C4^2.276C2^3 - GroupNames

G = C₄².₂₇₆C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₂₇₆C₂³, Q₁₆⋊₅(C₂×C₄), C₄.₇₈(C₄×D₄), C₄⋊C₄.₃₉₉D₄, (C₄×Q₁₆)⋊₂₈C₂, SD₁₆⋊₄(C₂×C₄), C₈.C₂²⋊₅C₄, C₈.₃(C₂²×C₄), C₂.₃(Q₈○D₈), Q₁₆⋊C₄⋊₂C₂, (C₄×SD₁₆)⋊₁₄C₂, C₄.₂₄(C₂³×C₄), D₄.₇(C₂²×C₄), C₂².₁₇(C₄×D₄), SD₁₆⋊C₄⋊₂C₂, Q₈.₇(C₂²×C₄), C₄⋊C₄.₃₆₄C₂³, C₈○₂M₄(2)⋊₅C₂, M₄(2)⋊₁₀(C₂×C₄), (C₂×C₄).₂₀₄C₂⁴, (C₂×C₈).₄₁₅C₂³, (C₄×C₈).₁₇₄C₂², C₂²⋊C₄.₁₈₆D₄, C₂.₅(D₄○SD₁₆), (C₄×D₄).₅₇C₂², C₂³.₄₃₆(C₂×D₄), (C₄×Q₈).₅₃C₂², (C₂×D₄).₃₇₃C₂³, (C₂×Q₈).₃₄₆C₂³, M₄(2)⋊C₄⋊₁₁C₂, C₂.D₈.₂₁₃C₂², C₈⋊C₄.₁₁₃C₂², C₄.Q₈.₁₂₇C₂², (C₂²×C₄).₉₂₅C₂³, (C₂²×C₈).₄₄₁C₂², (C₂×Q₁₆).₁₅₃C₂², C₂².₁₄₈(C₂²×D₄), D₄⋊C₄.₂₁₆C₂², C₂³.₃₂C₂³⋊₅C₂, C₄²⋊C₂.₈₁C₂², Q₈⋊C₄.₁₉₇C₂², (C₂×SD₁₆).₁₁₀C₂², C₂³.₂₄D₄.₁₁C₂, (C₂²×Q₈).₂₅₈C₂², (C₂×M₄(2)).₂₆₁C₂², C₂³.₃₃C₂³.₆C₂, C₂.₆₄(C₂×C₄×D₄), (C₂×Q₈)⋊₂₀(C₂×C₄), C₄.₁₂(C₂×C₄○D₄), C₄○D₄.₁₀(C₂×C₄), (C₂×C₄).₉₁₁(C₂×D₄), (C₂×Q₈⋊C₄)⋊₅₃C₂, (C₂×C₄).₇₁(C₂²×C₄), (C₂×C₈.C₂²).₉C₂, (C₂×C₄).₄₇₅(C₄○D₄), (C₂×C₄⋊C₄).₅₇₅C₂², (C₂×C₄○D₄).₈₈C₂², SmallGroup(128,1679)

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

Derived series

C₁ — C₄ — C₄².₂₇₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₄²⋊C₂ — C₂³.₃₂C₂³ — C₄².₂₇₆C₂³

Lower central

C₁ — C₂ — C₄ — C₄².₂₇₆C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₂₇₆C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₇₆C₂³

Generators and relations for C₄².₂₇₆C₂³
G = < a,b,c,d,e | a⁴=b⁴=e²=1, c²=a², d²=b², ab=ba, ac=ca, ad=da, eae=ab², cbc^-1=dbd^-1=b^-1, be=eb, dcd^-1=bc, ce=ec, de=ed >

Subgroups: 364 in 235 conjugacy classes, 140 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₄×C₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₄×Q₈, C₄×Q₈, C₂²×C₈, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂²×Q₈, C₂×C₄○D₄, C₈○₂M₄(2), C₂×Q₈⋊C₄, C₂³.₂₄D₄, M₄(2)⋊C₄, C₄×SD₁₆, C₄×Q₁₆, SD₁₆⋊C₄, Q₁₆⋊C₄, C₂³.₃₂C₂³, C₂³.₃₃C₂³, C₂×C₈.C₂², C₄².₂₇₆C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²×C₄, C₂×D₄, C₄○D₄, C₂⁴, C₄×D₄, C₂³×C₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄×D₄, D₄○SD₁₆, Q₈○D₈, C₄².₂₇₆C₂³

Smallest permutation representation of C₄².₂₇₆C₂³
►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 15 27 23)(2 16 28 24)(3 13 25 21)(4 14 26 22)(5 11 35 31)(6 12 36 32)(7 9 33 29)(8 10 34 30)(17 42 49 62)(18 43 50 63)(19 44 51 64)(20 41 52 61)(37 53 57 45)(38 54 58 46)(39 55 59 47)(40 56 60 48)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4L	4M	···	4Z	8A	8B	8C	8D	8E	···	8J
order	1	2	2	2	2	2	2	2	4	···	4	4	···	4	8	8	8	8	8	···	8
size	1	1	1	1	2	2	4	4	2	···	2	4	···	4	2	2	2	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+			-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	D₄	D₄	C₄○D₄	D₄○SD₁₆	Q₈○D₈
kernel	C₄².₂₇₆C₂³	C₈○₂M₄(2)	C₂×Q₈⋊C₄	C₂³.₂₄D₄	M₄(2)⋊C₄	C₄×SD₁₆	C₄×Q₁₆	SD₁₆⋊C₄	Q₁₆⋊C₄	C₂³.₃₂C₂³	C₂³.₃₃C₂³	C₂×C₈.C₂²	C₈.C₂²	C₂²⋊C₄	C₄⋊C₄	C₂×C₄	C₂	C₂
# reps	1	1	1	1	1	2	2	2	2	1	1	1	16	2	2	4	2	2

Matrix representation of C₄².₂₇₆C₂³ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	13	0	0	0	0
0	0	7	0	0	16
0	0	0	7	1	0
0	0	0	1	10	0
0	0	16	0	0	10

16	0	0	0	0	0
0	16	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

4	0	0	0	0	0
12	13	0	0	0	0
0	0	0	10	16	0
0	0	10	0	0	1
0	0	16	0	0	10
0	0	0	1	10	0

6	13	0	0	0	0
13	11	0	0	0	0
0	0	4	4	3	14
0	0	4	13	14	14
0	0	3	14	4	4
0	0	14	14	4	13

16	0	0	0	0	0
0	16	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

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

C₄².₂₇₆C₂³ in GAP, Magma, Sage, TeX

C_4^2._{276}C_2^3

% in TeX

G:=Group("C4^2.276C2^3");

// GroupNames label

G:=SmallGroup(128,1679);

// by ID

G=gap.SmallGroup(128,1679);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,521,2804,1411,172]);

// Polycyclic

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

// generators/relations

G = C42.276C23 order 128 = 27

137th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₇₆C₂³ order 128 = 2⁷

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