C4xC3^2:2C8 - GroupNames

G = C₄×C₃²⋊₂C₈ order 288 = 2⁵·3²

Direct product of C₄ and C₃²⋊₂C₈

direct product, metabelian, soluble, monomial, A-group

Aliases: C₄×C₃²⋊₂C₈, (C₃×C₁₂)⋊₃C₈, C₃²⋊₃(C₄×C₈), (C₆×C₁₂).₄C₄, C₃⋊Dic₃⋊₆C₈, C₆².₃(C₂×C₄), (C₃×C₆).₃C₄², C₂.₂(C₄×C₃²⋊C₄), (C₃×C₆).₂₄(C₂×C₈), C₂.₂(C₂×C₃²⋊₂C₈), C₂.₂(C₃⋊S₃⋊₃C₈), C₂².₈(C₂×C₃²⋊C₄), (C₂×C₄).₁₀(C₃²⋊C₄), C₃⋊Dic₃.₂₇(C₂×C₄), (C₂×C₃⋊Dic₃).₁₃C₄, (C₄×C₃⋊Dic₃).₁₄C₂, (C₂×C₃²⋊₂C₈).₁₁C₂, (C₂×C₃⋊Dic₃).₁₀₇C₂², SmallGroup(288,423)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₄×C₃²⋊₂C₈

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₂×C₃⋊Dic₃ — C₂×C₃²⋊₂C₈ — C₄×C₃²⋊₂C₈

Lower central

C₃² — C₄×C₃²⋊₂C₈

Upper central

C₁ — C₂×C₄

Generators and relations for C₄×C₃²⋊₂C₈
G = < a,b,c,d | a⁴=b³=c³=d⁸=1, ab=ba, ac=ca, ad=da, dcd^-1=bc=cb, dbd^-1=b^-1c >

Subgroups: 272 in 74 conjugacy classes, 30 normal (16 characteristic)
C₁, C₂, C₃, C₄, C₄, C₂², C₆, C₈, C₂×C₄, C₂×C₄, C₃², Dic₃, C₁₂, C₂×C₆, C₄², C₂×C₈, C₃×C₆, C₂×Dic₃, C₂×C₁₂, C₄×C₈, C₃⋊Dic₃, C₃×C₁₂, C₆², C₄×Dic₃, C₃²⋊₂C₈, C₂×C₃⋊Dic₃, C₆×C₁₂, C₄×C₃⋊Dic₃, C₂×C₃²⋊₂C₈, C₄×C₃²⋊₂C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₄², C₂×C₈, C₄×C₈, C₃²⋊C₄, C₃²⋊₂C₈, C₂×C₃²⋊C₄, C₃⋊S₃⋊₃C₈, C₄×C₃²⋊C₄, C₂×C₃²⋊₂C₈, C₄×C₃²⋊₂C₈

Smallest permutation representation of C₄×C₃²⋊₂C₈
►On 96 points

Generators in S₉₆

(1 77 47 15)(2 78 48 16)(3 79 41 9)(4 80 42 10)(5 73 43 11)(6 74 44 12)(7 75 45 13)(8 76 46 14)(17 56 88 27)(18 49 81 28)(19 50 82 29)(20 51 83 30)(21 52 84 31)(22 53 85 32)(23 54 86 25)(24 55 87 26)(33 69 57 89)(34 70 58 90)(35 71 59 91)(36 72 60 92)(37 65 61 93)(38 66 62 94)(39 67 63 95)(40 68 64 96)

(2 23 68)(4 70 17)(6 19 72)(8 66 21)(10 34 27)(12 29 36)(14 38 31)(16 25 40)(42 90 88)(44 82 92)(46 94 84)(48 86 96)(50 60 74)(52 76 62)(54 64 78)(56 80 58)

(1 22 67)(2 23 68)(3 69 24)(4 70 17)(5 18 71)(6 19 72)(7 65 20)(8 66 21)(9 33 26)(10 34 27)(11 28 35)(12 29 36)(13 37 30)(14 38 31)(15 32 39)(16 25 40)(41 89 87)(42 90 88)(43 81 91)(44 82 92)(45 93 83)(46 94 84)(47 85 95)(48 86 96)(49 59 73)(50 60 74)(51 75 61)(52 76 62)(53 63 77)(54 64 78)(55 79 57)(56 80 58)

(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)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,77,47,15)(2,78,48,16)(3,79,41,9)(4,80,42,10)(5,73,43,11)(6,74,44,12)(7,75,45,13)(8,76,46,14)(17,56,88,27)(18,49,81,28)(19,50,82,29)(20,51,83,30)(21,52,84,31)(22,53,85,32)(23,54,86,25)(24,55,87,26)(33,69,57,89)(34,70,58,90)(35,71,59,91)(36,72,60,92)(37,65,61,93)(38,66,62,94)(39,67,63,95)(40,68,64,96), (2,23,68)(4,70,17)(6,19,72)(8,66,21)(10,34,27)(12,29,36)(14,38,31)(16,25,40)(42,90,88)(44,82,92)(46,94,84)(48,86,96)(50,60,74)(52,76,62)(54,64,78)(56,80,58), (1,22,67)(2,23,68)(3,69,24)(4,70,17)(5,18,71)(6,19,72)(7,65,20)(8,66,21)(9,33,26)(10,34,27)(11,28,35)(12,29,36)(13,37,30)(14,38,31)(15,32,39)(16,25,40)(41,89,87)(42,90,88)(43,81,91)(44,82,92)(45,93,83)(46,94,84)(47,85,95)(48,86,96)(49,59,73)(50,60,74)(51,75,61)(52,76,62)(53,63,77)(54,64,78)(55,79,57)(56,80,58), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)>;

G:=Group( (1,77,47,15)(2,78,48,16)(3,79,41,9)(4,80,42,10)(5,73,43,11)(6,74,44,12)(7,75,45,13)(8,76,46,14)(17,56,88,27)(18,49,81,28)(19,50,82,29)(20,51,83,30)(21,52,84,31)(22,53,85,32)(23,54,86,25)(24,55,87,26)(33,69,57,89)(34,70,58,90)(35,71,59,91)(36,72,60,92)(37,65,61,93)(38,66,62,94)(39,67,63,95)(40,68,64,96), (2,23,68)(4,70,17)(6,19,72)(8,66,21)(10,34,27)(12,29,36)(14,38,31)(16,25,40)(42,90,88)(44,82,92)(46,94,84)(48,86,96)(50,60,74)(52,76,62)(54,64,78)(56,80,58), (1,22,67)(2,23,68)(3,69,24)(4,70,17)(5,18,71)(6,19,72)(7,65,20)(8,66,21)(9,33,26)(10,34,27)(11,28,35)(12,29,36)(13,37,30)(14,38,31)(15,32,39)(16,25,40)(41,89,87)(42,90,88)(43,81,91)(44,82,92)(45,93,83)(46,94,84)(47,85,95)(48,86,96)(49,59,73)(50,60,74)(51,75,61)(52,76,62)(53,63,77)(54,64,78)(55,79,57)(56,80,58), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96) );

G=PermutationGroup([[(1,77,47,15),(2,78,48,16),(3,79,41,9),(4,80,42,10),(5,73,43,11),(6,74,44,12),(7,75,45,13),(8,76,46,14),(17,56,88,27),(18,49,81,28),(19,50,82,29),(20,51,83,30),(21,52,84,31),(22,53,85,32),(23,54,86,25),(24,55,87,26),(33,69,57,89),(34,70,58,90),(35,71,59,91),(36,72,60,92),(37,65,61,93),(38,66,62,94),(39,67,63,95),(40,68,64,96)], [(2,23,68),(4,70,17),(6,19,72),(8,66,21),(10,34,27),(12,29,36),(14,38,31),(16,25,40),(42,90,88),(44,82,92),(46,94,84),(48,86,96),(50,60,74),(52,76,62),(54,64,78),(56,80,58)], [(1,22,67),(2,23,68),(3,69,24),(4,70,17),(5,18,71),(6,19,72),(7,65,20),(8,66,21),(9,33,26),(10,34,27),(11,28,35),(12,29,36),(13,37,30),(14,38,31),(15,32,39),(16,25,40),(41,89,87),(42,90,88),(43,81,91),(44,82,92),(45,93,83),(46,94,84),(47,85,95),(48,86,96),(49,59,73),(50,60,74),(51,75,61),(52,76,62),(53,63,77),(54,64,78),(55,79,57),(56,80,58)], [(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),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)]])

48 conjugacy classes

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	···	4L	6A	···	6F	8A	···	8P	12A	···	12H
order	1	2	2	2	3	3	4	4	4	4	4	···	4	6	···	6	8	···	8	12	···	12
size	1	1	1	1	4	4	1	1	1	1	9	···	9	4	···	4	9	···	9	4	···	4

48 irreducible representations

dim	1	1	1	1	1	1	1	1	4	4	4	4	4
type	+	+	+						+	-	+
image	C₁	C₂	C₂	C₄	C₄	C₄	C₈	C₈	C₃²⋊C₄	C₃²⋊₂C₈	C₂×C₃²⋊C₄	C₃⋊S₃⋊₃C₈	C₄×C₃²⋊C₄
kernel	C₄×C₃²⋊₂C₈	C₄×C₃⋊Dic₃	C₂×C₃²⋊₂C₈	C₃²⋊₂C₈	C₂×C₃⋊Dic₃	C₆×C₁₂	C₃⋊Dic₃	C₃×C₁₂	C₂×C₄	C₄	C₂²	C₂	C₂
# reps	1	1	2	8	2	2	8	8	2	4	2	4	4

Matrix representation of C₄×C₃²⋊₂C₈ ►in GL₅(𝔽₇₃)

46	0	0	0	0
0	27	0	0	0
0	0	27	0	0
0	0	0	27	0
0	0	0	0	27

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	72	72

1	0	0	0	0
0	72	72	0	0
0	1	0	0	0
0	0	0	0	1
0	0	0	72	72

63	0	0	0	0
0	0	0	1	0
0	0	0	0	1
0	3	31	0	0
0	28	70	0	0

G:=sub<GL(5,GF(73))| [46,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,1,72],[1,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,72],[63,0,0,0,0,0,0,0,3,28,0,0,0,31,70,0,1,0,0,0,0,0,1,0,0] >;

C₄×C₃²⋊₂C₈ in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_2C_8

% in TeX

G:=Group("C4xC3^2:2C8");

// GroupNames label

G:=SmallGroup(288,423);

// by ID

G=gap.SmallGroup(288,423);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,100,9413,691,12550,2372]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;

// generators/relations

G = C4×C32⋊2C8 order 288 = 25·32

Direct product of C4 and C32⋊2C8

G = C₄×C₃²⋊₂C₈ order 288 = 2⁵·3²

Direct product of C₄ and C₃²⋊₂C₈