C4^2.308C2^3 - GroupNames

G = C₄².₃₀₈C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₀₈C₂³, C₄.₁₈₁2₊¹⁺⁴, C₄.₁₂₈2_-¹⁺⁴, (C₈×D₄)⋊₅₂C₂, (C₈×Q₈)⋊₃₇C₂, C₈⋊₉D₄⋊₄₉C₂, C₈⋊₆D₄⋊₄₇C₂, C₈⋊₄Q₈⋊₄₇C₂, C₄.₂₃(C₈○D₄), C₄⋊D₄.₃₁C₄, C₂²⋊Q₈.₃₁C₄, C₄⋊C₈.₂₄₁C₂², (C₄×C₈).₃₄₇C₂², (C₂×C₄).₆₉₀C₂⁴, (C₂×C₈).₄₄₈C₂³, C₄²⋊₂C₂.₅C₄, C₄².₂₃₈(C₂×C₄), C₄.₄D₄.₂₄C₄, (C₄×D₄).₆₅C₂², C₄².C₂.₂₄C₄, C₄⋊M₄(2)⋊₃₉C₂, C₂³.₄₇(C₂²×C₄), (C₄×Q₈).₂₈₇C₂², C₄².₁₂C₄⋊₅₉C₂, C₈⋊C₄.₁₀₇C₂², C₂²⋊C₈.₁₅₀C₂², C₂.₃₈(Q₈○M₄(2)), C₂².₂₁₂(C₂³×C₄), (C₂²×C₄).₉₅₀C₂³, (C₂×C₄²).₇₉₇C₂², (C₂²×C₈).₄₅₅C₂², C₂².D₄.₁₂C₄, C₄².₆C₂²⋊₃₃C₂, C₄².₇C₂²⋊₃₀C₂, C₄²⋊C₂.₉₂C₂², (C₂×M₄(2)).₂₅₃C₂², C₂³.₃₆C₂³.₁₇C₂, C₂.₄₈(C₂³.₃₃C₂³), C₂.₃₈(C₂×C₈○D₄), C₄⋊C₄.₁₇₁(C₂×C₄), (C₂×D₄).₁₄₆(C₂×C₄), C₂²⋊C₄.₂₆(C₂×C₄), (C₂×C₄).₈₈(C₂²×C₄), (C₂×Q₈).₁₆₈(C₂×C₄), (C₂²×C₄).₃₆₈(C₂×C₄), SmallGroup(128,1725)

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

Derived series

C₁ — C₂² — C₄².₃₀₈C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₂×C₄² — C₂³.₃₆C₂³ — C₄².₃₀₈C₂³

Lower central

C₁ — C₂² — C₄².₃₀₈C₂³

Upper central

C₁ — C₂×C₄ — C₄².₃₀₈C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₀₈C₂³

Generators and relations for C₄².₃₀₈C₂³
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, dcd=b²c, ece=a²b²c, ede=a²b²d >

Subgroups: 252 in 177 conjugacy classes, 126 normal (52 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, 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, 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, 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), C₄².₆C₂², C₄².₁₂C₄, C₄².₇C₂², C₈×D₄, C₈⋊₉D₄, C₈⋊₆D₄, C₈⋊₆D₄, C₈×Q₈, C₈⋊₄Q₈, C₂³.₃₆C₂³, C₄².₃₀₈C₂³
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), C₄².₃₀₈C₂³

Smallest permutation representation of C₄².₃₀₈C₂³
►On 64 points

Generators in S₆₄

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

(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)

(2 6)(4 8)(9 60)(10 57)(11 62)(12 59)(13 64)(14 61)(15 58)(16 63)(18 22)(20 24)(26 30)(28 32)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)(50 54)(52 56)

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

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

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

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

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	1	1	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+								+	-
image	C₁	C₂	C₂	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	C₄².₃₀₈C₂³	C₄⋊M₄(2)	C₄².₆C₂²	C₄².₁₂C₄	C₄².₇C₂²	C₈×D₄	C₈⋊₉D₄	C₈⋊₆D₄	C₈×Q₈	C₈⋊₄Q₈	C₂³.₃₆C₂³	C₄⋊D₄	C₂²⋊Q₈	C₂².D₄	C₄.₄D₄	C₄².C₂	C₄²⋊₂C₂	C₄	C₄	C₄	C₂
# reps	1	1	2	1	2	1	2	3	1	1	1	2	2	4	2	2	4	8	1	1	2

Matrix representation of C₄².₃₀₈C₂³ ►in GL₆(𝔽₁₇)

0	2	0	0	0	0
9	0	0	0	0	0
0	0	0	4	0	0
0	0	4	0	0	0
0	0	15	10	4	13
0	0	8	8	0	13

4	0	0	0	0	0
0	4	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

0	4	0	0	0	0
1	0	0	0	0	0
0	0	7	13	15	0
0	0	14	2	15	2
0	0	14	16	14	13
0	0	2	16	13	11

1	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	4	1	0
0	0	3	4	2	16

0	8	0	0	0	0
15	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	5	13	16	1
0	0	9	8	0	1

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

C₄².₃₀₈C₂³ in GAP, Magma, Sage, TeX

C_4^2._{308}C_2^3

% in TeX

G:=Group("C4^2.308C2^3");

// GroupNames label

G:=SmallGroup(128,1725);

// by ID

G=gap.SmallGroup(128,1725);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,1018,80,124]);

// 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,d*c*d=b^2*c,e*c*e=a^2*b^2*c,e*d*e=a^2*b^2*d>;

// generators/relations

G = C42.308C23 order 128 = 27

169th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₀₈C₂³ order 128 = 2⁷

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