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G = C23.614C24  order 128 = 27

331st central stem extension by C23 of C24

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

Aliases: C23.614C24, C24.413C23, C22.3882+ (1+4), C4⋊C4⋊17D4, C23⋊2D4⋊43C2, C2.31(Q8⋊6D4), C23.83(C4○D4), C2.119(D4⋊5D4), C23.11D4⋊95C2, C23.10D4⋊96C2, C23.23D4⋊99C2, (C23×C4).469C22, (C2×C42).665C22, (C22×C4).192C23, C23.8Q8⋊115C2, C22.423(C22×D4), C24.3C22⋊88C2, (C22×D4).248C22, C24.C22⋊141C2, C2.72(C22.32C24), C2.63(C22.29C24), C23.63C23⋊141C2, C2.20(C22.54C24), C2.C42.320C22, C2.88(C22.47C24), (C2×C4⋊D4)⋊38C2, (C2×C4).112(C2×D4), (C2×C42⋊2C2)⋊22C2, (C2×C4).201(C4○D4), (C2×C4⋊C4).427C22, C22.476(C2×C4○D4), (C2×C22⋊C4).279C22, SmallGroup(128,1446)

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

Derived series
C1 — C23 — C23.614C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C24.3C22 — C23.614C24
Lower central
C1 — C23 — C23.614C24
Upper central
C1 — C23 — C23.614C24
Jennings
C1 — C23 — C23.614C24

Subgroups: 692 in 299 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×15], C22 [×7], C22 [×31], C2×C4 [×8], C2×C4 [×33], D4 [×24], C23, C23 [×2], C23 [×27], C42 [×3], C22⋊C4 [×22], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×11], C22×C4 [×4], C2×D4 [×25], C24 [×4], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×15], C2×C4⋊C4 [×5], C4⋊D4 [×4], C42⋊2C2 [×4], C23×C4, C22×D4 [×6], C23.8Q8, C23.23D4, C23.63C23, C24.C22, C24.3C22 [×3], C23⋊2D4 [×3], C23.10D4 [×2], C23.11D4, C2×C4⋊D4, C2×C42⋊2C2, C23.614C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×4], C22.29C24, C22.32C24 [×2], D4⋊5D4, Q8⋊6D4, C22.47C24, C22.54C24, C23.614C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=g2=a, f2=ba=ab, ac=ca, ede=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
200000
030000
000200
003000
000040
000004
,
040000
400000
001000
000100
000013
000004
,
200000
020000
000200
002000
000040
000041
,
040000
100000
000100
001000
000010
000001

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4P4Q4R4S
order12···2222224···4444
size11···1448884···4888

32 irreducible representations

dim111111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)
kernelC23.614C24C23.8Q8C23.23D4C23.63C23C24.C22C24.3C22C23⋊2D4C23.10D4C23.11D4C2×C4⋊D4C2×C42⋊2C2C4⋊C4C2×C4C23C22
# reps111113321114444

In GAP, Magma, Sage, TeX 

C_2^3._{614}C_2^4
 
% in TeX
 
G:=Group("C2^3.614C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1446);
 
// by ID
 
G=gap.SmallGroup(128,1446);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,268,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=g^2=a,f^2=b*a=a*b,a*c=c*a,e*d*e=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*g=g*f>;
 
// generators/relations
 

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