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G = C23.33C23order 64 = 26

6th non-split extension by C23 of C23 acting via C23/C22=C2

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

Aliases: C23.33C23, C22.13C24, C42.35C22, C2.22+ 1+4, C2.22- 1+4, D4(C4⋊C4), Q8(C4⋊C4), C4○D45C4, (C4×D4)⋊6C2, D48(C2×C4), (C4×Q8)⋊6C2, Q87(C2×C4), C2.9(C23×C4), C42⋊C27C2, C4⋊C4.81C22, (C2×C4).51C23, C4.21(C22×C4), (C2×D4).78C22, C22.3(C22×C4), (C2×Q8).73C22, C22⋊C4.29C22, (C22×C4).58C22, C4⋊C4(C2×D4), (C2×C4)⋊5(C2×C4), (C2×C4⋊C4)⋊13C2, C22⋊C4(C4⋊C4), (C2×C4○D4).9C2, SmallGroup(64,201)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C23.33C23
C1C2C22C2×C4C22×C4C2×C4○D4 — C23.33C23
C1C2 — C23.33C23
C1C22 — C23.33C23
C1C22 — C23.33C23

Generators and relations for C23.33C23
 G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=c, f2=b, eae=ab=ba, ac=ca, ad=da, af=fa, bc=cb, fdf-1=bd=db, fef-1=be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed >

Subgroups: 185 in 147 conjugacy classes, 121 normal (10 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C2×C4, C2×C4 [×23], C2×C4 [×6], D4 [×12], Q8 [×4], C23 [×3], C42 [×6], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C23.33C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4, 2- 1+4, C23.33C23

Smallest permutation representation of C23.33C23
On 32 points
Generators in S32
(5 20)(6 17)(7 18)(8 19)(21 29)(22 30)(23 31)(24 32)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 32)
(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 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 31)(2 32)(3 29)(4 30)(5 14)(6 15)(7 16)(8 13)(9 19)(10 20)(11 17)(12 18)(21 25)(22 26)(23 27)(24 28)
(1 15 27 11)(2 12 28 16)(3 13 25 9)(4 10 26 14)(5 22 20 30)(6 31 17 23)(7 24 18 32)(8 29 19 21)

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

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

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

C23.33C23 is a maximal subgroup of
C4≀C2⋊C4  C429(C2×C4)  C8.C22⋊C4  C8⋊C22⋊C4  2+ 1+45C4  2- 1+44C4  C4○D4.7Q8  C4○D4.8Q8  C42.275C23  C42.276C23  C42.280C23  C42.281C23  C42.18C23  C42.19C23  C42.22C23  C42.23C23  (C2×D4).301D4  (C2×D4).302D4  (C2×D4).303D4  (C2×D4).304D4  C42.353C23  C42.354C23  C42.358C23  C42.359C23  C22.14C25  C4×2+ 1+4  C4×2- 1+4  C22.77C25  C22.78C25  C22.81C25  C22.82C25  C22.83C25  C22.84C25  C22.93C25  C22.96C25  C22.101C25  C22.102C25  C22.104C25  C22.105C25  C22.110C25  C22.111C25  C22.113C25  C4○D4⋊C12  D5.2+ 1+4
 C2p.2+ 1+4: C22.92C25  C22.95C25  C22.100C25  C22.106C25  C23.144C24  C6.82+ 1+4  C42.91D6  C42.108D6 ...
C23.33C23 is a maximal quotient of
C24.524C23  D44C42  Q84C42  C24.542C23  C24.192C23  C23.199C24  C24.547C23  C23.201C24  C23.202C24  C24.195C23  C4213D4  C24.198C23  C23.211C24  C42.33Q8  C24.203C23  C24.204C23  C23.218C24  C24.205C23  C24.549C23  C23.223C24  C23.225C24  C24.208C23  C23.229C24  D4×C4⋊C4  Q8×C4⋊C4  C23.234C24  C23.236C24  C23.237C24  C23.241C24  C24.558C23  C24.215C23  C24.220C23  C23.252C24  C23.255C24  C24.225C23  C23.259C24  C24.227C23  C23.261C24  C23.264C24  C24.230C23  D5.2+ 1+4
 C42.D2p: C42.159D4  C42.160D4  C42.91D6  C42.108D6  C42.126D6  C42.91D10  C42.108D10  C42.126D10 ...
 C2p.2+ 1+4: C42.697C23  C42.698C23  D48M4(2)  Q87M4(2)  C42.307C23  C42.308C23  C42.309C23  C42.310C23 ...

34 conjugacy classes

class 1 2A2B2C2D···2I4A···4X
order12222···24···4
size11112···22···2

34 irreducible representations

dim111111144
type+++++++-
imageC1C2C2C2C2C2C42+ 1+42- 1+4
kernelC23.33C23C2×C4⋊C4C42⋊C2C4×D4C4×Q8C2×C4○D4C4○D4C2C2
# reps1336211611

Matrix representation of C23.33C23 in GL5(𝔽5)

40000
01000
00100
00340
01404
,
10000
04000
00400
00040
00004
,
40000
04000
00400
00040
00004
,
20000
00300
03000
00034
00002
,
40000
00440
04042
00013
00004
,
10000
00100
04000
00042
00041

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

C23.33C23 in GAP, Magma, Sage, TeX

C_2^3._{33}C_2^3
% in TeX

G:=Group("C2^3.33C2^3");
// GroupNames label

G:=SmallGroup(64,201);
// by ID

G=gap.SmallGroup(64,201);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,188,579,69]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=1,d^2=c,f^2=b,e*a*e=a*b=b*a,a*c=c*a,a*d=d*a,a*f=f*a,b*c=c*b,f*d*f^-1=b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d>;
// generators/relations

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