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G = C23.324C24order 128 = 27

41st central stem extension by C23 of C24

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

Aliases: C24.6C23, C25.39C22, C23.324C24, C22.1372+ 1+4, C2.12D42, (C2×D4)⋊31D4, C22⋊C435D4, C232D49C2, C221(C4⋊D4), (C23×C4)⋊22C22, (C2×C42)⋊21C22, C23.158(C2×D4), (C22×D4)⋊5C22, C2.24(D45D4), C23.Q85C2, C23.8Q833C2, C23.229(C4○D4), C23.23D431C2, C23.10D415C2, (C22×C4).794C23, C22.204(C22×D4), C2.C4222C22, C24.3C2232C2, C24.C2234C2, C2.11(C22.32C24), C2.24(C22.19C24), (C2×C4×D4)⋊26C2, (C2×C4⋊D4)⋊7C2, (C2×C22≀C2)⋊6C2, (C2×C4).313(C2×D4), C2.22(C2×C4⋊D4), (C2×C4⋊C4)⋊110C22, (C2×C22⋊C4)⋊16C22, (C22×C22⋊C4)⋊21C2, C22.203(C2×C4○D4), SmallGroup(128,1156)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.324C24
C1C2C22C23C24C25C22×C22⋊C4 — C23.324C24
C1C23 — C23.324C24
C1C23 — C23.324C24
C1C23 — C23.324C24

Generators and relations for C23.324C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=b, ab=ba, ac=ca, ede-1=gdg=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 >

Subgroups: 1044 in 461 conjugacy classes, 120 normal (82 characteristic)
C1, C2 [×7], C2 [×11], C4 [×15], C22 [×7], C22 [×4], C22 [×65], C2×C4 [×10], C2×C4 [×37], D4 [×32], C23, C23 [×12], C23 [×61], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×25], C4⋊C4 [×7], C22×C4 [×10], C22×C4 [×15], C2×D4 [×4], C2×D4 [×35], C24 [×5], C24 [×12], C2.C42 [×4], C2×C42, C2×C22⋊C4 [×15], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C4×D4 [×4], C22≀C2 [×8], C4⋊D4 [×8], C23×C4 [×3], C22×D4 [×7], C25, C23.8Q8, C23.23D4 [×2], C24.C22, C24.3C22, C232D4, C23.10D4 [×2], C23.Q8, C22×C22⋊C4, C2×C4×D4, C2×C22≀C2 [×2], C2×C4⋊D4 [×2], C23.324C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C4⋊D4 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C4⋊D4, C22.19C24, C22.32C24, D42 [×2], D45D4 [×2], C23.324C24

Smallest permutation representation of C23.324C24
On 32 points
Generators in S32
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(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)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)
(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 9 3 11)(2 6 4 8)(5 21 7 23)(10 22 12 24)(13 18 15 20)(14 32 16 30)(17 26 19 28)(25 31 27 29)
(1 3)(5 18)(6 17)(7 20)(8 19)(9 31)(10 30)(11 29)(12 32)(13 15)(21 23)(25 27)
(1 25)(2 14)(3 27)(4 16)(5 18)(6 32)(7 20)(8 30)(9 31)(10 19)(11 29)(12 17)(13 23)(15 21)(22 28)(24 26)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2Q2R4A4B4C4D4E···4P4Q4R4S
order12···222222···2244444···4444
size11···122224···4822224···4888

38 irreducible representations

dim1111111111112224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC23.324C24C23.8Q8C23.23D4C24.C22C24.3C22C232D4C23.10D4C23.Q8C22×C22⋊C4C2×C4×D4C2×C22≀C2C2×C4⋊D4C22⋊C4C2×D4C23C22
# reps1121112111228482

Matrix representation of C23.324C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
400000
000100
001000
000040
000004
,
300000
020000
004000
000100
000043
000001
,
400000
010000
001000
000100
000010
000044
,
400000
010000
001000
000400
000040
000004

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

C23.324C24 in GAP, Magma, Sage, TeX

C_2^3._{324}C_2^4
% in TeX

G:=Group("C2^3.324C2^4");
// GroupNames label

G:=SmallGroup(128,1156);
// by ID

G=gap.SmallGroup(128,1156);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,232,758,723,675]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=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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