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G = C23.226C24  order 128 = 27

79th central extension by C23 of C24

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

Aliases: C23.226C24, C24.553C23, C22.452- 1+4, C22.66(C4×D4), C4⋊2(C42⋊C2), C2.2(D4⋊6D4), C22⋊C4.183D4, C23.414(C2×D4), C23.289(C4○D4), C23.216(C22×C4), (C2×C42).427C22, C22.117(C23×C4), (C23×C4).300C22, (C22×C4).755C23, C22.104(C22×D4), C23.7Q8.29C2, C23.34D4.11C2, C23.65C23⋊18C2, C23.63C23⋊13C2, C2.C42.472C22, C2.2(C22.46C24), C2.14(C23.32C23), (C4×C4⋊C4)⋊32C2, (C2×C4⋊C4)⋊36C4, C2.30(C2×C4×D4), C22⋊C4○3(C4⋊C4), C4⋊C4.206(C2×C4), (C2×C4).885(C2×D4), (C22×C4⋊C4).27C2, (C4×C22⋊C4).25C2, (C2×C4).789(C4○D4), (C2×C4⋊C4).973C22, (C22×C4).309(C2×C4), (C2×C4).231(C22×C4), C2.27(C2×C42⋊C2), C22.111(C2×C4○D4), (C2×C42⋊C2).31C2, (C2×C22⋊C4).436C22, SmallGroup(128,1076)

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

Derived series
C1 — C22 — C23.226C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4⋊C4 — C23.226C24
Lower central
C1 — C22 — C23.226C24
Upper central
C1 — C23 — C23.226C24
Jennings
C1 — C23 — C23.226C24

Generators and relations for C23.226C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=c, g2=b, ab=ba, eae-1=ac=ca, ad=da, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 460 in 302 conjugacy classes, 156 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23×C4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C23.34D4, C23.63C23, C23.65C23, C22×C4⋊C4, C2×C42⋊C2, C23.226C24
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C42⋊C2, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2- 1+4, C2×C42⋊C2, C2×C4×D4, C23.32C23, D4⋊6D4, C22.46C24, C23.226C24

Smallest permutation representation of C23.226C24
►On 64 points
Generators in S64
(1 3)(2 50)(4 52)(5 34)(6 8)(7 36)(9 11)(10 24)(12 22)(13 15)(14 28)(16 26)(17 19)(18 32)(20 30)(21 23)(25 27)(29 31)(33 35)(37 39)(38 64)(40 62)(41 43)(42 56)(44 54)(45 47)(46 60)(48 58)(49 51)(53 55)(57 59)(61 63)

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

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4T4U···4AL
order12···222224···44···4
size11···122222···24···4

50 irreducible representations

dim11111111112224
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4D4C4○D4C4○D42- 1+4
kernelC23.226C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.34D4C23.63C23C23.65C23C22×C4⋊C4C2×C42⋊C2C2×C4⋊C4C22⋊C4C2×C4C23C22
# reps112124212164842

Matrix representation of C23.226C24 ►in GL6(𝔽5)

100000
040000
004000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
010000
100000
000100
004000
000014
000004
,
200000
020000
003000
000300
000023
000043
,
100000
010000
004000
000400
000032
000002

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,4,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,4,0,0,0,0,3,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,2,2] >;
 

C23.226C24 in GAP, Magma, Sage, TeX 

C_2^3._{226}C_2^4
 
% in TeX
 
G:=Group("C2^3.226C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1076);
 
// by ID
 
G=gap.SmallGroup(128,1076);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c,g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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