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G = C23.377C24  order 128 = 27

94th central stem extension by C23 of C24

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

Aliases: C23.377C24, C24.294C23, C22.1812+ (1+4), C22.1342- (1+4), C4⋊C4⋊45D4, C23⋊Q8⋊14C2, C2.58(D4⋊5D4), C2.31(Q8⋊5D4), C23.38(C4○D4), (C2×C42).36C22, C23.8Q8⋊57C2, C23.11D4⋊21C2, (C23×C4).365C22, (C22×C4).823C23, C22.257(C22×D4), C24.C22⋊57C2, C23.23D4.22C2, C23.10D4.10C2, (C22×D4).142C22, (C22×Q8).113C22, C23.63C23⋊55C2, C23.83C23⋊14C2, C23.67C23⋊47C2, C2.49(C22.19C24), C2.22(C22.45C24), C2.C42.133C22, C2.43(C23.36C23), C2.26(C22.36C24), C2.28(C22.46C24), (C4×C22⋊C4)⋊71C2, (C2×C4).345(C2×D4), (C2×C22⋊Q8)⋊17C2, (C2×C42⋊C2)⋊26C2, (C2×C4).372(C4○D4), (C2×C4⋊C4).853C22, C22.254(C2×C4○D4), (C2×C22⋊C4).146C22, SmallGroup(128,1209)

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

Derived series
C1 — C23 — C23.377C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.377C24
Lower central
C1 — C23 — C23.377C24
Upper central
C1 — C23 — C23.377C24
Jennings
C1 — C23 — C23.377C24

Subgroups: 516 in 270 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×18], C22 [×7], C22 [×20], C2×C4 [×10], C2×C4 [×46], D4 [×4], Q8 [×4], C23, C23 [×4], C23 [×12], C42 [×5], C22⋊C4 [×19], C4⋊C4 [×4], C4⋊C4 [×9], C22×C4 [×13], C22×C4 [×11], C2×D4 [×5], C2×Q8 [×5], C24 [×2], C2.C42 [×12], C2×C42 [×3], C2×C22⋊C4 [×10], C2×C4⋊C4 [×6], C42⋊C2 [×4], C22⋊Q8 [×4], C23×C4 [×2], C22×D4, C22×Q8, C4×C22⋊C4, C23.8Q8, C23.23D4 [×2], C23.63C23 [×2], C24.C22 [×2], C23.67C23, C23⋊Q8, C23.10D4, C23.11D4, C23.83C23, C2×C42⋊C2, C2×C22⋊Q8, C23.377C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C22.19C24, C23.36C23, C22.36C24, D4⋊5D4, Q8⋊5D4, C22.45C24, C22.46C24, C23.377C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=a, e2=b, g2=ba=ab, 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 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 33)(2 34)(3 35)(4 36)(5 30)(6 31)(7 32)(8 29)(9 56)(10 53)(11 54)(12 55)(13 43)(14 44)(15 41)(16 42)(17 52)(18 49)(19 50)(20 51)(21 47)(22 48)(23 45)(24 46)(25 63)(26 64)(27 61)(28 62)(37 57)(38 58)(39 59)(40 60)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
300000
220000
004000
000400
000040
000001
,
240000
330000
001000
000400
000020
000002
,
400000
040000
000100
001000
000001
000010
,
430000
110000
004000
000400
000020
000002

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[2,3,0,0,0,0,4,3,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.377C24C4×C22⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C23.67C23C23⋊Q8C23.10D4C23.11D4C23.83C23C2×C42⋊C2C2×C22⋊Q8C4⋊C4C2×C4C23C22C22
# reps111222111111148811

In GAP, Magma, Sage, TeX 

C_2^3._{377}C_2^4
 
% in TeX
 
G:=Group("C2^3.377C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1209);
 
// by ID
 
G=gap.SmallGroup(128,1209);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=a,e^2=b,g^2=b*a=a*b,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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