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G = C23.380C24  order 128 = 27

97th central stem extension by C23 of C24

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

Aliases: C25.44C22, C23.380C24, C24.297C23, C22.1832+ (1+4), (C2×C42)⋊5C22, C24⋊3C4.8C2, C22⋊C4.135D4, C23.181(C2×D4), C2.59(D4⋊5D4), (C22×C4).66C23, (C23×C4).94C22, C23.7Q8⋊51C2, C23.8Q8⋊59C2, C23.Q8⋊21C2, C22⋊2(C42⋊2C2), C23.144(C4○D4), C23.11D4⋊22C2, C22.260(C22×D4), C2.C42⋊55C22, C24.C22⋊59C2, C2.52(C22.19C24), C2.18(C22.32C24), C2.24(C22.45C24), (C2×C4⋊C4)⋊19C22, (C4×C22⋊C4)⋊13C2, (C2×C4).904(C2×D4), (C2×C42⋊2C2)⋊5C2, C2.10(C2×C42⋊2C2), C22.257(C2×C4○D4), (C22×C22⋊C4).23C2, (C2×C22⋊C4).148C22, SmallGroup(128,1212)

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

Derived series
C1 — C23 — C23.380C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.380C24
Lower central
C1 — C23 — C23.380C24
Upper central
C1 — C23 — C23.380C24
Jennings
C1 — C23 — C23.380C24

Subgroups: 724 in 332 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×8], C4 [×14], C22 [×7], C22 [×4], C22 [×48], C2×C4 [×4], C2×C4 [×42], C23, C23 [×10], C23 [×48], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×26], C4⋊C4 [×9], C22×C4 [×12], C22×C4 [×10], C24 [×3], C24 [×12], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×6], C42⋊2C2 [×4], C23×C4 [×2], C25, C4×C22⋊C4, C24⋊3C4 [×2], C23.7Q8, C23.8Q8 [×2], C24.C22 [×3], C23.Q8, C23.11D4 [×3], C22×C22⋊C4, C2×C42⋊2C2, C23.380C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C42⋊2C2 [×4], C22×D4, C2×C4○D4 [×4], 2+ (1+4) [×2], C22.19C24, C2×C42⋊2C2, C22.32C24, D4⋊5D4 [×2], C22.45C24 [×2], C23.380C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=ba=ab, f2=a, ac=ca, ede-1=gdg-1=ad=da, 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, eg=ge, fg=gf >

Smallest permutation representation
►On 32 points
Generators in S32
(1 20)(2 17)(3 18)(4 19)(5 13)(6 14)(7 15)(8 16)(9 32)(10 29)(11 30)(12 31)(21 28)(22 25)(23 26)(24 27)

(1 18)(2 19)(3 20)(4 17)(5 15)(6 16)(7 13)(8 14)(9 30)(10 31)(11 32)(12 29)(21 26)(22 27)(23 28)(24 25)

(1 22)(2 23)(3 24)(4 21)(5 29)(6 30)(7 31)(8 32)(9 16)(10 13)(11 14)(12 15)(17 26)(18 27)(19 28)(20 25)

(1 22)(2 26)(3 24)(4 28)(5 31)(6 9)(7 29)(8 11)(10 15)(12 13)(14 32)(16 30)(17 23)(18 27)(19 21)(20 25)

(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)

(1 6 20 14)(2 31 17 12)(3 8 18 16)(4 29 19 10)(5 28 13 21)(7 26 15 23)(9 24 32 27)(11 22 30 25)

(1 2 3 4)(5 30 7 32)(6 31 8 29)(9 13 11 15)(10 14 12 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)
 
G:=sub<Sym(32)| (1,20)(2,17)(3,18)(4,19)(5,13)(6,14)(7,15)(8,16)(9,32)(10,29)(11,30)(12,31)(21,28)(22,25)(23,26)(24,27), (1,18)(2,19)(3,20)(4,17)(5,15)(6,16)(7,13)(8,14)(9,30)(10,31)(11,32)(12,29)(21,26)(22,27)(23,28)(24,25), (1,22)(2,23)(3,24)(4,21)(5,29)(6,30)(7,31)(8,32)(9,16)(10,13)(11,14)(12,15)(17,26)(18,27)(19,28)(20,25), (1,22)(2,26)(3,24)(4,28)(5,31)(6,9)(7,29)(8,11)(10,15)(12,13)(14,32)(16,30)(17,23)(18,27)(19,21)(20,25), (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), (1,6,20,14)(2,31,17,12)(3,8,18,16)(4,29,19,10)(5,28,13,21)(7,26,15,23)(9,24,32,27)(11,22,30,25), (1,2,3,4)(5,30,7,32)(6,31,8,29)(9,13,11,15)(10,14,12,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)>;
 
G:=Group( (1,20)(2,17)(3,18)(4,19)(5,13)(6,14)(7,15)(8,16)(9,32)(10,29)(11,30)(12,31)(21,28)(22,25)(23,26)(24,27), (1,18)(2,19)(3,20)(4,17)(5,15)(6,16)(7,13)(8,14)(9,30)(10,31)(11,32)(12,29)(21,26)(22,27)(23,28)(24,25), (1,22)(2,23)(3,24)(4,21)(5,29)(6,30)(7,31)(8,32)(9,16)(10,13)(11,14)(12,15)(17,26)(18,27)(19,28)(20,25), (1,22)(2,26)(3,24)(4,28)(5,31)(6,9)(7,29)(8,11)(10,15)(12,13)(14,32)(16,30)(17,23)(18,27)(19,21)(20,25), (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), (1,6,20,14)(2,31,17,12)(3,8,18,16)(4,29,19,10)(5,28,13,21)(7,26,15,23)(9,24,32,27)(11,22,30,25), (1,2,3,4)(5,30,7,32)(6,31,8,29)(9,13,11,15)(10,14,12,16)(17,18,19,20)(21,22,23,24)(25,26,27,28) );
 
G=PermutationGroup([(1,20),(2,17),(3,18),(4,19),(5,13),(6,14),(7,15),(8,16),(9,32),(10,29),(11,30),(12,31),(21,28),(22,25),(23,26),(24,27)], [(1,18),(2,19),(3,20),(4,17),(5,15),(6,16),(7,13),(8,14),(9,30),(10,31),(11,32),(12,29),(21,26),(22,27),(23,28),(24,25)], [(1,22),(2,23),(3,24),(4,21),(5,29),(6,30),(7,31),(8,32),(9,16),(10,13),(11,14),(12,15),(17,26),(18,27),(19,28),(20,25)], [(1,22),(2,26),(3,24),(4,28),(5,31),(6,9),(7,29),(8,11),(10,15),(12,13),(14,32),(16,30),(17,23),(18,27),(19,21),(20,25)], [(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)], [(1,6,20,14),(2,31,17,12),(3,8,18,16),(4,29,19,10),(5,28,13,21),(7,26,15,23),(9,24,32,27),(11,22,30,25)], [(1,2,3,4),(5,30,7,32),(6,31,8,29),(9,13,11,15),(10,14,12,16),(17,18,19,20),(21,22,23,24),(25,26,27,28)])
 

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
440000
004000
000400
000010
000004
,
430000
110000
001000
000400
000001
000010
,
300000
030000
000100
001000
000002
000020
,
430000
110000
001000
000100
000001
000010

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4R4S4T4U4V
order12···22222222244444···44444
size11···12222444422224···48888

38 irreducible representations

dim1111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)
kernelC23.380C24C4×C22⋊C4C24⋊3C4C23.7Q8C23.8Q8C24.C22C23.Q8C23.11D4C22×C22⋊C4C2×C42⋊2C2C22⋊C4C23C22
# reps11212313114162

In GAP, Magma, Sage, TeX 

C_2^3._{380}C_2^4
 
% in TeX
 
G:=Group("C2^3.380C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1212);
 
// by ID
 
G=gap.SmallGroup(128,1212);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,100,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b*a=a*b,f^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^-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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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