C4^2.354C2^3 - GroupNames

G = C₄².₃₅₄C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₃₅₄C₂³, C₄⋊C₄.₃₄₇D₄, (C₄×Q₁₆)⋊₂₂C₂, (C₄×SD₁₆)⋊₇C₂, Q₈.Q₈⋊₂₀C₂, D₄.Q₈⋊₂₀C₂, Q₁₆⋊C₄⋊₉C₂, C₄⋊₂Q₁₆⋊₂₄C₂, D₄.D₄⋊₇C₂, C₄⋊C₈.₅₅C₂², C₄⋊C₄.₇₃C₂³, (C₂×C₈).₄₇C₂³, C₂.₁₇(Q₈○D₈), D₄.₁₁(C₄○D₄), (C₄×C₈).₁₁₁C₂², (C₂×C₄).₃₁₈C₂⁴, Q₈.₁₀(C₄○D₄), C₂²⋊C₄.₁₄₈D₄, D₄.₇D₄.₂C₂, (C₄×D₄).₈₁C₂², C₂³.₂₅₇(C₂×D₄), C₄⋊Q₈.₁₀₅C₂², SD₁₆⋊C₄⋊₁₄C₂, (C₂×Q₈).₈₁C₂³, (C₄×Q₈).₇₈C₂², C₈⋊C₄.₁₂C₂², C₂.D₈.₉₁C₂², C₂.₂₈(D₄○SD₁₆), (C₂×D₄).₄₀₉C₂³, C₂³.₄₇D₄⋊₆C₂, C₂²⋊C₈.₃₁C₂², (C₂×Q₁₆).₆₀C₂², C₄.Q₈.₁₅₅C₂², C₂²⋊Q₈.₂₇C₂², D₄⋊C₄.₃₆C₂², C₂³.₄₈D₄⋊₁₇C₂, C₄².₇C₂²⋊₃C₂, (C₂²×C₄).₂₉₁C₂³, Q₈⋊C₄.₃₆C₂², C₂².₅₇₈(C₂²×D₄), C₄².C₂.₁₃C₂², C₂².₃₅C₂⁴⋊₁C₂, (C₂×SD₁₆).₁₄₃C₂², C₄²⋊C₂.₁₂₉C₂², C₂.₁₁₉(C₂².₁₉C₂⁴), C₂³.₃₃C₂³.₁₀C₂, C₄.₂₀₃(C₂×C₄○D₄), (C₂×C₄).₅₀₂(C₂×D₄), (C₂×C₄⋊C₄).₆₁₅C₂², (C₂×C₄○D₄).₁₄₄C₂², SmallGroup(128,1852)

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

Derived series

C₁ — C₂×C₄ — C₄².₃₅₄C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄² — C₄×D₄ — C₂³.₃₃C₂³ — C₄².₃₅₄C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₃₅₄C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₃₅₄C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₃₅₄C₂³

Generators and relations for C₄².₃₅₄C₂³
G = < a,b,c,d,e | a⁴=b⁴=e²=1, c²=d²=b², ab=ba, ac=ca, dad^-1=ab², ae=ea, cbc^-1=ebe=b^-1, bd=db, dcd^-1=a²c, ece=bc, de=ed >

Subgroups: 324 in 186 conjugacy classes, 88 normal (84 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂×SD₁₆, C₂×Q₁₆, C₂×C₄○D₄, C₄².₇C₂², C₄×SD₁₆, C₄×Q₁₆, SD₁₆⋊C₄, Q₁₆⋊C₄, D₄.₇D₄, D₄.D₄, C₄⋊₂Q₁₆, D₄.Q₈, Q₈.Q₈, C₂³.₄₇D₄, C₂³.₄₈D₄, C₂³.₃₃C₂³, C₂².₃₅C₂⁴, C₄².₃₅₄C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, C₂².₁₉C₂⁴, D₄○SD₁₆, Q₈○D₈, C₄².₃₅₄C₂³

Smallest permutation representation of C₄².₃₅₄C₂³
►On 64 points

Generators in S₆₄

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

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

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

(5 10)(6 11)(7 12)(8 9)(13 61)(14 62)(15 63)(16 64)(17 23)(18 24)(19 21)(20 22)(33 38)(34 39)(35 40)(36 37)(45 54)(46 55)(47 56)(48 53)(49 58)(50 59)(51 60)(52 57)

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	···	4F	4G	···	4O	4P	4Q	4R	4S	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	2	2	4	···	4	4	···	4	4	4	4	4	8	8	8	8	8	8
size	1	1	1	1	4	4	4	2	···	2	4	···	4	8	8	8	8	4	4	4	4	8	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+				-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	C₄○D₄	C₄○D₄	D₄○SD₁₆	Q₈○D₈
kernel	C₄².₃₅₄C₂³	C₄².₇C₂²	C₄×SD₁₆	C₄×Q₁₆	SD₁₆⋊C₄	Q₁₆⋊C₄	D₄.₇D₄	D₄.D₄	C₄⋊₂Q₁₆	D₄.Q₈	Q₈.Q₈	C₂³.₄₇D₄	C₂³.₄₈D₄	C₂³.₃₃C₂³	C₂².₃₅C₂⁴	C₂²⋊C₄	C₄⋊C₄	D₄	Q₈	C₂	C₂
# reps	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	2	2	4	4	2	2

Matrix representation of C₄².₃₅₄C₂³ ►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	1	0	0	0
0	0	0	1	0	0

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

8	10	0	0	0	0
9	9	0	0	0	0
0	0	5	12	0	0
0	0	12	12	0	0
0	0	0	0	5	12
0	0	0	0	12	12

1	2	0	0	0	0
0	16	0	0	0	0
0	0	16	0	6	0
0	0	0	16	0	6
0	0	11	0	1	0
0	0	0	11	0	1

16	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	0	1	0
0	0	0	0	0	16

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

C₄².₃₅₄C₂³ in GAP, Magma, Sage, TeX

C_4^2._{354}C_2^3

% in TeX

G:=Group("C4^2.354C2^3");

// GroupNames label

G:=SmallGroup(128,1852);

// by ID

G=gap.SmallGroup(128,1852);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,1018,80,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.354C23 order 128 = 27

215th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₃₅₄C₂³ order 128 = 2⁷

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