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G = C23.397C24  order 128 = 27

114th central stem extension by C23 of C24

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

Aliases: C23.397C24, C24.307C23, C22.1962+ (1+4), C22⋊C4.7Q8, C42⋊9C4⋊20C2, C23.18(C2×Q8), C2.20(D4⋊3Q8), (C2×C42).46C22, (C22×C4).77C23, C4.28(C42⋊2C2), C23.Q8.7C2, C22.86(C22×Q8), (C23×C4).381C22, C23.7Q8.48C2, C23.65C23⋊73C2, C23.83C23⋊24C2, C23.63C23⋊64C2, C24.C22.22C2, C2.C42.149C22, C2.17(C22.34C24), C2.35(C22.47C24), C2.15(C22.49C24), C2.20(C23.37C23), (C4×C4⋊C4)⋊74C2, (C2×C4).39(C2×Q8), (C4×C22⋊C4).50C2, (C2×C4).125(C4○D4), (C2×C4⋊C4).266C22, C2.16(C2×C42⋊2C2), C22.274(C2×C4○D4), (C2×C22⋊C4).499C22, SmallGroup(128,1229)

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

Derived series
C1 — C23 — C23.397C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.397C24
Lower central
C1 — C23 — C23.397C24
Upper central
C1 — C23 — C23.397C24
Jennings
C1 — C23 — C23.397C24

Subgroups: 388 in 214 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×16], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], C23, C23 [×2], C23 [×6], C42 [×6], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×18], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C24, C2.C42 [×2], C2.C42 [×10], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×8], C23×C4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8 [×2], C42⋊9C4, C23.63C23 [×2], C24.C22 [×2], C23.65C23 [×2], C23.Q8 [×2], C23.83C23 [×2], C23.397C24

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C42⋊2C2 [×4], C22×Q8, C2×C4○D4 [×4], 2+ (1+4) [×2], C2×C42⋊2C2, C23.37C23, C22.34C24, C22.47C24, D4⋊3Q8 [×2], C22.49C24, C23.397C24

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

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

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

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

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

(2 9)(4 11)(5 21)(6 40)(7 23)(8 38)(14 32)(16 30)(17 51)(18 61)(19 49)(20 63)(22 25)(24 27)(26 37)(28 39)(33 62)(34 50)(35 64)(36 52)(41 54)(43 56)(46 57)(48 59)

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
200000
030000
003300
004200
000020
000002
,
040000
100000
004000
002100
000043
000001
,
100000
010000
001000
003400
000010
000044
,
010000
400000
002000
001300
000010
000001

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim1111111111224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2Q8C4○D42+ (1+4)
kernelC23.397C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C42⋊9C4C23.63C23C24.C22C23.65C23C23.Q8C23.83C23C22⋊C4C2×C4C22
# reps11121222224162

In GAP, Magma, Sage, TeX 

C_2^3._{397}C_2^4
 
% in TeX
 
G:=Group("C2^3.397C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1229);
 
// by ID
 
G=gap.SmallGroup(128,1229);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=a*b*c,e^2=b*a=a*b,g^2=a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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