	9	0	0	0	0
0	0	0	0	1	0
0	0	9	8	14	13
0	0	16	0	0	0
0	0	6	12	9	9

16	0	0	0	0	0
0	16	0	0	0	0
0	0	4	1	0	0
0	0	2	13	0	0
0	0	9	8	1	13
0	0	11	9	0	16

0	4	0	0	0	0
13	0	0	0	0	0
0	0	16	4	0	0
0	0	0	1	0	0
0	0	15	2	13	16
0	0	10	3	15	4

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

C₄².₃₀₇C₂³ in GAP, Magma, Sage, TeX

C_4^2._{307}C_2^3

% in TeX

G:=Group("C4^2.307C2^3");

// GroupNames label

G:=SmallGroup(128,1724);

// by ID

G=gap.SmallGroup(128,1724);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,80,124]);

// Polycyclic

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

// generators/relations

G = C42.307C23 order 128 = 27

168th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₀₇C₂³ order 128 = 2⁷

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